Optimal. Leaf size=39 \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
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Rubi [A] time = 0.0155377, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{2/3}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^4}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (-1+x-x^2+x^3+\frac{1}{1+x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 \sqrt [3]{x}+\frac{3 x^{2/3}}{2}-x+\frac{3 x^{4/3}}{4}+3 \log \left (1+\sqrt [3]{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0132745, size = 39, normalized size = 1. \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.7 \begin{align*} -3\,\sqrt [3]{x}+{\frac{3}{2}{x}^{{\frac{2}{3}}}}-x+{\frac{3}{4}{x}^{{\frac{4}{3}}}}+3\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01202, size = 57, normalized size = 1.46 \begin{align*} \frac{3}{4} \,{\left (x^{\frac{1}{3}} + 1\right )}^{4} - 4 \,{\left (x^{\frac{1}{3}} + 1\right )}^{3} + 9 \,{\left (x^{\frac{1}{3}} + 1\right )}^{2} - 12 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) - 12 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5236, size = 81, normalized size = 2.08 \begin{align*} \frac{3}{4} \,{\left (x - 4\right )} x^{\frac{1}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.206077, size = 34, normalized size = 0.87 \begin{align*} \frac{3 x^{\frac{4}{3}}}{4} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} - x + 3 \log{\left (\sqrt [3]{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19845, size = 36, normalized size = 0.92 \begin{align*} \frac{3}{4} \, x^{\frac{4}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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