Optimal. Leaf size=83 \[ -\log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 129, normalized size of antiderivative = 1.55, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2011, 59} \begin {gather*} -\frac {x^{2/3} \sqrt [3]{x+1} \log (x)}{2 \sqrt [3]{x^3+x^2}}-\frac {3 x^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3+x^2}}-\frac {\sqrt {3} 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 )}{\sqrt [3]{x^3+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 2011
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{x^2+x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{1+x}} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=-\frac {\sqrt {3} 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 )}{\sqrt [3]{x^2+x^3}}-\frac {x^{2/3} \sqrt [3]{1+x} \log (x)}{2 \sqrt [3]{x^2+x^3}}-\frac {3 x^{2/3} \sqrt [3]{1+x} \log \left (-1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.41 \begin {gather*} \frac {3 x \sqrt [3]{x+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-x\right )}{\sqrt [3]{x^2 (x+1)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 83, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 84, normalized size = 1.01 \begin {gather*} -\sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.03, size = 55, normalized size = 0.66 \begin {gather*} -\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}{2} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \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 = 0.31, size = 15, normalized size = 0.18 \begin {gather*} 3 x^{\frac {1}{3}} \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x \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 {1}{{\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 [B] time = 0.95, size = 25, normalized size = 0.30 \begin {gather*} \frac {3\,x\,{\left (x+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ -x\right )}{{\left (x^3+x^2\right )}^{1/3}} \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 {1}{\sqrt [3]{x^{3} + x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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