Optimal. Leaf size=108 \[ \frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0610106, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
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^{7/2}}{1+x^2} \, dx &=\frac{2 x^{5/2}}{5}-\int \frac{x^{3/2}}{1+x^2} \, dx\\ &=-2 \sqrt{x}+\frac{2 x^{5/2}}{5}+\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=-2 \sqrt{x}+\frac{2 x^{5/2}}{5}+2 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-2 \sqrt{x}+\frac{2 x^{5/2}}{5}+\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{2 x^{5/2}}{5}+\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{2 x^{5/2}}{5}-\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{2 x^{5/2}}{5}-\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 = 0.0229026, size = 108, normalized size = 1. \[ \frac{2 x^{5/2}}{5}-2 \sqrt{x}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}+\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 72, normalized size = 0.7 \begin{align*}{\frac{2}{5}{x}^{{\frac{5}{2}}}}-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 = 3.18871, size = 113, normalized size = 1.05 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \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 = 1.33545, size = 382, normalized size = 3.54 \begin{align*} \frac{2}{5} \,{\left (x^{2} - 5\right )} \sqrt{x} - \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.34837, size = 105, normalized size = 0.97 \begin{align*} \frac{2 x^{\frac{5}{2}}}{5} - 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 = 2.95182, size = 113, normalized size = 1.05 \begin{align*} \frac{2}{5} \, x^{\frac{5}{2}} + \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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