Optimal. Leaf size=111 \[ \frac {\left (x^3+x^2\right )^{2/3} (3 x-4)}{6 x}-\frac {2}{9} \log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{9} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 164, normalized size of antiderivative = 1.48, number of steps used = 4, 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 {2 \left (x^3+x^2\right )^{2/3}}{3 x}+\frac {1}{2} \left (x^3+x^2\right )^{2/3}-\frac {x^{2/3} \sqrt [3]{x+1} \log (x)}{9 \sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x}}-1\right )}{3 \sqrt [3]{x^3+x^2}}-\frac {2 x^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{3 \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^2}{\sqrt [3]{x^2+x^3}} \, dx &=\frac {1}{2} \left (x^2+x^3\right )^{2/3}-\frac {2}{3} \int \frac {x}{\sqrt [3]{x^2+x^3}} \, dx\\ &=\frac {1}{2} \left (x^2+x^3\right )^{2/3}-\frac {2 \left (x^2+x^3\right )^{2/3}}{3 x}+\frac {2}{9} \int \frac {1}{\sqrt [3]{x^2+x^3}} \, dx\\ &=\frac {1}{2} \left (x^2+x^3\right )^{2/3}-\frac {2 \left (x^2+x^3\right )^{2/3}}{3 x}+\frac {\left (2 x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x}} \, dx}{9 \sqrt [3]{x^2+x^3}}\\ &=\frac {1}{2} \left (x^2+x^3\right )^{2/3}-\frac {2 \left (x^2+x^3\right )^{2/3}}{3 x}-\frac {2 x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{3 \sqrt {3} \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log (x)}{9 \sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{x}}\right )}{3 \sqrt [3]{x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.34 \begin {gather*} \frac {3 x^3 \sqrt [3]{x+1} \, _2F_1\left (\frac {1}{3},\frac {7}{3};\frac {10}{3};-x\right )}{7 \sqrt [3]{x^2 (x+1)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 111, normalized size = 1.00 \begin {gather*} \frac {\left (x^3+x^2\right )^{2/3} (3 x-4)}{6 x}-\frac {2}{9} \log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{9} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 108, normalized size = 0.97 \begin {gather*} -\frac {4 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 4 \, x \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 2 \, 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 ) - 3 \, {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x - 4\right )}}{18 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 79, normalized size = 0.71 \begin {gather*} -\frac {1}{6} \, {\left (4 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{2} - \frac {2}{9} \, \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}{9} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \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 = 0.29, size = 36, normalized size = 0.32 \begin {gather*} \frac {\left (-4+3 x \right ) x \left (1+x \right )}{6 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}+\frac {2 x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x \right )}{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^{2}}{{\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^2}{{\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^{2}}{\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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