Optimal. Leaf size=129 \[ \frac{\sqrt{x}}{16 \left (x^2+1\right )}-\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.0679411, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt{x}}{16 \left (x^2+1\right )}-\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{3 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{3 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (1+x^2\right )^3} \, dx &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{1}{8} \int \frac{1}{\sqrt{x} \left (1+x^2\right )^2} \, dx\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}+\frac{3}{32} \int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}+\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}+\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{3}{64} \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 )}{64 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}-\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=-\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{\sqrt{x}}{16 \left (1+x^2\right )}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0301475, size = 121, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{8 \sqrt{x}}{x^2+1}-\frac{32 \sqrt{x}}{\left (x^2+1\right )^2}-3 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+3 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 82, normalized size = 0.6 \begin{align*} 2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( 1/32\,{x}^{5/2}-{\frac{3\,\sqrt{x}}{32}} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{3\,\sqrt{2}}{128}\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 = 2.93074, size = 131, normalized size = 1.02 \begin{align*} \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{3}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{3}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{5}{2}} - 3 \, \sqrt{x}}{16 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50976, size = 516, normalized size = 4. \begin{align*} -\frac{12 \, \sqrt{2}{\left (x^{4} + 2 \, 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^{4} + 2 \, 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^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + 3 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (x^{2} - 3\right )} \sqrt{x}}{128 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 17.4848, size = 481, normalized size = 3.73 \begin{align*} \frac{8 x^{\frac{5}{2}}}{128 x^{4} + 256 x^{2} + 128} - \frac{24 \sqrt{x}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} x^{4} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} x^{4} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{6 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{12 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{3 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{3 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{6 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.77861, size = 124, normalized size = 0.96 \begin{align*} \frac{3}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{3}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{3}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{3}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{x^{\frac{5}{2}} - 3 \, \sqrt{x}}{16 \,{\left (x^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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