Optimal. Leaf size=51 \[ -\frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x}}{3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.0146102, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5033, 50, 63, 203} \[ -\frac{x^{5/2}}{15}+\frac{x^{3/2}}{9}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{\sqrt{x}}{3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 5033
Rule 50
Rule 63
Rule 203
Rubi steps
\begin{align*} \int x^2 \tan ^{-1}\left (\sqrt{x}\right ) \, dx &=\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{x^{5/2}}{1+x} \, dx\\ &=-\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{6} \int \frac{x^{3/2}}{1+x} \, dx\\ &=\frac{x^{3/2}}{9}-\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )-\frac{1}{6} \int \frac{\sqrt{x}}{1+x} \, dx\\ &=-\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}-\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{6} \int \frac{1}{\sqrt{x} (1+x)} \, dx\\ &=-\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}-\frac{x^{5/2}}{15}+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{x}}{3}+\frac{x^{3/2}}{9}-\frac{x^{5/2}}{15}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x}\right )+\frac{1}{3} x^3 \tan ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0132214, size = 34, normalized size = 0.67 \[ \frac{1}{45} \left (\sqrt{x} \left (-3 x^2+5 x-15\right )+15 \left (x^3+1\right ) \tan ^{-1}\left (\sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 32, normalized size = 0.6 \begin{align*}{\frac{1}{9}{x}^{{\frac{3}{2}}}}-{\frac{1}{15}{x}^{{\frac{5}{2}}}}+{\frac{1}{3}\arctan \left ( \sqrt{x} \right ) }+{\frac{{x}^{3}}{3}\arctan \left ( \sqrt{x} \right ) }-{\frac{1}{3}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5056, size = 42, normalized size = 0.82 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\sqrt{x}\right ) - \frac{1}{15} \, x^{\frac{5}{2}} + \frac{1}{9} \, x^{\frac{3}{2}} - \frac{1}{3} \, \sqrt{x} + \frac{1}{3} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21294, size = 88, normalized size = 1.73 \begin{align*} \frac{1}{3} \,{\left (x^{3} + 1\right )} \arctan \left (\sqrt{x}\right ) - \frac{1}{45} \,{\left (3 \, x^{2} - 5 \, x + 15\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.15935, size = 39, normalized size = 0.76 \begin{align*} - \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left (\sqrt{x} \right )}}{3} + \frac{\operatorname{atan}{\left (\sqrt{x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1379, size = 42, normalized size = 0.82 \begin{align*} \frac{1}{3} \, x^{3} \arctan \left (\sqrt{x}\right ) - \frac{1}{15} \, x^{\frac{5}{2}} + \frac{1}{9} \, x^{\frac{3}{2}} - \frac{1}{3} \, \sqrt{x} + \frac{1}{3} \, \arctan \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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