Optimal. Leaf size=200 \[ 2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )+\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]
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Rubi [A] time = 0.377868, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {1593, 341, 321, 294, 634, 618, 204, 628, 31} \[ 2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}-\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2 \sqrt [3]{x}+\sqrt{5} \sqrt [6]{x}+\sqrt [6]{x}+2\right )+\frac{3}{5} \sqrt{2 \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 \sqrt [6]{x}-\sqrt{5}+1}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (4 \sqrt [6]{x}+\sqrt{5}+1\right )\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 341
Rule 321
Rule 294
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{-\frac{1}{\sqrt [3]{x}}+\sqrt{x}} \, dx &=\int \frac{\sqrt [3]{x}}{-1+x^{5/6}} \, dx\\ &=6 \operatorname{Subst}\left (\int \frac{x^7}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}+6 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^5} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}-\frac{6}{5} \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,\sqrt [6]{x}\right )-\frac{12}{5} \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-1-\sqrt{5}\right )+\frac{1}{4} \left (1+\sqrt{5}\right ) x}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{12}{5} \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-1+\sqrt{5}\right )+\frac{1}{4} \left (1-\sqrt{5}\right ) x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )}{\sqrt{5}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )}{\sqrt{5}}-\frac{1}{10} \left (3 \left (1-\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1+\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1+\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )-\frac{1}{10} \left (3 \left (1+\sqrt{5}\right )\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (1-\sqrt{5}\right )+2 x}{1+\frac{1}{2} \left (1-\sqrt{5}\right ) x+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=2 \sqrt{x}+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac{6 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5-\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1-\sqrt{5}\right )+2 \sqrt [6]{x}\right )}{\sqrt{5}}+\frac{6 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{2} \left (-5+\sqrt{5}\right )-x^2} \, dx,x,\frac{1}{2} \left (1+\sqrt{5}\right )+2 \sqrt [6]{x}\right )}{\sqrt{5}}\\ &=2 \sqrt{x}+6 \sqrt{\frac{2}{5 \left (5+\sqrt{5}\right )}} \tan ^{-1}\left (\frac{1-\sqrt{5}+4 \sqrt [6]{x}}{\sqrt{2 \left (5+\sqrt{5}\right )}}\right )-\frac{3}{5} \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{1}{2} \sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (1+\sqrt{5}+4 \sqrt [6]{x}\right )\right )+\frac{6}{5} \log \left (1-\sqrt [6]{x}\right )-\frac{3}{10} \left (1+\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}-\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )-\frac{3}{10} \left (1-\sqrt{5}\right ) \log \left (2+\sqrt [6]{x}+\sqrt{5} \sqrt [6]{x}+2 \sqrt [3]{x}\right )\\ \end{align*}
Mathematica [C] time = 0.0053364, size = 22, normalized size = 0.11 \[ -2 \sqrt{x} \left (\text{Hypergeometric2F1}\left (\frac{3}{5},1,\frac{8}{5},x^{5/6}\right )-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 175, normalized size = 0.9 \begin{align*} 2\,\sqrt{x}-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }+{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}+\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{12\,\sqrt{5}}{5\,\sqrt{10-2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10-2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}+\sqrt{5} \right ) } \right ) }-{\frac{3\,\sqrt{5}}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }-{\frac{3}{10}\ln \left ( 2+\sqrt [6]{x}+2\,\sqrt [3]{x}-\sqrt [6]{x}\sqrt{5} \right ) }+{\frac{12\,\sqrt{5}}{5\,\sqrt{10+2\,\sqrt{5}}}\arctan \left ({\frac{1}{\sqrt{10+2\,\sqrt{5}}} \left ( 1+4\,\sqrt [6]{x}-\sqrt{5} \right ) } \right ) }+{\frac{6}{5}\ln \left ( -1+\sqrt [6]{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.43988, size = 367, normalized size = 1.84 \begin{align*} -\frac{6}{5} \, \left (-1\right )^{\frac{3}{5}} \log \left (\left (-1\right )^{\frac{1}{5}} + x^{\frac{1}{6}}\right ) - \frac{6 \, \sqrt{5} \left (-1\right )^{\frac{3}{5}} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10} + \left (-1\right )^{\frac{1}{5}} - 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{2 \, \sqrt{5} - 10}} + \frac{6 \, \sqrt{5} \left (-1\right )^{\frac{3}{5}} \log \left (\frac{\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}{\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10} - \left (-1\right )^{\frac{1}{5}} + 4 \, x^{\frac{1}{6}}}\right )}{5 \, \sqrt{-2 \, \sqrt{5} - 10}} + 2 \, \sqrt{x} + \frac{6 \, \log \left (-x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} + \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{2}{5}} + \left (-1\right )^{\frac{2}{5}}\right )}} - \frac{6 \, \log \left (x^{\frac{1}{6}}{\left (\sqrt{5} \left (-1\right )^{\frac{1}{5}} - \left (-1\right )^{\frac{1}{5}}\right )} + 2 \, \left (-1\right )^{\frac{2}{5}} + 2 \, x^{\frac{1}{3}}\right )}{5 \,{\left (\sqrt{5} \left (-1\right )^{\frac{2}{5}} - \left (-1\right )^{\frac{2}{5}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 12.9079, size = 2130, normalized size = 10.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{x}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac{2}{3}} + \sqrt [3]{x} + \sqrt{x} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39809, size = 188, normalized size = 0.94 \begin{align*} \frac{3}{5} \, \sqrt{-2 \, \sqrt{5} + 10} \arctan \left (-\frac{\sqrt{5} - 4 \, x^{\frac{1}{6}} - 1}{\sqrt{2 \, \sqrt{5} + 10}}\right ) - \frac{3}{5} \, \sqrt{2 \, \sqrt{5} + 10} \arctan \left (\frac{\sqrt{5} + 4 \, x^{\frac{1}{6}} + 1}{\sqrt{-2 \, \sqrt{5} + 10}}\right ) + \frac{3}{10} \, \sqrt{5} \log \left (\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} + 1\right )} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{10} \, \sqrt{5} \log \left (-\frac{1}{2} \, x^{\frac{1}{6}}{\left (\sqrt{5} - 1\right )} + x^{\frac{1}{3}} + 1\right ) + 2 \, \sqrt{x} - \frac{3}{10} \, \log \left (x^{\frac{2}{3}} + \sqrt{x} + x^{\frac{1}{3}} + x^{\frac{1}{6}} + 1\right ) + \frac{6}{5} \, \log \left ({\left | x^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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