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How do you (x^3+12*x)*(x/(2*(x^3+12*x))+(-12-3*x^2)*(x^2+4)/(4*(x^3+12*x)^2))/(x^2+4) in partial fractions?

An expression to simplify:

The solution

You have entered [src]
            /                /         2\ / 2    \\
/ 3       \ |      x         \-12 - 3*x /*\x  + 4/|
\x  + 12*x/*|------------- + ---------------------|
            |  / 3       \                    2   |
            |2*\x  + 12*x/         / 3       \    |
            \                    4*\x  + 12*x/    /
---------------------------------------------------
                        2                          
                       x  + 4                      
$$\frac{\left(x^{3} + 12 x\right) \left(\frac{x}{2 \left(x^{3} + 12 x\right)} + \frac{\left(- 3 x^{2} - 12\right) \left(x^{2} + 4\right)}{4 \left(x^{3} + 12 x\right)^{2}}\right)}{x^{2} + 4}$$
((x^3 + 12*x)*(x/((2*(x^3 + 12*x))) + ((-12 - 3*x^2)*(x^2 + 4))/((4*(x^3 + 12*x)^2))))/(x^2 + 4)
General simplification [src]
      /      4\      
     -\48 + x /      
---------------------
    /      4       2\
4*x*\48 + x  + 16*x /
$$- \frac{x^{4} + 48}{4 x \left(x^{4} + 16 x^{2} + 48\right)}$$
-(48 + x^4)/(4*x*(48 + x^4 + 16*x^2))
Fraction decomposition [src]
-1/(4*x) + x/(2*(4 + x^2)) - x/(2*(12 + x^2))
$$- \frac{x}{2 \left(x^{2} + 12\right)} + \frac{x}{2 \left(x^{2} + 4\right)} - \frac{1}{4 x}$$
   1        x             x     
- --- + ---------- - -----------
  4*x     /     2\     /      2\
        2*\4 + x /   2*\12 + x /
Numerical answer [src]
(x^3 + 12.0*x)*(x/(2.0*x^3 + 24.0*x) + 0.00173611111111111*(4.0 + x^2)*(-12.0 - 3.0*x^2)/(x + 0.0833333333333333*x^3)^2)/(4.0 + x^2)
(x^3 + 12.0*x)*(x/(2.0*x^3 + 24.0*x) + 0.00173611111111111*(4.0 + x^2)*(-12.0 - 3.0*x^2)/(x + 0.0833333333333333*x^3)^2)/(4.0 + x^2)
Combining rational expressions [src]
   2 /      2\     /      2\ /     2\
2*x *\12 + x / + 3*\-4 - x /*\4 + x /
-------------------------------------
            /     2\ /      2\       
        4*x*\4 + x /*\12 + x /       
$$\frac{2 x^{2} \left(x^{2} + 12\right) + 3 \left(- x^{2} - 4\right) \left(x^{2} + 4\right)}{4 x \left(x^{2} + 4\right) \left(x^{2} + 12\right)}$$
(2*x^2*(12 + x^2) + 3*(-4 - x^2)*(4 + x^2))/(4*x*(4 + x^2)*(12 + x^2))
Combinatorics [src]
      /      4\       
     -\48 + x /       
----------------------
    /     2\ /      2\
4*x*\4 + x /*\12 + x /
$$- \frac{x^{4} + 48}{4 x \left(x^{2} + 4\right) \left(x^{2} + 12\right)}$$
-(48 + x^4)/(4*x*(4 + x^2)*(12 + x^2))
Common denominator [src]
     /      4\      
    -\48 + x /      
--------------------
   5       3        
4*x  + 64*x  + 192*x
$$- \frac{x^{4} + 48}{4 x^{5} + 64 x^{3} + 192 x}$$
-(48 + x^4)/(4*x^5 + 64*x^3 + 192*x)
Trigonometric part [src]
            /              /         2\ /     2\\
/ 3       \ |     x        \-12 - 3*x /*\4 + x /|
\x  + 12*x/*|----------- + ---------------------|
            |   3                           2   |
            |2*x  + 24*x         / 3       \    |
            \                  4*\x  + 12*x/    /
-------------------------------------------------
                           2                     
                      4 + x                      
$$\frac{\left(x^{3} + 12 x\right) \left(\frac{x}{2 x^{3} + 24 x} + \frac{\left(- 3 x^{2} - 12\right) \left(x^{2} + 4\right)}{4 \left(x^{3} + 12 x\right)^{2}}\right)}{x^{2} + 4}$$
(x^3 + 12*x)*(x/(2*x^3 + 24*x) + (-12 - 3*x^2)*(4 + x^2)/(4*(x^3 + 12*x)^2))/(4 + x^2)
Powers [src]
            /              /         2\ /     2\\
/ 3       \ |     x        \-12 - 3*x /*\4 + x /|
\x  + 12*x/*|----------- + ---------------------|
            |   3                           2   |
            |2*x  + 24*x         / 3       \    |
            \                  4*\x  + 12*x/    /
-------------------------------------------------
                           2                     
                      4 + x                      
$$\frac{\left(x^{3} + 12 x\right) \left(\frac{x}{2 x^{3} + 24 x} + \frac{\left(- 3 x^{2} - 12\right) \left(x^{2} + 4\right)}{4 \left(x^{3} + 12 x\right)^{2}}\right)}{x^{2} + 4}$$
(x^3 + 12*x)*(x/(2*x^3 + 24*x) + (-12 - 3*x^2)*(4 + x^2)/(4*(x^3 + 12*x)^2))/(4 + x^2)
Assemble expression [src]
            /              /         2\ /     2\\
/ 3       \ |     x        \-12 - 3*x /*\4 + x /|
\x  + 12*x/*|----------- + ---------------------|
            |   3                           2   |
            |2*x  + 24*x         / 3       \    |
            \                  4*\x  + 12*x/    /
-------------------------------------------------
                           2                     
                      4 + x                      
$$\frac{\left(x^{3} + 12 x\right) \left(\frac{x}{2 x^{3} + 24 x} + \frac{\left(- 3 x^{2} - 12\right) \left(x^{2} + 4\right)}{4 \left(x^{3} + 12 x\right)^{2}}\right)}{x^{2} + 4}$$
(x^3 + 12*x)*(x/(2*x^3 + 24*x) + (-12 - 3*x^2)*(4 + x^2)/(4*(x^3 + 12*x)^2))/(4 + x^2)
Rational denominator [src]
               2                                      
    / 3       \    /         2\ /     2\ /   3       \
4*x*\x  + 12*x/  + \-12 - 3*x /*\4 + x /*\2*x  + 24*x/
------------------------------------------------------
           /     2\ / 3       \ /   3       \         
         4*\4 + x /*\x  + 12*x/*\2*x  + 24*x/         
$$\frac{4 x \left(x^{3} + 12 x\right)^{2} + \left(- 3 x^{2} - 12\right) \left(x^{2} + 4\right) \left(2 x^{3} + 24 x\right)}{4 \left(x^{2} + 4\right) \left(x^{3} + 12 x\right) \left(2 x^{3} + 24 x\right)}$$
(4*x*(x^3 + 12*x)^2 + (-12 - 3*x^2)*(4 + x^2)*(2*x^3 + 24*x))/(4*(4 + x^2)*(x^3 + 12*x)*(2*x^3 + 24*x))
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