Optimal. Leaf size=38 \[ \frac{x^2}{10}+\frac{2}{5} x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )+\frac{x}{5}+\frac{1}{5} \log (1-x) \]
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Rubi [A] time = 0.0177347, antiderivative size = 38, 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 = {6098, 43} \[ \frac{x^2}{10}+\frac{2}{5} x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )+\frac{x}{5}+\frac{1}{5} \log (1-x) \]
Antiderivative was successfully verified.
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Rule 6098
Rule 43
Rubi steps
\begin{align*} \int x^{3/2} \coth ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{2}{5} x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{5} \int \frac{x^2}{1-x} \, dx\\ &=\frac{2}{5} x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )-\frac{1}{5} \int \left (-1+\frac{1}{1-x}-x\right ) \, dx\\ &=\frac{x}{5}+\frac{x^2}{10}+\frac{2}{5} x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )+\frac{1}{5} \log (1-x)\\ \end{align*}
Mathematica [A] time = 0.0154204, size = 31, normalized size = 0.82 \[ \frac{1}{10} \left (4 x^{5/2} \coth ^{-1}\left (\sqrt{x}\right )+(x+2) x+2 \log (1-x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 35, normalized size = 0.9 \begin{align*}{\frac{2}{5}{x}^{{\frac{5}{2}}}{\rm arccoth} \left (\sqrt{x}\right )}+{\frac{{x}^{2}}{10}}+{\frac{x}{5}}+{\frac{1}{5}\ln \left ( -1+\sqrt{x} \right ) }+{\frac{1}{5}\ln \left ( 1+\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956905, size = 32, normalized size = 0.84 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} \operatorname{arcoth}\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 = 1.57908, size = 111, normalized size = 2.92 \begin{align*} \frac{1}{5} \, x^{\frac{5}{2}} \log \left (\frac{x + 2 \, \sqrt{x} + 1}{x - 1}\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 = 9.51341, size = 121, normalized size = 3.18 \begin{align*} \frac{4 x^{\frac{7}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{10 x - 10} - \frac{4 x^{\frac{5}{2}} \operatorname{acoth}{\left (\sqrt{x} \right )}}{10 x - 10} + \frac{x^{3}}{10 x - 10} + \frac{x^{2}}{10 x - 10} + \frac{4 x \log{\left (\sqrt{x} + 1 \right )}}{10 x - 10} - \frac{4 x \operatorname{acoth}{\left (\sqrt{x} \right )}}{10 x - 10} - \frac{4 \log{\left (\sqrt{x} + 1 \right )}}{10 x - 10} + \frac{4 \operatorname{acoth}{\left (\sqrt{x} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{\frac{3}{2}} \operatorname{arcoth}\left (\sqrt{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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