Optimal. Leaf size=36 \[ \frac{x^2}{10}+\frac{2}{5} x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )-\frac{x}{5}+\frac{1}{5} \log (x+1) \]
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Rubi [A] time = 0.0137792, antiderivative size = 36, 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 = {5034, 43} \[ \frac{x^2}{10}+\frac{2}{5} x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )-\frac{x}{5}+\frac{1}{5} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 5034
Rule 43
Rubi steps
\begin{align*} \int x^{3/2} \cot ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{2}{5} x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{5} \int \frac{x^2}{1+x} \, dx\\ &=\frac{2}{5} x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{5} \int \left (-1+x+\frac{1}{1+x}\right ) \, dx\\ &=-\frac{x}{5}+\frac{x^2}{10}+\frac{2}{5} x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )+\frac{1}{5} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0161584, size = 29, normalized size = 0.81 \[ \frac{1}{10} \left (4 x^{5/2} \cot ^{-1}\left (\sqrt{x}\right )+(x-2) x+2 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 25, normalized size = 0.7 \begin{align*} -{\frac{x}{5}}+{\frac{{x}^{2}}{10}}+{\frac{2}{5}{x}^{{\frac{5}{2}}}{\rm arccot} \left (\sqrt{x}\right )}+{\frac{\ln \left ( x+1 \right ) }{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973942, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{10} \, x^{2} - \frac{1}{5} \, x + \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20899, size = 88, normalized size = 2.44 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \operatorname{arccot}\left (\sqrt{x}\right ) + \frac{1}{10} \, x^{2} - \frac{1}{5} \, x + \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.88749, size = 85, normalized size = 2.36 \begin{align*} \frac{4 x^{\frac{7}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{10 x + 10} + \frac{4 x^{\frac{5}{2}} \operatorname{acot}{\left (\sqrt{x} \right )}}{10 x + 10} + \frac{x^{3}}{10 x + 10} - \frac{x^{2}}{10 x + 10} + \frac{2 x \log{\left (x + 1 \right )}}{10 x + 10} + \frac{2 \log{\left (x + 1 \right )}}{10 x + 10} + \frac{2}{10 x + 10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09629, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \arctan \left (\frac{1}{\sqrt{x}}\right ) + \frac{1}{10} \, x^{2} - \frac{1}{5} \, x + \frac{1}{5} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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