Fraction decomposition
[src]
-10/27 + 1/(4*(1 + x)) + 11/(27*(-1 + 3*x)^2) + 31/(108*(-1 + 3*x))
$$- \frac{10}{27} + \frac{31}{108 \left(3 x - 1\right)} + \frac{11}{27 \left(3 x - 1\right)^{2}} + \frac{1}{4 \left(x + 1\right)}$$
10 1 11 31
- -- + --------- + -------------- + --------------
27 4*(1 + x) 2 108*(-1 + 3*x)
27*(-1 + 3*x)
General simplification
[src]
/ 2 \
2*x*\2 - 5*x + 3*x/
-----------------------
2 3
3 - 15*x + 9*x + 27*x
$$\frac{2 x \left(- 5 x^{2} + 3 x + 2\right)}{27 x^{3} + 9 x^{2} - 15 x + 3}$$
2*x*(2 - 5*x^2 + 3*x)/(3 - 15*x + 9*x^2 + 27*x^3)
-0.333333333333333*(1.0 + (1.0 + x)/(1.0 - 3.0*x))*(1.0 + (1.0 + (1.0 + x)/(1.0 - 3.0*x))/(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x))))*(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x)))
-0.333333333333333*(1.0 + (1.0 + x)/(1.0 - 3.0*x))*(1.0 + (1.0 + (1.0 + x)/(1.0 - 3.0*x))/(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x))))*(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x)))
/ 1 + x \
| 1 + ------- |
/ 1 + x \ | 1 - 3*x | / 3 \
-|1 + -------|*|1 + ---------------------|*|1 - -----------------|
\ 1 - 3*x/ | 3 | \ (1 + x)*(1 - 3*x)/
| 1 - -----------------|
\ (1 + x)*(1 - 3*x)/
-------------------------------------------------------------------
3
$$- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3}$$
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
Assemble expression
[src]
/ 1 + x \
| 1 + ------- |
/ 1 + x \ | 1 - 3*x | / 3 \
-|1 + -------|*|1 + ---------------------|*|1 - -----------------|
\ 1 - 3*x/ | 3 | \ (1 + x)*(1 - 3*x)/
| 1 - -----------------|
\ (1 + x)*(1 - 3*x)/
-------------------------------------------------------------------
3
$$- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3}$$
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
2
10 10 - 14*x + 84*x
- -- + ---------------------------
27 2 3
27 - 135*x + 81*x + 243*x
$$\frac{84 x^{2} - 14 x + 10}{243 x^{3} + 81 x^{2} - 135 x + 27} - \frac{10}{27}$$
-10/27 + (10 - 14*x + 84*x^2)/(27 - 135*x + 81*x^2 + 243*x^3)
/ 1 + x \ / / 3 \\
| 1 + -------| | |-----||
| 1 - 3*x| | \1 + x/| / 1 1 + x \
|1 + -----------|*|1 - -------|*|- - - -----------|
| / 3 \| \ 1 - 3*x/ \ 3 3*(1 - 3*x)/
| |-----||
| \1 + x/|
| 1 - -------|
\ 1 - 3*x/
$$\left(- \frac{1}{3} - \frac{x + 1}{3 \left(1 - 3 x\right)}\right) \left(1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}}\right)$$
(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/(1 + x)/(1 - 3*x)))*(1 - 3/(1 + x)/(1 - 3*x))*(-1/3 - (1 + x)/(3*(1 - 3*x)))
/ 1 + x \
| 1 + ------- |
/ 1 + x \ | 1 - 3*x | / 3 \
-|1 + -------|*|1 + ---------------------|*|1 - -----------------|
\ 1 - 3*x/ | 3 | \ (1 + x)*(1 - 3*x)/
| 1 - -----------------|
\ (1 + x)*(1 - 3*x)/
-------------------------------------------------------------------
3
$$- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3}$$
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
Rational denominator
[src]
(2 - 2*x)*((1 - 3*x)*(-3 + (1 + x)*(1 - 3*x)) + (1 + x)*(1 - 3*x)*(2 - 2*x))
----------------------------------------------------------------------------
2
(1 + x)*(1 - 3*x) *(-3 + 9*x)
$$\frac{\left(2 - 2 x\right) \left(\left(1 - 3 x\right) \left(2 - 2 x\right) \left(x + 1\right) + \left(1 - 3 x\right) \left(\left(1 - 3 x\right) \left(x + 1\right) - 3\right)\right)}{\left(1 - 3 x\right)^{2} \left(x + 1\right) \left(9 x - 3\right)}$$
(2 - 2*x)*((1 - 3*x)*(-3 + (1 + x)*(1 - 3*x)) + (1 + x)*(1 - 3*x)*(2 - 2*x))/((1 + x)*(1 - 3*x)^2*(-3 + 9*x))
-2*x*(-1 + x)*(2 + 5*x)
-----------------------
2
3*(1 + x)*(-1 + 3*x)
$$- \frac{2 x \left(x - 1\right) \left(5 x + 2\right)}{3 \left(x + 1\right) \left(3 x - 1\right)^{2}}$$
-2*x*(-1 + x)*(2 + 5*x)/(3*(1 + x)*(-1 + 3*x)^2)
Combining rational expressions
[src]
-2*(1 - x)*(-3 + (1 + x)*(1 - 3*x) + 2*(1 + x)*(1 - x))
-------------------------------------------------------
2
3*(1 + x)*(1 - 3*x)
$$- \frac{2 \left(1 - x\right) \left(\left(1 - 3 x\right) \left(x + 1\right) + 2 \left(1 - x\right) \left(x + 1\right) - 3\right)}{3 \left(1 - 3 x\right)^{2} \left(x + 1\right)}$$
-2*(1 - x)*(-3 + (1 + x)*(1 - 3*x) + 2*(1 + x)*(1 - x))/(3*(1 + x)*(1 - 3*x)^2)