Optimal. Leaf size=113 \[ -\frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0587743, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{x^{3/2}}{2 \left (x^2+1\right )}+\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\left (1+x^2\right )^2} \, dx &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}+\frac{3}{4} \int \frac{\sqrt{x}}{1+x^2} \, dx\\ &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2}}\\ &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}+\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{8 \sqrt{2}}-\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{8 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}\\ &=-\frac{x^{3/2}}{2 \left (1+x^2\right )}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{4 \sqrt{2}}+\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{8 \sqrt{2}}-\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0105864, size = 30, normalized size = 0.27 \[ 2 x^{3/2} \left (\, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-x^2\right )-\frac{1}{x^2+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 74, normalized size = 0.7 \begin{align*} -{\frac{1}{2\,{x}^{2}+2}{x}^{{\frac{3}{2}}}}+{\frac{3\,\sqrt{2}}{8}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{8}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{16}\ln \left ({ \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 4.39211, size = 116, normalized size = 1.03 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37106, size = 447, normalized size = 3.96 \begin{align*} -\frac{12 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 12 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) + 3 \, \sqrt{2}{\left (x^{2} + 1\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 3 \, \sqrt{2}{\left (x^{2} + 1\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + 8 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.90666, size = 264, normalized size = 2.34 \begin{align*} - \frac{8 x^{\frac{3}{2}}}{16 x^{2} + 16} + \frac{3 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{3 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} + \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} - \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{16 x^{2} + 16} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{16 x^{2} + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.44266, size = 116, normalized size = 1.03 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{3}{16} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{3}{16} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{x^{\frac{3}{2}}}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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