Optimal. Leaf size=99 \[ 2 \sqrt{x}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0550577, antiderivative size = 99, 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 = {321, 329, 211, 1165, 628, 1162, 617, 204} \[ 2 \sqrt{x}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{1+x^2} \, dx &=2 \sqrt{x}-\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=2 \sqrt{x}-2 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x}-\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )-\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=2 \sqrt{x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2}}\\ &=2 \sqrt{x}+\frac{\log \left (1-\sqrt{2} \sqrt{x}+x\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} \sqrt{x}+x\right )}{2 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}\\ &=2 \sqrt{x}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\log \left (1-\sqrt{2} \sqrt{x}+x\right )}{2 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} \sqrt{x}+x\right )}{2 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0139088, size = 99, normalized size = 1. \[ 2 \sqrt{x}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 67, normalized size = 0.7 \begin{align*} 2\,\sqrt{x}-{\frac{\sqrt{2}}{2}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.23855, size = 107, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + 2 \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37285, size = 366, normalized size = 3.7 \begin{align*} \sqrt{2} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) - \frac{1}{4} \, \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + 2 \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.922942, size = 97, normalized size = 0.98 \begin{align*} 2 \sqrt{x} + \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} - \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.4465, size = 107, normalized size = 1.08 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{1}{4} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{1}{4} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + 2 \, \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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