\(\int \frac {(-1+x^5)^{2/3} (3+2 x^5) (-2+x^3+2 x^5)}{x^6 (-1+x^3+x^5)} \, dx\) [1511]

Optimal result

Integrand size = 40, antiderivative size = 105 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]

[Out]

3/10*(x^5-1)^(2/3)*(4*x^5-5*x^3-4)/x^5-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^5-1)^(1/3)))-ln(x+(x^5-1)^(1/3))+1/2*
ln(x^2-x*(x^5-1)^(1/3)+(x^5-1)^(2/3))

Rubi [F]

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

[In]

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

[Out]

(6*(-1 + x^5)^(2/3))/5 - (6*(-1 + x^5)^(2/3))/(5*x^5) - (3*(-1 + x^5)^(2/3)*Hypergeometric2F1[-2/3, -2/5, 3/5,
x^5])/(2*x^2*(1 - x^5)^(2/3)) - 3*Defer[Int][(-1 + x^5)^(2/3)/(-1 + x^3 + x^5), x] - 5*Defer[Int][(x^2*(-1 +
x^5)^(2/3))/(-1 + x^3 + x^5), x]

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

Mathematica [A] (verified)

Time = 2.48 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]

[In]

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

[Out]

(3*(-1 + x^5)^(2/3)*(-4 - 5*x^3 + 4*x^5))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(-1 + x^5)^(1/3))] - Lo
g[x + (-1 + x^5)^(1/3)] + Log[x^2 - x*(-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/2

Maple [A] (verified)

Time = 11.33 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {-10 \ln \left (\frac {x +\left (x^{5}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (12 x^{5}-15 x^{3}-12\right ) \left (x^{5}-1\right )^{\frac {2}{3}}+5 x^{5} \left (-2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{5}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {x^{2}-x \left (x^{5}-1\right )^{\frac {1}{3}}+\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{10 x^{5}}\)	\(103\)
risch	\(\frac {\frac {6}{5} x^{10}-\frac {3}{2} x^{8}-\frac {12}{5} x^{5}+\frac {3}{2} x^{3}+\frac {6}{5}}{x^{5} \left (x^{5}-1\right )^{\frac {1}{3}}}-\ln \left (\frac {x^{5} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}-1\right )^{\frac {2}{3}} x +2 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-2 x^{5} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+x^{5}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{x^{5}+x^{3}-1}\right )\)	\(260\)
trager	\(\text {Expression too large to display}\)	\(601\)

[In]

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

[Out]

1/10*(-10*ln((x+(x^5-1)^(1/3))/x)*x^5+(12*x^5-15*x^3-12)*(x^5-1)^(2/3)+5*x^5*(-2*3^(1/2)*arctan(1/3*(x-2*(x^5-
1)^(1/3))*3^(1/2)/x)+ln((x^2-x*(x^5-1)^(1/3)+(x^5-1)^(2/3))/x^2)))/x^5

Fricas [A] (verification not implemented)

none

Time = 3.85 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1092 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 2002 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (121 \, x^{5} + 576 \, x^{3} - 121\right )}}{3 \, {\left (1331 \, x^{5} - 216 \, x^{3} - 1331\right )}}\right ) + 5 \, x^{5} \log \left (\frac {x^{5} + x^{3} + 3 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) - 3 \, {\left (4 \, x^{5} - 5 \, x^{3} - 4\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]

[In]

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

[Out]

-1/10*(10*sqrt(3)*x^5*arctan(1/3*(1092*sqrt(3)*(x^5 - 1)^(1/3)*x^2 + 2002*sqrt(3)*(x^5 - 1)^(2/3)*x + sqrt(3)*
(121*x^5 + 576*x^3 - 121))/(1331*x^5 - 216*x^3 - 1331)) + 5*x^5*log((x^5 + x^3 + 3*(x^5 - 1)^(1/3)*x^2 + 3*(x^
5 - 1)^(2/3)*x - 1)/(x^5 + x^3 - 1)) - 3*(4*x^5 - 5*x^3 - 4)*(x^5 - 1)^(2/3))/x^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int \frac {{\left (x^5-1\right )}^{2/3}\,\left (2\,x^5+3\right )\,\left (2\,x^5+x^3-2\right )}{x^6\,\left (x^5+x^3-1\right )} \,d x \]

[In]

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

[Out]

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