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

Optimal. Leaf size=90 \[ \log \left (\sqrt [3]{x^5+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5+1}+x}\right )+\frac {3 \left (x^5+1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (\sqrt [3]{x^5+1} x+\left (x^5+1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.76, 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 )}{x^3 \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))/(x^3*(1 - x^3 + x^5)),x]

[Out]

(3*Hypergeometric2F1[-2/3, -2/5, 3/5, -x^5])/(2*x^2) - 3*Defer[Int][(1 + x^5)^(2/3)/(1 - x^3 + x^5), x] + 5*De
fer[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 )}{x^3 \left (1-x^3+x^5\right )} \, dx &=\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^3}+\frac {\left (-3+5 x^2\right ) \left (1+x^5\right )^{2/3}}{1-x^3+x^5}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^5\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (-3+5 x^2\right ) \left (1+x^5\right )^{2/3}}{1-x^3+x^5} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}+\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 {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}-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 = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \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))/(x^3*(1 - x^3 + x^5)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.50, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \tan ^{-1}\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]

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

[Out]

(3*(1 + x^5)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^5)^(1/3))] + Log[-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 = 5.27, size = 135, normalized size = 1.50 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {67616276 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 10249526 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (1423013 \, x^{5} + 37509888 \, x^{3} + 1423013\right )}}{300763 \, x^{5} - 86350888 \, x^{3} + 300763}\right ) - x^{2} \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 (x^{5} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*x^2*arctan((67616276*sqrt(3)*(x^5 + 1)^(1/3)*x^2 + 10249526*sqrt(3)*(x^5 + 1)^(2/3)*x + sqrt(3
)*(1423013*x^5 + 37509888*x^3 + 1423013))/(300763*x^5 - 86350888*x^3 + 300763)) - x^2*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*(x^5 + 1)^(2/3))/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{{\left (x^{5} - x^{3} + 1\right )} x^{3}}\,{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)/x^3/(x^5-x^3+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 10.58, size = 289, normalized size = 3.21

method	result	size
risch	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+x^{5}+3 \left (x^{5}+1\right )^{\frac {2}{3}} x +3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}+x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{5}-x^{3}+1}\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+1\right )^{\frac {2}{3}} x +3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{x^{5}-x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+1\right )^{\frac {2}{3}} x +3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{x^{5}-x^{3}+1}\right )\)	\(289\)
trager	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}-3 \ln \left (\frac {970550226 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-1941100452 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-116825222 x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-1321025892 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-305544126 \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-175237833 x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-116825222}{x^{5}-x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\ln \left (-\frac {444836727 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-889673454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-175237833 x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x -1469304801 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x -305544126 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-58412611 x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-175237833}{x^{5}-x^{3}+1}\right )-\ln \left (\frac {970550226 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-1941100452 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-116825222 x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-1321025892 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-305544126 \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-175237833 x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-116825222}{x^{5}-x^{3}+1}\right )\)	\(594\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^5+1)^(2/3)/x^2+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))*RootO
f(_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} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{{\left (x^{5} - x^{3} + 1\right )} x^{3}}\,{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)/x^3/(x^5-x^3+1),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 + 1)^(2/3)/((x^5 - x^3 + 1)*x^3), 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 )}{x^3\,\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*(x^5 - x^3 + 1)),x)

[Out]

int(((x^5 + 1)^(2/3)*(2*x^5 - 3))/(x^3*(x^5 - x^3 + 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 )}{x^{3} \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)/x**3/(x**5-x**3+1),x)

[Out]

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

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