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

Optimal result

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

[Out]

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)

Rubi [F]

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

[In]

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]

[Out]

(3*Hypergeometric2F1[-1/7, 2/3, 6/7, x^7])/(2*x) - (x^2*Hypergeometric2F1[2/7, 2/3, 9/7, x^7])/4 - (x^6*Hyperg
eometric2F1[2/3, 6/7, 13/7, x^7])/3 - (7*Defer[Int][x/((1 - x^7)^(2/3)*(-4 + x^3 + 4*x^7)), x])/2 + Defer[Int]
[x^4/((1 - x^7)^(2/3)*(-4 + x^3 + 4*x^7)), x]/2

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx \\ & = -\int \left (\frac {3}{2 x^2 \left (1-x^7\right )^{2/3}}+\frac {x}{2 \left (1-x^7\right )^{2/3}}+\frac {2 x^5}{\left (1-x^7\right )^{2/3}}-\frac {x \left (-7+x^3\right )}{2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {x}{\left (1-x^7\right )^{2/3}} \, dx\right )+\frac {1}{2} \int \frac {x \left (-7+x^3\right )}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {3}{2} \int \frac {1}{x^2 \left (1-x^7\right )^{2/3}} \, dx-2 \int \frac {x^5}{\left (1-x^7\right )^{2/3}} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{7},\frac {2}{3},\frac {6}{7},x^7\right )}{2 x}-\frac {1}{4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{7},\frac {2}{3},\frac {9}{7},x^7\right )-\frac {1}{3} x^6 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {6}{7},\frac {13}{7},x^7\right )+\frac {1}{2} \int \left (-\frac {7 x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}+\frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{7},\frac {2}{3},\frac {6}{7},x^7\right )}{2 x}-\frac {1}{4} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{7},\frac {2}{3},\frac {9}{7},x^7\right )-\frac {1}{3} x^6 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {6}{7},\frac {13}{7},x^7\right )+\frac {1}{2} \int \frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {7}{2} \int \frac {x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 9.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx=\frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \]

[In]

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]

[Out]

(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))

Maple [C] (warning: unable to verify)

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

Time = 46.13 (sec) , antiderivative size = 1486, normalized size of antiderivative = 10.32

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

[In]

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)

[Out]

-3/2*(x^7-1)/x/(-x^7+1)^(2/3)+(1/4*RootOf(_Z^3+2)*ln((4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^
14+4*RootOf(_Z^3+2)*x^14+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^10+RootOf(Ro
otOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^10-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4
*_Z^2)*x^10-3*(x^14-2*x^7+1)^(1/3)*RootOf(_Z^3+2)^2*x^8-RootOf(_Z^3+2)*x^10-8*RootOf(RootOf(_Z^3+2)^2+2*_Z*Roo
tOf(_Z^3+2)+4*_Z^2)*x^7-8*RootOf(_Z^3+2)*x^7-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3
+2)^2*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-3*RootOf(RootOf(_Z^3+2)^2+2
*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(x^14-2*x^7+1)^(2/3)*x^2+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+
2)+4*_Z^2)*x^3+3*(x^14-2*x^7+1)^(1/3)*RootOf(_Z^3+2)^2*x+RootOf(_Z^3+2)*x^3+4*RootOf(RootOf(_Z^3+2)^2+2*_Z*Roo
tOf(_Z^3+2)+4*_Z^2)+4*RootOf(_Z^3+2))/(x^6+x^5+x^4+x^3+x^2+x+1)/(4*x^7+x^3-4)/(-1+x))-1/4*ln((-4*RootOf(RootOf
(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^14-4*RootOf(_Z^3+2)*x^14+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+
4*_Z^2)^2*RootOf(_Z^3+2)^2*x^10+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^10-6*Ro
otOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^14-2*x^7+1)^(1/3)*x^8+8*RootOf(RootOf(_Z^3
+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^7+8*RootOf(_Z^3+2)*x^7-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)
^2*RootOf(_Z^3+2)^2*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3+3*RootOf(Root
Of(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(x^14-2*x^7+1)^(2/3)*x^2+6*RootOf(RootOf(_Z^3+2)^2+2
*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^14-2*x^7+1)^(1/3)*x-3*(x^14-2*x^7+1)^(2/3)*x^2-4*RootOf(RootOf(_Z
^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-4*RootOf(_Z^3+2))/(x^6+x^5+x^4+x^3+x^2+x+1)/(4*x^7+x^3-4)/(-1+x))*RootOf(_
Z^3+2)-1/2*ln((-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^14-4*RootOf(_Z^3+2)*x^14+RootOf(RootOf
(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^10+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_
Z^2)*RootOf(_Z^3+2)^3*x^10-6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^14-2*x^7+1)
^(1/3)*x^8+8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^7+8*RootOf(_Z^3+2)*x^7-RootOf(RootOf(_Z^3+2
)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*Roo
tOf(_Z^3+2)^3*x^3+3*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*(x^14-2*x^7+1)^(2/3)*
x^2+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*(x^14-2*x^7+1)^(1/3)*x-3*(x^14-2*x^7+
1)^(2/3)*x^2-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-4*RootOf(_Z^3+2))/(x^6+x^5+x^4+x^3+x^2+x+1)
/(4*x^7+x^3-4)/(-1+x))*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(-x^7+1)^(2/3)*((x^7-1)^2)^(1/3)

Fricas [F(-1)]

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

[In]

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")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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)

[Out]

Timed out

Maxima [F]

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

[In]

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")

[Out]

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]

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

[In]

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")

[Out]

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

[In]

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)

[Out]

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)