Optimal. Leaf size=37 \[ -\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{1}{3 x}-\frac{\log (x)}{3}+\frac{1}{3} \log (x+1) \]
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Rubi [A] time = 0.0137589, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 44} \[ -\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{1}{3 x}-\frac{\log (x)}{3}+\frac{1}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 44
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{x^{5/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}+\frac{1}{3} \int \frac{1}{x^2 (1+x)} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}+\frac{1}{3} \int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx\\ &=-\frac{1}{3 x}-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{\log (x)}{3}+\frac{1}{3} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0179356, size = 31, normalized size = 0.84 \[ \frac{1}{3} \left (-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{1}{x}-\log (x)+\log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 26, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,x}}-{\frac{2}{3}\arctan \left ( \sqrt{x} \right ){x}^{-{\frac{3}{2}}}}-{\frac{\ln \left ( x \right ) }{3}}+{\frac{\ln \left ( x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995832, size = 34, normalized size = 0.92 \begin{align*} -\frac{2 \, \arctan \left (\sqrt{x}\right )}{3 \, x^{\frac{3}{2}}} - \frac{1}{3 \, x} + \frac{1}{3} \, \log \left (x + 1\right ) - \frac{1}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19221, size = 96, normalized size = 2.59 \begin{align*} \frac{x^{2} \log \left (x + 1\right ) - x^{2} \log \left (x\right ) - 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - x}{3 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 8.49852, size = 143, normalized size = 3.86 \begin{align*} - \frac{2 x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{3 x^{3} + 3 x^{2}} - \frac{2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )}}{3 x^{3} + 3 x^{2}} - \frac{x^{3} \log{\left (x \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{3} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{3}}{3 x^{3} + 3 x^{2}} - \frac{x^{2} \log{\left (x \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{2} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} - \frac{x}{3 x^{3} + 3 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15805, size = 38, normalized size = 1.03 \begin{align*} \frac{x - 1}{3 \, x} - \frac{2 \, \arctan \left (\sqrt{x}\right )}{3 \, x^{\frac{3}{2}}} + \frac{1}{3} \, \log \left (x + 1\right ) - \frac{1}{3} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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