Optimal. Leaf size=58 \[ \frac{6 x^{5/6}}{5}-3 \sqrt [3]{x}-3 \log \left (\sqrt [6]{x}+1\right )+\log \left (\sqrt{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.02957, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {341, 50, 56, 618, 204, 31} \[ \frac{6 x^{5/6}}{5}-3 \sqrt [3]{x}-3 \log \left (\sqrt [6]{x}+1\right )+\log \left (\sqrt{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
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Rule 341
Rule 50
Rule 56
Rule 618
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{x}}{1+\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{5/3}}{1+x} \, dx,x,\sqrt{x}\right )\\ &=\frac{6 x^{5/6}}{5}-2 \operatorname{Subst}\left (\int \frac{x^{2/3}}{1+x} \, dx,x,\sqrt{x}\right )\\ &=-3 \sqrt [3]{x}+\frac{6 x^{5/6}}{5}+2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{x} (1+x)} \, dx,x,\sqrt{x}\right )\\ &=-3 \sqrt [3]{x}+\frac{6 x^{5/6}}{5}+\log \left (1+\sqrt{x}\right )-3 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sqrt [6]{x}\right )+3 \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=-3 \sqrt [3]{x}+\frac{6 x^{5/6}}{5}-3 \log \left (1+\sqrt [6]{x}\right )+\log \left (1+\sqrt{x}\right )-6 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 \sqrt [6]{x}\right )\\ &=-3 \sqrt [3]{x}+\frac{6 x^{5/6}}{5}-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right )-3 \log \left (1+\sqrt [6]{x}\right )+\log \left (1+\sqrt{x}\right )\\ \end{align*}
Mathematica [C] time = 0.0078134, size = 35, normalized size = 0.6 \[ \frac{3}{5} \sqrt [3]{x} \left (5 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\sqrt{x}\right )+2 \sqrt{x}-5\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 49, normalized size = 0.8 \begin{align*}{\frac{6}{5}{x}^{{\frac{5}{6}}}}-3\,\sqrt [3]{x}+\ln \left ( \sqrt [3]{x}-\sqrt [6]{x}+1 \right ) +2\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,\sqrt [6]{x}-1 \right ) \sqrt{3} \right ) -2\,\ln \left ( 1+\sqrt [6]{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51049, size = 65, normalized size = 1.12 \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3426, size = 174, normalized size = 3. \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} x^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.40805, size = 138, normalized size = 2.38 \begin{align*} \frac{16 x^{\frac{5}{6}} \Gamma \left (\frac{8}{3}\right )}{5 \Gamma \left (\frac{11}{3}\right )} - \frac{8 \sqrt [3]{x} \Gamma \left (\frac{8}{3}\right )}{\Gamma \left (\frac{11}{3}\right )} - \frac{16 e^{- \frac{2 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 \log{\left (- \sqrt [6]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 e^{\frac{2 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0984, size = 65, normalized size = 1.12 \begin{align*} 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 3 \, x^{\frac{1}{3}} + \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 2 \, \log \left (x^{\frac{1}{6}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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