Optimal. Leaf size=105 \[ \frac {1}{2} \log \left (x^2 \left (x^2+1\right )^{2/3}\right )-\frac {1}{2} \log \left (\left (x^2+1\right )^{2/3} x^2+\sqrt [3]{x^2+1} x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{x^2+1}}{\sqrt [3]{x^2+1} x+2}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{x^2+1}\right ) \]
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Rubi [F] time = 0.55, 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 {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx &=\int \left (\frac {3 x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}+\frac {5 x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}\right ) \, dx\\ &=3 \int \frac {x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx+5 \int \frac {x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.56, size = 105, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (x^2 \left (x^2+1\right )^{2/3}\right )-\frac {1}{2} \log \left (\left (x^2+1\right )^{2/3} x^2+\sqrt [3]{x^2+1} x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt [3]{x^2+1}}{\sqrt [3]{x^2+1} x+2}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.80, size = 103, normalized size = 0.98 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{5} + x^{3}\right )}}{x^{5} + x^{3} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{5} + x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{2} + 3\right )} x}{{\left (x^{5} + x^{3} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.62, size = 282, normalized size = 2.69 \begin {gather*} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{2} \left (x^{2}+1\right )^{\frac {2}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x -x \left (x^{2}+1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{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 )^{2} x^{5}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+2 x^{2} \left (x^{2}+1\right )^{\frac {2}{3}}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x -x^{3}-x \left (x^{2}+1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{5}+x^{3}-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{2} + 3\right )} x}{{\left (x^{5} + x^{3} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (5\,x^2+3\right )}{{\left (x^2+1\right )}^{1/3}\,\left (x^5+x^3-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (5 x^{2} + 3\right )}{\sqrt [3]{x^{2} + 1} \left (x^{5} + x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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