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

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

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Rubi [F]  time = 1.23, 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} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x} \, dx,x,x^5\right )+\frac {6}{5} \operatorname {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*}

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Mathematica [F]  time = 0.29, size = 0, normalized size = 0.00 \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]

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]

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

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IntegrateAlgebraic [A]  time = 3.50, size = 105, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^5-1}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-1}-x}\right )+\frac {3 \left (x^5-1\right )^{2/3} \left (4 x^5-5 x^3-4\right )}{10 x^5}+\frac {1}{2} \log \left (-\sqrt [3]{x^5-1} x+\left (x^5-1\right )^{2/3}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-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))] - L
og[x + (-1 + x^5)^(1/3)] + Log[x^2 - x*(-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/2

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fricas [A]  time = 6.14, 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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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)

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maple [C]  time = 3.53, size = 326, normalized size = 3.10 \begin {gather*} \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}}}+\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 ) \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}+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 ) \end {gather*}

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)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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)

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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)

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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}} \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)

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