Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^5-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^5-x^2} x+\left (x^5-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x^2}-x}\right ) \]
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Rubi [F] time = 1.01, 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 {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^3}\right ) \int \frac {1+2 x^3}{x^{2/3} \sqrt [3]{-1+x^3} \left (-1+x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x^9}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [3]{-1+x^9}}+\frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (6 x^{2/3} \sqrt [3]{1-x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}-\frac {2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {6 x \sqrt [3]{1-x^3} \, _2F_1\left (\frac {1}{9},\frac {1}{3};\frac {10}{9};x^3\right )}{\sqrt [3]{-x^2+x^5}}-\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.05, size = 93, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^5-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^5-x^2} x+\left (x^5-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x^2}-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 124, normalized size = 1.33 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} - x}\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 {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.37, size = 385, normalized size = 4.14 \begin {gather*} -\ln \left (\frac {1462231700 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-9984360926 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-5117810950 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-1758826818 x^{4}-1462231700 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -17923671861 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-14667651144 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x +6431745091 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+9984360926 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +6512041434 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+35847343722 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}+1256304870 x^{2}+1758826818 x}{x \left (x^{3}+x -1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {251260974 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+7105231670 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-879413409 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+29244634000 x^{4}-251260974 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+6512041434 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x -19968721852 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-7105231670 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-71694687444 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}+11697853600 x^{2}-29244634000 x}{x \left (x^{3}+x -1\right )}\right )}{2} \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 {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}}\,{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 {2\,x^3+1}{{\left (x^5-x^2\right )}^{1/3}\,\left (x^3+x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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