Optimal. Leaf size=110 \[ \frac {\left (x^3-x^2\right )^{2/3}}{x}-\frac {1}{3} \log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{6} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 152, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2024, 2011, 59} \begin {gather*} \frac {\left (x^3-x^2\right )^{2/3}}{x}-\frac {\sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{6 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{x^3-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2011
Rule 2024
Rubi steps
\begin {align*} \int \frac {x}{\sqrt [3]{-x^2+x^3}} \, dx &=\frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {1}{3} \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx\\ &=\frac {\left (-x^2+x^3\right )^{2/3}}{x}+\frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{3 \sqrt [3]{-x^2+x^3}}\\ &=\frac {\left (-x^2+x^3\right )^{2/3}}{x}-\frac {\sqrt [3]{-1+x} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{6 \sqrt [3]{-x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.32 \begin {gather*} \frac {3 \left ((x-1) x^2\right )^{2/3} \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};1-x\right )}{2 x^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 110, normalized size = 1.00 \begin {gather*} \frac {\left (x^3-x^2\right )^{2/3}}{x}-\frac {1}{3} \log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{6} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 113, normalized size = 1.03 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 2 \, x \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - x \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 6 \, {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{6 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 74, normalized size = 0.67 \begin {gather*} x {\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 41, normalized size = 0.37 \begin {gather*} \frac {x \left (-1+x \right )}{\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}+\frac {\left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\mathrm {signum}\left (-1+x \right )^{\frac {1}{3}}} \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 {x}{{\left (x^{3} - x^{2}\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 (x^3-x^2\right )}^{1/3}} \,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}{\sqrt [3]{x^{2} \left (x - 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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