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 = 0.01, antiderivative size = 37, 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 = {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 44
Rule 5033
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.48, size = 33, normalized size = 0.89 \[ \frac {x^{2} \log \left (x + 1\right ) - x^{2} \log \relax (x) - 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - x}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 28, normalized size = 0.76 \[ \frac {x - 1}{3 \, x} - \frac {2 \, \arctan \left (\sqrt {x}\right )}{3 \, x^{\frac {3}{2}}} + \frac {1}{3} \, \log \left (x + 1\right ) - \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, normalized size = 0.70 \[ -\frac {1}{3 x}-\frac {2 \arctan \left (\sqrt {x}\right )}{3 x^{\frac {3}{2}}}-\frac {\ln \relax (x )}{3}+\frac {\ln \left (x +1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 25, normalized size = 0.68 \[ -\frac {2 \, \arctan \left (\sqrt {x}\right )}{3 \, x^{\frac {3}{2}}} - \frac {1}{3 \, x} + \frac {1}{3} \, \log \left (x + 1\right ) - \frac {1}{3} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 27, normalized size = 0.73 \[ \frac {\ln \left (x+1\right )}{3}-\frac {2\,\ln \left (\sqrt {x}\right )}{3}-\frac {2\,\mathrm {atan}\left (\sqrt {x}\right )}{3\,x^{3/2}}-\frac {1}{3\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.90, size = 143, normalized size = 3.86 \[ - \frac {2 x^{\frac {3}{2}} \operatorname {atan}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} - \frac {2 \sqrt {x} \operatorname {atan}{\left (\sqrt {x} \right )}}{3 x^{3} + 3 x^{2}} - \frac {x^{3} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} + \frac {x^{3} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} - \frac {x^{2} \log {\relax (x )}}{3 x^{3} + 3 x^{2}} + \frac {x^{2} \log {\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} - \frac {x^{2}}{3 x^{3} + 3 x^{2}} - \frac {x}{3 x^{3} + 3 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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