Optimal. Leaf size=42 \[ -\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0098464, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5033, 50, 63, 203} \[ -\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{\sqrt{x}}{2}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int x \tan ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{x^{3/2}}{1+x} \, dx\\ &=-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{4} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{4} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{2}-\frac{x^{3/2}}{6}-\frac{1}{2} \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{2} x^2 \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.01061, size = 28, normalized size = 0.67 \[ \frac{1}{6} \left (3 \left (x^2-1\right ) \tan ^{-1}\left (\sqrt{x}\right )-(x-3) \sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 27, normalized size = 0.6 \begin{align*} -{\frac{1}{6}{x}^{{\frac{3}{2}}}}-{\frac{1}{2}\arctan \left ( \sqrt{x} \right ) }+{\frac{{x}^{2}}{2}\arctan \left ( \sqrt{x} \right ) }+{\frac{1}{2}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49418, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \, x^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20084, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{2} \,{\left (x^{2} - 1\right )} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \,{\left (x - 3\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.02125, size = 32, normalized size = 0.76 \begin{align*} - \frac{x^{\frac{3}{2}}}{6} + \frac{\sqrt{x}}{2} + \frac{x^{2} \operatorname{atan}{\left (\sqrt{x} \right )}}{2} - \frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11712, size = 35, normalized size = 0.83 \begin{align*} \frac{1}{2} \, x^{2} \arctan \left (\sqrt{x}\right ) - \frac{1}{6} \, x^{\frac{3}{2}} + \frac{1}{2} \, \sqrt{x} - \frac{1}{2} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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