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

Optimal. Leaf size=105 \[ \frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]

[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]
time = 0.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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} \, _2F_1\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]
time = 2.07, size = 102, normalized size = 0.97 \begin {gather*} \frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}+\sqrt {3} \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 20.01, size = 333, normalized size = 3.17

method	result	size
risch	\(\frac {-\frac {3}{2} x^{8}+\frac {3}{2} x^{3}-\frac {12}{5} x^{5}+\frac {6}{5}+\frac {6}{5} x^{10}}{x^{5} \left (x^{5}-1\right )^{\frac {1}{3}}}-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{5}+x^{3}-1}\right )+\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )\)	\(333\)
trager	\(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}} \left (4 x^{5}-5 x^{3}-4\right )}{10 x^{5}}-\ln \left (\frac {1973169492632104665894772909105152 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-97346550098888531109714265679328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-7646031783949405580342245022782464 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+714468692923457785263401407273 x^{5}-105550256795540691231071445123936 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x +19277590060830940057788253380576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+127681310055207165244114369615680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+1300290071420537825925621859422 \left (x^{5}-1\right )^{\frac {2}{3}} x +1099481841620215533656994220041 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-530089675394823518098652657009 x^{3}-1973169492632104665894772909105152 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+97346550098888531109714265679328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-714468692923457785263401407273}{x^{5}+x^{3}-1}\right )+96 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (-\frac {1012432749095564771328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}+22999020922536659328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-3923176902745313488896 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-219255606241635155 x^{5}-29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -65139908061224566944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+38915943346923997728 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-369746647037727189 \left (x^{5}-1\right )^{\frac {2}{3}} x +308794061933362050 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-76262819562307880 x^{3}-1012432749095564771328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-22999020922536659328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+219255606241635155}{x^{5}+x^{3}-1}\right )\)	\(414\)

Verification of antiderivative is not currently implemented for this CAS.

[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]

3/10*(4*x^10-5*x^8-8*x^5+5*x^3+4)/x^5/(x^5-1)^(1/3)-ln(-(RootOf(_Z^2-_Z+1)*x^5-RootOf(_Z^2-_Z+1)^2*x^3-2*x^5+2
*RootOf(_Z^2-_Z+1)*x^3+3*(x^5-1)^(2/3)*x-3*(x^5-1)^(1/3)*x^2-RootOf(_Z^2-_Z+1)+2)/(x^5+x^3-1))*RootOf(_Z^2-_Z+
1)+RootOf(_Z^2-_Z+1)*ln(-(-RootOf(_Z^2-_Z+1)*x^5-RootOf(_Z^2-_Z+1)^2*x^3-x^5+3*(x^5-1)^(2/3)*x-3*(x^5-1)^(1/3)
*x^2+x^3+RootOf(_Z^2-_Z+1)+1)/(x^5+x^3-1))+ln(-(RootOf(_Z^2-_Z+1)*x^5-RootOf(_Z^2-_Z+1)^2*x^3-2*x^5+2*RootOf(_
Z^2-_Z+1)*x^3+3*(x^5-1)^(2/3)*x-3*(x^5-1)^(1/3)*x^2-RootOf(_Z^2-_Z+1)+2)/(x^5+x^3-1))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]
time = 3.46, size = 144, normalized size = 1.37 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________