\(\int \frac {\sqrt [3]{1-x^7} (-2+x^3+2 x^7) (3+4 x^7)}{x^2 (-1+x^7) (-4+x^3+4 x^7)} \, dx\) [2026]

Optimal result

Integrand size = 51, antiderivative size = 144 \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \] Output:

3/2*(-x^7+1)^(1/3)/x-1/4*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(2/3)*(-x^7+1)^(1 
/3)))*2^(1/3)-1/4*ln(-x+2^(2/3)*(-x^7+1)^(1/3))*2^(1/3)+1/8*ln(x^2+2^(2/3) 
*x*(-x^7+1)^(1/3)+2*2^(1/3)*(-x^7+1)^(2/3))*2^(1/3)
 

Mathematica [A] (verified)

Time = 9.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \] Input:

Integrate[((1 - x^7)^(1/3)*(-2 + x^3 + 2*x^7)*(3 + 4*x^7))/(x^2*(-1 + x^7) 
*(-4 + x^3 + 4*x^7)),x]
 

Output:

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

\(\displaystyle \int \frac {\sqrt [3]{1-x^7} \left (2 x^7+x^3-2\right ) \left (4 x^7+3\right )}{x^2 \left (x^7-1\right ) \left (4 x^7+x^3-4\right )} \, dx\)

\(\Big \downarrow \) 281

\(\displaystyle -\int \frac {\left (-2 x^7-x^3+2\right ) \left (4 x^7+3\right )}{x^2 \left (-4 x^7-x^3+4\right ) \left (1-x^7\right )^{2/3}}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\int \left (\frac {2 x^5}{\left (1-x^7\right )^{2/3}}+\frac {x}{2 \left (1-x^7\right )^{2/3}}-\frac {\left (x^3-7\right ) x}{2 \left (1-x^7\right )^{2/3} \left (4 x^7+x^3-4\right )}+\frac {3}{2 \left (1-x^7\right )^{2/3} x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7}{2} \int \frac {x}{\left (1-x^7\right )^{2/3} \left (4 x^7+x^3-4\right )}dx+\frac {1}{2} \int \frac {x^4}{\left (1-x^7\right )^{2/3} \left (4 x^7+x^3-4\right )}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{7},\frac {2}{3},\frac {6}{7},x^7\right )}{2 x}-\frac {1}{3} x^6 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {6}{7},\frac {13}{7},x^7\right )-\frac {1}{4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{7},\frac {2}{3},\frac {9}{7},x^7\right )\)

Input:

Int[((1 - x^7)^(1/3)*(-2 + x^3 + 2*x^7)*(3 + 4*x^7))/(x^2*(-1 + x^7)*(-4 + 
 x^3 + 4*x^7)),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 46.15 (sec) , antiderivative size = 1070, normalized size of antiderivative = 7.43

\[\text {Expression too large to display}\]

Input:

int((-x^7+1)^(1/3)*(2*x^7+x^3-2)*(4*x^7+3)/x^2/(x^7-1)/(4*x^7+x^3-4),x)
 

Output:

-3/2*(x^7-1)/x/(-x^7+1)^(2/3)-(1/2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^ 
3-2)+4*_Z^2)*ln(-(-4*RootOf(_Z^3-2)*x^14+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro 
otOf(_Z^3-2)+4*_Z^2)*x^14+RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z 
^2)*RootOf(_Z^3-2)^3*x^10-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4* 
_Z^2)^2*RootOf(_Z^3-2)^2*x^10-6*(x^14-2*x^7+1)^(1/3)*RootOf(_Z^3-2)^2*x^8- 
12*(x^14-2*x^7+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2 
)*RootOf(_Z^3-2)*x^8+RootOf(_Z^3-2)*x^10-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro 
otOf(_Z^3-2)+4*_Z^2)*x^10+8*RootOf(_Z^3-2)*x^7-16*RootOf(RootOf(_Z^3-2)^2+ 
2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^7-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2 
)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3- 
2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-6*(x^14-2*x^7+1)^(2/3)*RootOf(RootOf(_Z^ 
3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x^2+6*(x^14-2*x^7+1)^( 
1/3)*RootOf(_Z^3-2)^2*x+12*(x^14-2*x^7+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2* 
_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-RootOf(_Z^3-2)*x^3+2*RootOf(Roo 
tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-4*RootOf(_Z^3-2)+8*RootOf(Ro 
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(-1+x)/(x^6+x^5+x^4+x^3+x^2+x+ 
1)/(4*x^7+x^3-4))+1/4*RootOf(_Z^3-2)*ln(-(4*RootOf(_Z^3-2)*x^14-8*RootOf(R 
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^14+RootOf(RootOf(_Z^3-2)^2+2 
*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^10-2*RootOf(RootOf(_Z^3-2)^2 
+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^10+6*(x^14-2*x^7+1)^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\text {Timed out} \] Input:

integrate((-x^7+1)^(1/3)*(2*x^7+x^3-2)*(4*x^7+3)/x^2/(x^7-1)/(4*x^7+x^3-4) 
,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\text {Timed out} \] Input:

integrate((-x**7+1)**(1/3)*(2*x**7+x**3-2)*(4*x**7+3)/x**2/(x**7-1)/(4*x** 
7+x**3-4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\int { \frac {{\left (4 \, x^{7} + 3\right )} {\left (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}} \,d x } \] Input:

integrate((-x^7+1)^(1/3)*(2*x^7+x^3-2)*(4*x^7+3)/x^2/(x^7-1)/(4*x^7+x^3-4) 
,x, algorithm="maxima")
 

Output:

integrate((4*x^7 + 3)*(2*x^7 + x^3 - 2)*(-x^7 + 1)^(1/3)/((4*x^7 + x^3 - 4 
)*(x^7 - 1)*x^2), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\int { \frac {{\left (4 \, x^{7} + 3\right )} {\left (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}} \,d x } \] Input:

integrate((-x^7+1)^(1/3)*(2*x^7+x^3-2)*(4*x^7+3)/x^2/(x^7-1)/(4*x^7+x^3-4) 
,x, algorithm="giac")
 

Output:

integrate((4*x^7 + 3)*(2*x^7 + x^3 - 2)*(-x^7 + 1)^(1/3)/((4*x^7 + x^3 - 4 
)*(x^7 - 1)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\int -\frac {\left (4\,x^7+3\right )\,\left (2\,x^7+x^3-2\right )}{x^2\,{\left (1-x^7\right )}^{2/3}\,\left (4\,x^7+x^3-4\right )} \,d x \] Input:

int(-((4*x^7 + 3)*(x^3 + 2*x^7 - 2))/(x^2*(1 - x^7)^(2/3)*(x^3 + 4*x^7 - 4 
)),x)
 

Output:

int(-((4*x^7 + 3)*(x^3 + 2*x^7 - 2))/(x^2*(1 - x^7)^(2/3)*(x^3 + 4*x^7 - 4 
)), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=-6 \left (\int \frac {\left (-x^{7}+1\right )^{\frac {1}{3}}}{4 x^{16}+x^{12}-8 x^{9}-x^{5}+4 x^{2}}d x \right )+8 \left (\int \frac {\left (-x^{7}+1\right )^{\frac {1}{3}} x^{12}}{4 x^{14}+x^{10}-8 x^{7}-x^{3}+4}d x \right )+4 \left (\int \frac {\left (-x^{7}+1\right )^{\frac {1}{3}} x^{8}}{4 x^{14}+x^{10}-8 x^{7}-x^{3}+4}d x \right )-2 \left (\int \frac {\left (-x^{7}+1\right )^{\frac {1}{3}} x^{5}}{4 x^{14}+x^{10}-8 x^{7}-x^{3}+4}d x \right )+3 \left (\int \frac {\left (-x^{7}+1\right )^{\frac {1}{3}} x}{4 x^{14}+x^{10}-8 x^{7}-x^{3}+4}d x \right ) \] Input:

int((-x^7+1)^(1/3)*(2*x^7+x^3-2)*(4*x^7+3)/x^2/(x^7-1)/(4*x^7+x^3-4),x)
 

Output:

 - 6*int(( - x**7 + 1)**(1/3)/(4*x**16 + x**12 - 8*x**9 - x**5 + 4*x**2),x 
) + 8*int((( - x**7 + 1)**(1/3)*x**12)/(4*x**14 + x**10 - 8*x**7 - x**3 + 
4),x) + 4*int((( - x**7 + 1)**(1/3)*x**8)/(4*x**14 + x**10 - 8*x**7 - x**3 
 + 4),x) - 2*int((( - x**7 + 1)**(1/3)*x**5)/(4*x**14 + x**10 - 8*x**7 - x 
**3 + 4),x) + 3*int((( - x**7 + 1)**(1/3)*x)/(4*x**14 + x**10 - 8*x**7 - x 
**3 + 4),x)