Mister Exam

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Integral of d{x}:
Integral of cosx
Integral of ln|x|
Integral of x*sin(x)^2
Integral of x^n*e^(-x)
y/(x^5+1)
Identical expressions
y/(x^ five + one)
y divide by (x to the power of 5 plus 1)
y divide by (x to the power of five plus one)
y/(x5+1)
y/x5+1
y/(x⁵+1)
y/x^5+1
y divide by (x^5+1)
y/(x^5+1)dx
Similar expressions
y/(x^5-1)

Integral of y/(x^5+1) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |    y      
 |  ------ dx
 |   5       
 |  x  + 1   
 |           
/            
0

$$\int\limits_{0}^{1} \frac{y}{x^{5} + 1}\, dx$$

Integral(y/(x^5 + 1), (x, 0, 1))

The answer (Indefinite) [src]

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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     ____________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    ____________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ____________________________                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                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                     |                    /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  ___________                                                                                                                                     ___________                                                           \                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  ___________                                                                                                                                     ___________                                                           \                                                                                                                                                                                   /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ___________                                                                                                                                      ___________                                                            \          /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ___________                                                                                                                                      ___________                                                            \                                                                                                                                                                                                                                                                                                                    /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ___________                                                                                                                                       ___________                                                            \          /                ___________      /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ___________                                                                                                                                       ___________                                                            \|
                     |                   /         ____   /       ___       |                                                                                                                                                                                                                                                                                                                                                                     ___                                                                                                                                  ____   /       ___                                                                                                                               ___   /       ___                                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                /         ____   /       ___       |                                                                                                                                                                                                                                                                                                                                                                     ___                                                                                                                                  ____   /       ___                                                                                                                               ___   /       ___                                                            |                                                                                                                                                                                  /         ____   /       ___       |                                                                                                                                                                                                                                                                                                                                                                          ___                                                                                                                                      ___   /       ___                                                                                                                               ____   /       ___                                                             |         /         ____   /       ___       |                                                                                                                                                                                                                         ___                                                                                                                                                                                                                                                                                      ____   /       ___                                                                                                                                ___   /       ___                                                             |          ____________                                                                                                                                      ____________                                                                                                                                           /         ____   /       ___       |                                                                                                                                                                                                                             ___                                                                                                                                                                                                                                                                                        ____   /       ___                                                                                                                                 ___   /       ___                                                             |         /         ____   /       ___       |                                                                                                                                                                                                                             ___                                                                                                                                                                                                                                                                                         ___   /       ___                                                                                                                                ____   /       ___                                                             ||
                     |                  /   1    \/ 10 *\/  3 + \/ 5        |                                                                     18                                                                                                                                            32*x                                                                                                                                         22*\/ 5                                                                                                                                 \/ 10 *\/  3 + \/ 5                                                                                                                           15*\/ 2 *\/  3 + \/ 5                                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               /   1    \/ 10 *\/  3 - \/ 5        |                                                                     18                                                                                                                                            32*x                                                                                                                                         22*\/ 5                                                                                                                                 \/ 10 *\/  3 - \/ 5                                                                                                                           15*\/ 2 *\/  3 - \/ 5                                                             |                                                                                                                                                                                 /   3    \/ 10 *\/  3 + \/ 5        |                                                                      94                                                                                                                                              44*x                                                                                                                                           92*\/ 5                                                                                                                                  75*\/ 2 *\/  3 + \/ 5                                                                                                                            6*\/ 10 *\/  3 + \/ 5                                                              |        /   3    \/ 10 *\/  3 - \/ 5        |                                                                      94                                                                                                                                            92*\/ 5                                                                                                                                            44*x                                                                                                                                   6*\/ 10 *\/  3 - \/ 5                                                                                                                            75*\/ 2 *\/  3 - \/ 5                                                              |         /        ___      /                  ____                                    ___                                       ___              \         /        ___      /                   ___                                       ____                                        ___               \        /   3    \/ 10 *\/  3 + \/ 5        |                                                                         96                                                                                                                                             42*\/ 5                                                                                                                                             44*x                                                                                                                                    8*\/ 10 *\/  3 + \/ 5                                                                                                                             45*\/ 2 *\/  3 + \/ 5                                                              |        /   3    \/ 10 *\/  3 - \/ 5        |                                                                         96                                                                                                                                             42*\/ 5                                                                                                                                             44*x                                                                                                                                    45*\/ 2 *\/  3 - \/ 5                                                                                                                             8*\/ 10 *\/  3 - \/ 5                                                              ||
                     |             2*  /    -- - --------------------- *atan|-------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------|              /                           /                              ___________               ___________\                  ___________                 ___________\                  /                      /                             ___________             ___________\                ___________               ___________\                  /                      /                           ___________               ___________\                ___________               ___________\            /                            /                            ___________               ___________\                 ___________                  ___________\            /                            /                            ___________               ___________\                  ___________                 ___________\   2*  /    -- - --------------------- *atan|-------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------|              /                           /                              ___________               ___________\                 ___________                  ___________\   4*  /    -- - --------------------- *atan|---------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------|   4*  /    -- - --------------------- *atan|---------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------|        /  1    \/ 5       |                \/ 10                                   \/ 2                                  4*x*\/ 2               |        /  1    \/ 5       |                 \/ 2                                      \/ 10                                   4*x*\/ 2                |   8*  /    -- - --------------------- *atan|- ----------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------------------------------------------|   8*  /    -- - --------------------- *atan|- ----------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------------------------------------------------------------------------------||
                     |               \/     20            200               |           ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________|              |                   ___     |            ___        ___   /       ___        ____   /       ___ |          ____   /       ___           ___   /       ___ |   /      ___\    |              ___     |           ___        ___   /       ___      ____   /       ___ |        ____   /       ___         ___   /       ___ |   /      ___\    |              ___     |           ___     ____   /       ___         ___   /       ___ |        ____   /       ___         ___   /       ___ |            |                    ___     |          ___        ___   /       ___        ____   /       ___ |          ___   /       ___           ____   /       ___ |            |                    ___     |          ___       ____   /       ___         ___   /       ___ |          ____   /       ___           ___   /       ___ |     \/     20            200               |           ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________              ____________________________               ____________________________                              ____________________________|              |                   ___     |            ___       ____   /       ___         ___   /       ___ |          ___   /       ___           ____   /       ___ |     \/     20             50               |      ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________|     \/     20             50               |      ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________         ______________________________               ______________________________                               ______________________________|   6*  /   -- - ----- *atan|------------------------------------- - ------------------------------------- + -------------------------------------|   6*  /   -- + ----- *atan|--------------------------------------- + --------------------------------------- - ---------------------------------------|     \/     20             50               |         ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________|     \/     20             50               |         ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________          ______________________________               ______________________________                               ______________________________||
  /                  |                                                      |          /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________ |       ___    |5217    2   2213*\/ 5      |  48   21*\/ 5    45*\/ 2 *\/  3 - \/ 5     4*\/ 10 *\/  3 - \/ 5  |   1381*\/ 10 *\/  3 - \/ 5     3045*\/ 2 *\/  3 - \/ 5  |   |1   \/ 5 |    |71    2   9*\/ 5      |  9   11*\/ 5    15*\/ 2 *\/  3 - \/ 5     \/ 10 *\/  3 - \/ 5  |   37*\/ 10 *\/  3 - \/ 5     35*\/ 2 *\/  3 - \/ 5  |   |1   \/ 5 |    |71    2   9*\/ 5      |  9   11*\/ 5    \/ 10 *\/  3 + \/ 5     15*\/ 2 *\/  3 + \/ 5  |   37*\/ 10 *\/  3 + \/ 5     35*\/ 2 *\/  3 + \/ 5  |     ___    |12387    2   4287*\/ 5      |47   46*\/ 5    75*\/ 2 *\/  3 + \/ 5     3*\/ 10 *\/  3 + \/ 5  |   4645*\/ 2 *\/  3 + \/ 5     3929*\/ 10 *\/  3 + \/ 5  |     ___    |12387    2   4287*\/ 5      |47   46*\/ 5    3*\/ 10 *\/  3 - \/ 5     75*\/ 2 *\/  3 - \/ 5  |   3929*\/ 10 *\/  3 - \/ 5     4645*\/ 2 *\/  3 - \/ 5  |                                            |          /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________              /                ___________               /                ___________               ___________    /                ___________ |       ___    |5217    2   2213*\/ 5      |  48   21*\/ 5    4*\/ 10 *\/  3 + \/ 5     45*\/ 2 *\/  3 + \/ 5  |   3045*\/ 2 *\/  3 + \/ 5     1381*\/ 10 *\/  3 + \/ 5  |                                            |     /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________ |                                            |     /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________         /                  ___________               /                  ___________                ___________    /                  ___________ |     \/    10     50       |   ___________            ___________      ___________            ___________      ___________            ___________|     \/    10     50       |     ___________            ___________        ___________            ___________        ___________            ___________|                                            |        /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________ |                                            |        /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________          /                  ___________               /                  ___________                ___________    /                  ___________ ||
 |                   |                                                      |  ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___  |   2*\/ 5 *log|---- + x  - ---------- + x*|- -- + -------- - ----------------------- + -----------------------| - -------------------------- + -------------------------|   |- - -----|*log|-- + x  - ------- + x*|- - + -------- - ----------------------- - ---------------------| - ------------------------ + -----------------------|   |- + -----|*log|-- + x  + ------- + x*|- - - -------- - --------------------- + -----------------------| - ------------------------ - -----------------------|   \/ 5 *log|----- + x  + ---------- + x*|-- + -------- - ----------------------- - -----------------------| - ------------------------- - --------------------------|   \/ 5 *log|----- + x  - ---------- + x*|-- - -------- - ----------------------- + -----------------------| - -------------------------- + -------------------------|                                            |  ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___       ____   /         ____   /       ___         ___   /         ____   /       ___         ___   /       ___    /         ____   /       ___  |   2*\/ 5 *log|---- + x  + ---------- + x*|- -- - -------- + ----------------------- + -----------------------| - ------------------------- - --------------------------|                                            |    /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___  |                                            |    /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___         /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___  |                           |  /       ___      ___   /       ___      /       ___      ___   /       ___      /       ___      ___   /       ___ |                           |    /       ___      ___   /       ___        /       ___      ___   /       ___        /       ___      ___   /       ___ |                                            |       /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___  |                                            |       /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___          /           ____   /       ___         ___   /           ____   /       ___         ____   /       ___    /           ____   /       ___  ||
 |   y               |log(1 + x)                                            \\/ 10 *\/   10 - \/ 10 *\/  3 + \/ 5    - 3*\/ 2 *\/   10 - \/ 10 *\/  3 + \/ 5    - 2*\/ 5 *\/  3 + \/ 5  *\/   10 - \/ 10 *\/  3 + \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 + \/ 5    - 3*\/ 2 *\/   10 - \/ 10 *\/  3 + \/ 5    - 2*\/ 5 *\/  3 + \/ 5  *\/   10 - \/ 10 *\/  3 + \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 + \/ 5    - 3*\/ 2 *\/   10 - \/ 10 *\/  3 + \/ 5    - 2*\/ 5 *\/  3 + \/ 5  *\/   10 - \/ 10 *\/  3 + \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 + \/ 5    - 3*\/ 2 *\/   10 - \/ 10 *\/  3 + \/ 5    - 2*\/ 5 *\/  3 + \/ 5  *\/   10 - \/ 10 *\/  3 + \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 + \/ 5    - 3*\/ 2 *\/   10 - \/ 10 *\/  3 + \/ 5    - 2*\/ 5 *\/  3 + \/ 5  *\/   10 - \/ 10 *\/  3 + \/ 5   /              \242            242         \  11      11                 22                        11          /              484                          484           /   \4     20 /    \16           8        \  8      8                  16                       16         /              64                         64          /   \4     20 /    \16           8        \  8      8                 16                       16          /              64                         64          /            \ 242            242         \11      11                 22                        11          /              484                         484            /            \ 242            242         \11      11                 11                        22          /              484                          484           /                                            \\/ 10 *\/   10 - \/ 10 *\/  3 - \/ 5    + 3*\/ 2 *\/   10 - \/ 10 *\/  3 - \/ 5    + 2*\/ 5 *\/  3 - \/ 5  *\/   10 - \/ 10 *\/  3 - \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 - \/ 5    + 3*\/ 2 *\/   10 - \/ 10 *\/  3 - \/ 5    + 2*\/ 5 *\/  3 - \/ 5  *\/   10 - \/ 10 *\/  3 - \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 - \/ 5    + 3*\/ 2 *\/   10 - \/ 10 *\/  3 - \/ 5    + 2*\/ 5 *\/  3 - \/ 5  *\/   10 - \/ 10 *\/  3 - \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 - \/ 5    + 3*\/ 2 *\/   10 - \/ 10 *\/  3 - \/ 5    + 2*\/ 5 *\/  3 - \/ 5  *\/   10 - \/ 10 *\/  3 - \/ 5      \/ 10 *\/   10 - \/ 10 *\/  3 - \/ 5    + 3*\/ 2 *\/   10 - \/ 10 *\/  3 - \/ 5    + 2*\/ 5 *\/  3 - \/ 5  *\/   10 - \/ 10 *\/  3 - \/ 5   /              \242            242         \  11      11                 11                        22          /              484                         484            /                                            \8*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 5*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      8*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 5*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      8*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 5*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      8*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 5*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      8*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 5*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5   /                                            \8*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 5*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      8*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 5*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      8*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 5*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      8*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 5*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      8*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 6*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 5*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5   /                           \\/  5 - \/ 5   + \/ 5 *\/  5 - \/ 5     \/  5 - \/ 5   + \/ 5 *\/  5 - \/ 5     \/  5 - \/ 5   + \/ 5 *\/  5 - \/ 5  /                           \- \/  5 + \/ 5   + \/ 5 *\/  5 + \/ 5     - \/  5 + \/ 5   + \/ 5 *\/  5 + \/ 5     - \/  5 + \/ 5   + \/ 5 *\/  5 + \/ 5  /                                            \  18*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 3*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      18*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 3*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      18*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 3*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      18*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 3*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5      18*\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 + \/ 5    + 3*\/ 10 *\/  3 + \/ 5  *\/   15 - 2*\/ 10 *\/  3 + \/ 5   /                                            \  18*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 3*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      18*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 3*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      18*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 3*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      18*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 3*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5      18*\/   15 - 2*\/ 10 *\/  3 - \/ 5    - 8*\/ 5 *\/   15 - 2*\/ 10 *\/  3 - \/ 5    + 3*\/ 10 *\/  3 - \/ 5  *\/   15 - 2*\/ 10 *\/  3 - \/ 5   /|
 | ------ dx = C + y*|---------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
 |  5                \    5                                                                                                                                                                                                                                                                                                                                                                                                 5                                                                                                                                                                                                                                                                                                                                                                                                                                                                              25                                                                                                                                                                   5                                                                                                                                                                5                                                                                                                                                                   25                                                                                                                                                                    25                                                                                                                                                                                                                                                                                                                                                                                                                                                                            5                                                                                                                                                                                                                                                                                                                                                                                                                                                                              25                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   5                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               5                                                                                                                                                                                                                                                                                                                                                                                                                                                                        5                                                                                                                                                    5                                                                                                                                                                                                                                                                                                                                                                                                                                                                              5                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      5                                                                                                                                                                                                                                                                                                                                                                                                  /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
 |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   
/

$$\int \frac{y}{x^{5} + 1}\, dx = C + y \left(\frac{\log{\left(x + 1 \right)}}{5} + \frac{2 \sqrt{5} \log{\left(x^{2} + x \left(- \frac{48}{11} - \frac{21 \sqrt{5}}{11} + \frac{4 \sqrt{10} \sqrt{\sqrt{5} + 3}}{11} + \frac{45 \sqrt{2} \sqrt{\sqrt{5} + 3}}{22}\right) - \frac{1381 \sqrt{10} \sqrt{\sqrt{5} + 3}}{484} - \frac{3045 \sqrt{2} \sqrt{\sqrt{5} + 3}}{484} + \frac{2213 \sqrt{5}}{242} + \frac{5217}{242} \right)}}{25} - \frac{2 \sqrt{5} \log{\left(x^{2} + x \left(- \frac{48}{11} - \frac{45 \sqrt{2} \sqrt{3 - \sqrt{5}}}{22} + \frac{4 \sqrt{10} \sqrt{3 - \sqrt{5}}}{11} + \frac{21 \sqrt{5}}{11}\right) - \frac{2213 \sqrt{5}}{242} - \frac{1381 \sqrt{10} \sqrt{3 - \sqrt{5}}}{484} + \frac{3045 \sqrt{2} \sqrt{3 - \sqrt{5}}}{484} + \frac{5217}{242} \right)}}{25} + \frac{\sqrt{5} \log{\left(x^{2} + x \left(- \frac{46 \sqrt{5}}{11} - \frac{3 \sqrt{10} \sqrt{3 - \sqrt{5}}}{11} + \frac{75 \sqrt{2} \sqrt{3 - \sqrt{5}}}{22} + \frac{47}{11}\right) - \frac{4287 \sqrt{5}}{242} - \frac{3929 \sqrt{10} \sqrt{3 - \sqrt{5}}}{484} + \frac{4645 \sqrt{2} \sqrt{3 - \sqrt{5}}}{484} + \frac{12387}{242} \right)}}{25} - \frac{\left(\frac{\sqrt{5}}{20} + \frac{1}{4}\right) \log{\left(x^{2} + x \left(- \frac{11 \sqrt{5}}{8} - \frac{9}{8} - \frac{\sqrt{10} \sqrt{\sqrt{5} + 3}}{16} + \frac{15 \sqrt{2} \sqrt{\sqrt{5} + 3}}{16}\right) - \frac{37 \sqrt{10} \sqrt{\sqrt{5} + 3}}{64} - \frac{35 \sqrt{2} \sqrt{\sqrt{5} + 3}}{64} + \frac{9 \sqrt{5}}{8} + \frac{71}{16} \right)}}{5} - \frac{\left(\frac{1}{4} - \frac{\sqrt{5}}{20}\right) \log{\left(x^{2} + x \left(- \frac{15 \sqrt{2} \sqrt{3 - \sqrt{5}}}{16} - \frac{9}{8} - \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{16} + \frac{11 \sqrt{5}}{8}\right) - \frac{9 \sqrt{5}}{8} - \frac{37 \sqrt{10} \sqrt{3 - \sqrt{5}}}{64} + \frac{35 \sqrt{2} \sqrt{3 - \sqrt{5}}}{64} + \frac{71}{16} \right)}}{5} - \frac{\sqrt{5} \log{\left(x^{2} + x \left(- \frac{75 \sqrt{2} \sqrt{\sqrt{5} + 3}}{22} - \frac{3 \sqrt{10} \sqrt{\sqrt{5} + 3}}{11} + \frac{47}{11} + \frac{46 \sqrt{5}}{11}\right) - \frac{3929 \sqrt{10} \sqrt{\sqrt{5} + 3}}{484} - \frac{4645 \sqrt{2} \sqrt{\sqrt{5} + 3}}{484} + \frac{4287 \sqrt{5}}{242} + \frac{12387}{242} \right)}}{25} + \frac{6 \sqrt{\frac{1}{10} - \frac{\sqrt{5}}{50}} \operatorname{atan}{\left(\frac{4 \sqrt{2} x}{\sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}}} - \frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}}} + \frac{\sqrt{10}}{\sqrt{5 - \sqrt{5}} + \sqrt{5} \sqrt{5 - \sqrt{5}}} \right)}}{5} + \frac{6 \sqrt{\frac{\sqrt{5}}{50} + \frac{1}{10}} \operatorname{atan}{\left(- \frac{4 \sqrt{2} x}{- \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{\sqrt{5} + 5}} + \frac{\sqrt{2}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{\sqrt{5} + 5}} + \frac{\sqrt{10}}{- \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{\sqrt{5} + 5}} \right)}}{5} + \frac{2 \sqrt{- \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{200} + \frac{1}{20}} \operatorname{atan}{\left(- \frac{32 x}{\sqrt{10} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 2 \sqrt{5} \sqrt{3 - \sqrt{5}} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10}} - \frac{22 \sqrt{5}}{\sqrt{10} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 2 \sqrt{5} \sqrt{3 - \sqrt{5}} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10}} + \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{\sqrt{10} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 2 \sqrt{5} \sqrt{3 - \sqrt{5}} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10}} + \frac{18}{\sqrt{10} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 2 \sqrt{5} \sqrt{3 - \sqrt{5}} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10}} + \frac{15 \sqrt{2} \sqrt{3 - \sqrt{5}}}{\sqrt{10} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 2 \sqrt{5} \sqrt{3 - \sqrt{5}} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10} + 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{3 - \sqrt{5}} + 10}} \right)}}{5} + \frac{8 \sqrt{- \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{50} + \frac{3}{20}} \operatorname{atan}{\left(\frac{44 x}{- 8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 3 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} - \frac{96}{- 8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 3 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} - \frac{45 \sqrt{2} \sqrt{3 - \sqrt{5}}}{- 8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 3 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} + \frac{8 \sqrt{10} \sqrt{3 - \sqrt{5}}}{- 8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 3 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} + \frac{42 \sqrt{5}}{- 8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 3 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} \right)}}{5} + \frac{4 \sqrt{- \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{50} + \frac{3}{20}} \operatorname{atan}{\left(\frac{44 x}{- 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 8 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 5 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} - \frac{92 \sqrt{5}}{- 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 8 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 5 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} - \frac{6 \sqrt{10} \sqrt{3 - \sqrt{5}}}{- 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 8 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 5 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} + \frac{75 \sqrt{2} \sqrt{3 - \sqrt{5}}}{- 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 8 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 5 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} + \frac{94}{- 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 8 \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15} + 5 \sqrt{10} \sqrt{3 - \sqrt{5}} \sqrt{- 2 \sqrt{10} \sqrt{3 - \sqrt{5}} + 15}} \right)}}{5} + \frac{8 \sqrt{- \frac{\sqrt{10} \sqrt{\sqrt{5} + 3}}{50} + \frac{3}{20}} \operatorname{atan}{\left(\frac{44 x}{8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 3 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} - \frac{96}{8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 3 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} - \frac{42 \sqrt{5}}{8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 3 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} + \frac{8 \sqrt{10} \sqrt{\sqrt{5} + 3}}{8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 3 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} + \frac{45 \sqrt{2} \sqrt{\sqrt{5} + 3}}{8 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 18 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 3 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} \right)}}{5} - \frac{2 \sqrt{- \frac{\sqrt{10} \sqrt{\sqrt{5} + 3}}{200} + \frac{1}{20}} \operatorname{atan}{\left(- \frac{32 x}{- 2 \sqrt{5} \sqrt{\sqrt{5} + 3} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} - 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} + \sqrt{10} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10}} + \frac{22 \sqrt{5}}{- 2 \sqrt{5} \sqrt{\sqrt{5} + 3} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} - 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} + \sqrt{10} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10}} + \frac{18}{- 2 \sqrt{5} \sqrt{\sqrt{5} + 3} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} - 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} + \sqrt{10} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10}} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 3}}{- 2 \sqrt{5} \sqrt{\sqrt{5} + 3} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} - 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} + \sqrt{10} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10}} - \frac{15 \sqrt{2} \sqrt{\sqrt{5} + 3}}{- 2 \sqrt{5} \sqrt{\sqrt{5} + 3} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} - 3 \sqrt{2} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10} + \sqrt{10} \sqrt{- \sqrt{10} \sqrt{\sqrt{5} + 3} + 10}} \right)}}{5} + \frac{4 \sqrt{- \frac{\sqrt{10} \sqrt{\sqrt{5} + 3}}{50} + \frac{3}{20}} \operatorname{atan}{\left(\frac{44 x}{8 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 5 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} - \frac{75 \sqrt{2} \sqrt{\sqrt{5} + 3}}{8 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 5 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} - \frac{6 \sqrt{10} \sqrt{\sqrt{5} + 3}}{8 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 5 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} + \frac{94}{8 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 5 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} + \frac{92 \sqrt{5}}{8 \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 6 \sqrt{5} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15} + 5 \sqrt{10} \sqrt{\sqrt{5} + 3} \sqrt{- 2 \sqrt{10} \sqrt{\sqrt{5} + 3} + 15}} \right)}}{5}\right)$$

Simplify

The answer [src]

  /log(2)          /     4        3       2                               \\            /     4        3       2                           \
y*|------ + RootSum\625*t  + 125*t  + 25*t  + 5*t + 1, t -> t*log(1 + 5*t)/| - y*RootSum\625*t  + 125*t  + 25*t  + 5*t + 1, t -> t*log(5*t)/
  \  5                                                                     /

$$y \left(\operatorname{RootSum} {\left(625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left( t \mapsto t \log{\left(5 t + 1 \right)} \right)\right)} + \frac{\log{\left(2 \right)}}{5}\right) - y \operatorname{RootSum} {\left(625 t^{4} + 125 t^{3} + 25 t^{2} + 5 t + 1, \left( t \mapsto t \log{\left(5 t \right)} \right)\right)}$$

  /log(2)          /     4        3       2                               \\            /     4        3       2                           \
y*|------ + RootSum\625*t  + 125*t  + 25*t  + 5*t + 1, t -> t*log(1 + 5*t)/| - y*RootSum\625*t  + 125*t  + 25*t  + 5*t + 1, t -> t*log(5*t)/
  \  5                                                                     /

y*(log(2)/5 + RootSum(625*_t^4 + 125*_t^3 + 25*_t^2 + 5*_t + 1, Lambda(_t, _t*log(1 + 5*_t)))) - y*RootSum(625*_t^4 + 125*_t^3 + 25*_t^2 + 5*_t + 1, Lambda(_t, _t*log(5*_t)))

Use the examples entering the upper and lower limits of integration.

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Integral of y/(x^5+1) dx

Limits of integration:

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Piecewise:

The solution