\(\int \frac {(2+5 x^7) \sqrt [3]{-x-x^3+x^8}}{(-1+x^7) (-1+x^2+x^7)} \, dx\) [2697]

Optimal result

Integrand size = 41, antiderivative size = 245 \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x-x^3+x^8}}\right )-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{-x-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{-x-x^3+x^8}\right )+\sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{-x-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x-x^3+x^8}+\left (-x-x^3+x^8\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{-x-x^3+x^8}-\sqrt [3]{2} \left (-x-x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \] Output:

3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^8-x^3-x)^(1/3)))-2^(1/3)*3^(1/2)*arctan( 
3^(1/2)*x/(-x+2^(2/3)*(x^8-x^3-x)^(1/3)))-ln(x+(x^8-x^3-x)^(1/3))+2^(1/3)* 
ln(2*x+2^(2/3)*(x^8-x^3-x)^(1/3))+1/2*ln(x^2-x*(x^8-x^3-x)^(1/3)+(x^8-x^3- 
x)^(2/3))-1/2*ln(-2*x^2+2^(2/3)*x*(x^8-x^3-x)^(1/3)-2^(1/3)*(x^8-x^3-x)^(2 
/3))*2^(1/3)
 

Mathematica [A] (verified)

Time = 3.66 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.22 \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\frac {x^{2/3} \left (-1-x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-x^2+x^7}}\right )+2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{-1-x^2+x^7}}\right )-2 \log \left (x^{2/3}+\sqrt [3]{-1-x^2+x^7}\right )+2 \sqrt [3]{2} \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{-1-x^2+x^7}\right )+\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-x^2+x^7}+\left (-1-x^2+x^7\right )^{2/3}\right )-\sqrt [3]{2} \log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{-1-x^2+x^7}-\sqrt [3]{2} \left (-1-x^2+x^7\right )^{2/3}\right )\right )}{2 \left (x \left (-1-x^2+x^7\right )\right )^{2/3}} \] Input:

Integrate[((2 + 5*x^7)*(-x - x^3 + x^8)^(1/3))/((-1 + x^7)*(-1 + x^2 + x^7 
)),x]
 

Output:

(x^(2/3)*(-1 - x^2 + x^7)^(2/3)*(-2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2 
/3) - 2*(-1 - x^2 + x^7)^(1/3))] + 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/ 
3))/(x^(2/3) - 2^(2/3)*(-1 - x^2 + x^7)^(1/3))] - 2*Log[x^(2/3) + (-1 - x^ 
2 + x^7)^(1/3)] + 2*2^(1/3)*Log[2*x^(2/3) + 2^(2/3)*(-1 - x^2 + x^7)^(1/3) 
] + Log[x^(4/3) - x^(2/3)*(-1 - x^2 + x^7)^(1/3) + (-1 - x^2 + x^7)^(2/3)] 
 - 2^(1/3)*Log[-2*x^(4/3) + 2^(2/3)*x^(2/3)*(-1 - x^2 + x^7)^(1/3) - 2^(1/ 
3)*(-1 - x^2 + x^7)^(2/3)]))/(2*(x*(-1 - x^2 + x^7))^(2/3))
 

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.

\(\displaystyle \int \frac {\left (5 x^7+2\right ) \sqrt [3]{x^8-x^3-x}}{\left (x^7-1\right ) \left (x^7+x^2-1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [3]{x^8-x^3-x} \int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1} \left (5 x^7+2\right )}{\left (1-x^7\right ) \left (-x^7-x^2+1\right )}dx}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \int \frac {x \sqrt [3]{x^7-x^2-1} \left (5 x^7+2\right )}{\left (1-x^7\right ) \left (-x^7-x^2+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \int \left (\frac {\sqrt [3]{x^7-x^2-1} \left (-\sqrt [3]{x}-2\right )}{3 \left (x^{2/3}+\sqrt [3]{x}+1\right )}+\frac {\sqrt [3]{x^7-x^2-1}}{3 \left (\sqrt [3]{x}-1\right )}+\frac {\left (-x^{5/3}-2 x^{4/3}+4 x+3 x^{2/3}+2 \sqrt [3]{x}+1\right ) \sqrt [3]{x^7-x^2-1}}{3 \left (x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1\right )}+\frac {\left (x^{11/3}+11 x^3-10 x^{8/3}+8 x^2-7 x^{5/3}+x^{4/3}+5 x-3 \sqrt [3]{x}+2\right ) \sqrt [3]{x^7-x^2-1}}{3 \left (x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1\right )}-\frac {x \left (7 x^5+2\right ) \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sqrt [3]{x^8-x^3-x} \left (\frac {1}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{\sqrt [3]{x}-1}d\sqrt [3]{x}-\frac {1}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^7-x^2-1}}{2 \sqrt [3]{x}-i \sqrt {3}+1}d\sqrt [3]{x}-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^7-x^2-1}}{2 \sqrt [3]{x}+i \sqrt {3}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {2}{3} \int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\int \frac {x^{2/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {4}{3} \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {2}{3} \int \frac {x^{4/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {1}{3} \int \frac {x^{5/3} \sqrt [3]{x^7-x^2-1}}{x^2+x^{5/3}+x^{4/3}+x+x^{2/3}+\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {2}{3} \int \frac {\sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\int \frac {\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {5}{3} \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {x^{4/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {7}{3} \int \frac {x^{5/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {8}{3} \int \frac {x^2 \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-\frac {10}{3} \int \frac {x^{8/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {11}{3} \int \frac {x^3 \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}+\frac {1}{3} \int \frac {x^{11/3} \sqrt [3]{x^7-x^2-1}}{x^4-x^{11/3}+x^3-x^{8/3}+x^2-x^{4/3}+x-\sqrt [3]{x}+1}d\sqrt [3]{x}-2 \int \frac {x \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}d\sqrt [3]{x}-7 \int \frac {x^6 \sqrt [3]{x^7-x^2-1}}{x^7+x^2-1}d\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{x^7-x^2-1}}\)

Input:

Int[((2 + 5*x^7)*(-x - x^3 + x^8)^(1/3))/((-1 + x^7)*(-1 + x^2 + x^7)),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (5 x^{7}+2\right ) \left (x^{8}-x^{3}-x \right )^{\frac {1}{3}}}{\left (x^{7}-1\right ) \left (x^{7}+x^{2}-1\right )}d x\]

Input:

int((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x)
 

Output:

int((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x, algorithm="fr 
icas")
                                                                                    
                                                                                    
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (residue poly has multiple non-linear fac 
tors)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\text {Timed out} \] Input:

integrate((5*x**7+2)*(x**8-x**3-x)**(1/3)/(x**7-1)/(x**7+x**2-1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int { \frac {{\left (x^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}} \,d x } \] Input:

integrate((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x, algorithm="ma 
xima")
 

Output:

integrate((x^8 - x^3 - x)^(1/3)*(5*x^7 + 2)/((x^7 + x^2 - 1)*(x^7 - 1)), x 
)
 

Giac [F]

\[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int { \frac {{\left (x^{8} - x^{3} - x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2\right )}}{{\left (x^{7} + x^{2} - 1\right )} {\left (x^{7} - 1\right )}} \,d x } \] Input:

integrate((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x, algorithm="gi 
ac")
 

Output:

integrate((x^8 - x^3 - x)^(1/3)*(5*x^7 + 2)/((x^7 + x^2 - 1)*(x^7 - 1)), x 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=\int \frac {\left (5\,x^7+2\right )\,{\left (x^8-x^3-x\right )}^{1/3}}{\left (x^7-1\right )\,\left (x^7+x^2-1\right )} \,d x \] Input:

int(((5*x^7 + 2)*(x^8 - x^3 - x)^(1/3))/((x^7 - 1)*(x^2 + x^7 - 1)),x)
 

Output:

int(((5*x^7 + 2)*(x^8 - x^3 - x)^(1/3))/((x^7 - 1)*(x^2 + x^7 - 1)), x)
 

Reduce [F]

\[ \int \frac {\left (2+5 x^7\right ) \sqrt [3]{-x-x^3+x^8}}{\left (-1+x^7\right ) \left (-1+x^2+x^7\right )} \, dx=5 \left (\int \frac {x^{\frac {22}{3}} \left (x^{7}-x^{2}-1\right )^{\frac {1}{3}}}{x^{14}+x^{9}-2 x^{7}-x^{2}+1}d x \right )+2 \left (\int \frac {x^{\frac {1}{3}} \left (x^{7}-x^{2}-1\right )^{\frac {1}{3}}}{x^{14}+x^{9}-2 x^{7}-x^{2}+1}d x \right ) \] Input:

int((5*x^7+2)*(x^8-x^3-x)^(1/3)/(x^7-1)/(x^7+x^2-1),x)
 

Output:

5*int((x**(1/3)*(x**7 - x**2 - 1)**(1/3)*x**7)/(x**14 + x**9 - 2*x**7 - x* 
*2 + 1),x) + 2*int((x**(1/3)*(x**7 - x**2 - 1)**(1/3))/(x**14 + x**9 - 2*x 
**7 - x**2 + 1),x)