Optimal. Leaf size=117 \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0771903, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {341, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 341
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{1+x^{2/3}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^{7/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{6 x^{5/6}}{5}-3 \operatorname{Subst}\left (\int \frac{x^{3/2}}{1+x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+6 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+3 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )+3 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt [6]{x}\right )\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt [6]{x}\right )}{2 \sqrt{2}}\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}\\ &=-6 \sqrt [6]{x}+\frac{6 x^{5/6}}{5}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt [6]{x}+\sqrt [3]{x}\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0243979, size = 117, normalized size = 1. \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 76, normalized size = 0.7 \begin{align*}{\frac{6}{5}{x}^{{\frac{5}{6}}}}-6\,\sqrt [6]{x}+{\frac{3\,\sqrt{2}}{2}\arctan \left ( -1+\sqrt [6]{x}\sqrt{2} \right ) }+{\frac{3\,\sqrt{2}}{4}\ln \left ({ \left ( 1+\sqrt [3]{x}+\sqrt [6]{x}\sqrt{2} \right ) \left ( 1+\sqrt [3]{x}-\sqrt [6]{x}\sqrt{2} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{2}\arctan \left ( 1+\sqrt [6]{x}\sqrt{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44915, size = 119, normalized size = 1.02 \begin{align*} \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57731, size = 424, normalized size = 3.62 \begin{align*} -3 \, \sqrt{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1} - \sqrt{2} x^{\frac{1}{6}} - 1\right ) - 3 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4} - \sqrt{2} x^{\frac{1}{6}} + 1\right ) + \frac{3}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} x^{\frac{1}{6}} + 4 \, x^{\frac{1}{3}} + 4\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.10253, size = 187, normalized size = 1.6 \begin{align*} \frac{27 x^{\frac{5}{6}} \Gamma \left (\frac{9}{4}\right )}{10 \Gamma \left (\frac{13}{4}\right )} - \frac{27 \sqrt [6]{x} \Gamma \left (\frac{9}{4}\right )}{2 \Gamma \left (\frac{13}{4}\right )} - \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{3 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} - \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{7 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24713, size = 119, normalized size = 1.02 \begin{align*} \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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