Optimal. Leaf size=52 \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.125848, antiderivative size = 52, 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 = {6729, 1593, 6725, 329, 212, 206, 203, 15, 298} \[ \frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6729
Rule 1593
Rule 6725
Rule 329
Rule 212
Rule 206
Rule 203
Rule 15
Rule 298
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x}+\sqrt{x^3}} \, dx &=\int \frac{\sqrt{x}-\sqrt{x^3}}{x-x^3} \, dx\\ &=\int \frac{\sqrt{x}-\sqrt{x^3}}{x \left (1-x^2\right )} \, dx\\ &=\int \left (-\frac{1}{\sqrt{x} \left (-1+x^2\right )}+\frac{\sqrt{x^3}}{x \left (-1+x^2\right )}\right ) \, dx\\ &=-\int \frac{1}{\sqrt{x} \left (-1+x^2\right )} \, dx+\int \frac{\sqrt{x^3}}{x \left (-1+x^2\right )} \, dx\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\sqrt{x}\right )\right )+\frac{\sqrt{x^3} \int \frac{\sqrt{x}}{-1+x^2} \, dx}{x^{3/2}}\\ &=\frac{\left (2 \sqrt{x^3}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt{x}\right )}{x^{3/2}}+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )+\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\tanh ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x^3} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}+\frac{\sqrt{x^3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )}{x^{3/2}}\\ &=\tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x^3} \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}+\tanh ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x^3} \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0274004, size = 49, normalized size = 0.94 \[ \frac{\left (x^{3/2}+\sqrt{x^3}\right ) \tan ^{-1}\left (\sqrt{x}\right )+\left (x^{3/2}-\sqrt{x^3}\right ) \tanh ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 30, normalized size = 0.6 \begin{align*} 2\,{\arctan \left ( \sqrt{x}\sqrt{{\frac{\sqrt{{x}^{3}}}{{x}^{3/2}}}} \right ){\frac{1}{\sqrt{{\frac{\sqrt{{x}^{3}}}{{x}^{3/2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71265, size = 8, normalized size = 0.15 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99446, size = 26, normalized size = 0.5 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \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{1}{\sqrt{x} + \sqrt{x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09506, size = 8, normalized size = 0.15 \begin{align*} 2 \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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