Optimal. Leaf size=36 \[ \frac {2}{5} x^{5/2} \cot ^{-1}\left (\sqrt {x}\right )+\frac {x^2}{10}-\frac {x}{5}+\frac {1}{5} \log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 36, 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 = {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 43
Rule 5034
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.46, size = 24, normalized size = 0.67 \[ \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{10} \, x^{2} - \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 39, normalized size = 1.08 \[ \frac {2}{5} \, x^{\frac {5}{2}} \arctan \left (\frac {1}{\sqrt {x}}\right ) - \frac {1}{10} \, x^{2} {\left (\frac {2}{x} - \frac {3}{x^{2}} - 1\right )} + \frac {1}{5} \, \log \relax (x) + \frac {1}{5} \, \log \left (\frac {1}{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 0.69 \[ -\frac {x}{5}+\frac {x^{2}}{10}+\frac {2 x^{\frac {5}{2}} \mathrm {arccot}\left (\sqrt {x}\right )}{5}+\frac {\ln \left (x +1\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 24, normalized size = 0.67 \[ \frac {2}{5} \, x^{\frac {5}{2}} \operatorname {arccot}\left (\sqrt {x}\right ) + \frac {1}{10} \, x^{2} - \frac {1}{5} \, x + \frac {1}{5} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 24, normalized size = 0.67 \[ \frac {\ln \left (x+1\right )}{5}-\frac {x}{5}+\frac {2\,x^{5/2}\,\mathrm {acot}\left (\sqrt {x}\right )}{5}+\frac {x^2}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.96, size = 85, normalized size = 2.36 \[ \frac {4 x^{\frac {7}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{10 x + 10} + \frac {4 x^{\frac {5}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{10 x + 10} + \frac {x^{3}}{10 x + 10} - \frac {x^{2}}{10 x + 10} + \frac {2 x \log {\left (x + 1 \right )}}{10 x + 10} + \frac {2 \log {\left (x + 1 \right )}}{10 x + 10} + \frac {2}{10 x + 10} \]
Verification of antiderivative is not currently implemented for this CAS.
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