Optimal. Leaf size=38 \[ \frac {2}{5} x^{5/2} \coth ^{-1}\left (\sqrt {x}\right )+\frac {x^2}{10}+\frac {x}{5}+\frac {1}{5} \log (1-x) \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 43
Rule 6098
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.27, size = 35, normalized size = 0.92 \[ \frac {1}{5} \, x^{\frac {5}{2}} \log \left (\frac {x + 2 \, \sqrt {x} + 1}{x - 1}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{\frac {3}{2}} \operatorname {arcoth}\left (\sqrt {x}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 0.92 \[ \frac {2 x^{\frac {5}{2}} \mathrm {arccoth}\left (\sqrt {x}\right )}{5}+\frac {x^{2}}{10}+\frac {x}{5}+\frac {\ln \left (-1+\sqrt {x}\right )}{5}+\frac {\ln \left (1+\sqrt {x}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 24, normalized size = 0.63 \[ \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {arcoth}\left (\sqrt {x}\right ) + \frac {1}{10} \, x^{2} + \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 24, normalized size = 0.63 \[ \frac {x}{5}+\frac {\ln \left (x-1\right )}{5}+\frac {2\,x^{5/2}\,\mathrm {acoth}\left (\sqrt {x}\right )}{5}+\frac {x^2}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.35, size = 121, normalized size = 3.18 \[ \frac {4 x^{\frac {7}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 x^{\frac {5}{2}} \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} + \frac {x^{3}}{10 x - 10} + \frac {x^{2}}{10 x - 10} + \frac {4 x \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} - \frac {4 x \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {4 \log {\left (\sqrt {x} + 1 \right )}}{10 x - 10} + \frac {4 \operatorname {acoth}{\left (\sqrt {x} \right )}}{10 x - 10} - \frac {2}{10 x - 10} \]
Verification of antiderivative is not currently implemented for this CAS.
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