\(\int \frac {(2+x^3) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx\) [696]

Optimal result

Integrand size = 42, antiderivative size = 54 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=-\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=-\text {RootSum}\left [-1+\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x-x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{1+2 \text {$\#$1}^3}\&\right ] \] Input:

Integrate[((2 + x^3)*(x - x^4)^(1/3))/(1 + x^2 - 2*x^3 - x^4 - x^5 + x^6), 
x]
 

Output:

-RootSum[-1 + #1^3 + #1^6 & , (-(Log[x]*#1) + Log[(x - x^4)^(1/3) - x*#1]* 
#1)/(1 + 2*#1^3) & ]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (x^3+2\right ) \sqrt [3]{x-x^4}}{x^6-x^5-x^4-2 x^3+x^2+1} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x-x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^3} \left (x^3+2\right )}{x^6-x^5-x^4-2 x^3+x^2+1}dx}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{x-x^4} \int \frac {x \sqrt [3]{1-x^3} \left (x^3+2\right )}{x^6-x^5-x^4-2 x^3+x^2+1}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {3 \sqrt [3]{x-x^4} \int \left (\frac {\sqrt [3]{1-x^3} x^4}{x^6-x^5-x^4-2 x^3+x^2+1}+\frac {2 \sqrt [3]{1-x^3} x}{x^6-x^5-x^4-2 x^3+x^2+1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x-x^4} \left (2 \int \frac {x \sqrt [3]{1-x^3}}{x^6-x^5-x^4-2 x^3+x^2+1}d\sqrt [3]{x}+\int \frac {x^4 \sqrt [3]{1-x^3}}{x^6-x^5-x^4-2 x^3+x^2+1}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^3}}\)

Input:

Int[((2 + x^3)*(x - x^4)^(1/3))/(1 + x^2 - 2*x^3 - x^4 - x^5 + x^6),x]
 

Output:

$Aborted
 

Maple [N/A] (verified)

Time = 131.94 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+\textit {\_Z}^{3}-1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (-x^{4}+x \right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}+1}\right )\)	\(45\)
trager	\(\text {Expression too large to display}\)	\(17892\)

Input:

int((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x,method=_RETURNVERBO 
SE)
 

Output:

-sum(_R*ln((-_R*x+(-x^4+x)^(1/3))/x)/(2*_R^3+1),_R=RootOf(_Z^6+_Z^3-1))
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="f 
ricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (trace 0)
 

Sympy [N/A]

Not integrable

Time = 2.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {\sqrt [3]{- x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} + 2\right )}{x^{6} - x^{5} - x^{4} - 2 x^{3} + x^{2} + 1}\, dx \] Input:

integrate((x**3+2)*(-x**4+x)**(1/3)/(x**6-x**5-x**4-2*x**3+x**2+1),x)
 

Output:

Integral((-x*(x - 1)*(x**2 + x + 1))**(1/3)*(x**3 + 2)/(x**6 - x**5 - x**4 
 - 2*x**3 + x**2 + 1), x)
 

Maxima [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \] Input:

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="m 
axima")
 

Output:

integrate((-x^4 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 - x^4 - 2*x^3 + x^2 + 1), 
x)
 

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int { \frac {{\left (-x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{x^{6} - x^{5} - x^{4} - 2 \, x^{3} + x^{2} + 1} \,d x } \] Input:

integrate((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x, algorithm="g 
iac")
 

Output:

integrate((-x^4 + x)^(1/3)*(x^3 + 2)/(x^6 - x^5 - x^4 - 2*x^3 + x^2 + 1), 
x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {{\left (x-x^4\right )}^{1/3}\,\left (x^3+2\right )}{x^6-x^5-x^4-2\,x^3+x^2+1} \,d x \] Input:

int(((x - x^4)^(1/3)*(x^3 + 2))/(x^2 - 2*x^3 - x^4 - x^5 + x^6 + 1),x)
 

Output:

int(((x - x^4)^(1/3)*(x^3 + 2))/(x^2 - 2*x^3 - x^4 - x^5 + x^6 + 1), x)
 

Reduce [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.54 \[ \int \frac {\left (2+x^3\right ) \sqrt [3]{x-x^4}}{1+x^2-2 x^3-x^4-x^5+x^6} \, dx=\int \frac {x^{\frac {10}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}}{x^{6}-x^{5}-x^{4}-2 x^{3}+x^{2}+1}d x +2 \left (\int \frac {x^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}}{x^{6}-x^{5}-x^{4}-2 x^{3}+x^{2}+1}d x \right ) \] Input:

int((x^3+2)*(-x^4+x)^(1/3)/(x^6-x^5-x^4-2*x^3+x^2+1),x)
 

Output:

int((x**(1/3)*( - x**3 + 1)**(1/3)*x**3)/(x**6 - x**5 - x**4 - 2*x**3 + x* 
*2 + 1),x) + 2*int((x**(1/3)*( - x**3 + 1)**(1/3))/(x**6 - x**5 - x**4 - 2 
*x**3 + x**2 + 1),x)