Optimal. Leaf size=108 \[ -\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0573483, 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 = {325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{\sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \left (1+x^2\right )} \, dx &=-\frac{2}{5 x^{5/2}}-\int \frac{1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+\int \frac{\sqrt{x}}{1+x^2} \, dx\\ &=-\frac{2}{5 x^{5/2}}+\frac{2}{\sqrt{x}}+2 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2}{5 x^{5/2}}+\frac{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 )\\ &=-\frac{2}{5 x^{5/2}}+\frac{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}}\\ &=-\frac{2}{5 x^{5/2}}+\frac{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}}\\ &=-\frac{2}{5 x^{5/2}}+\frac{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 [C] time = 0.0049442, size = 22, normalized size = 0.2 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-x^2\right )}{5 x^{5/2}} \]
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\,{\frac{1}{\sqrt{x}}}+{\frac{\sqrt{2}}{2}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 4.31251, size = 116, normalized size = 1.07 \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 ) + \frac{2 \,{\left (5 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31501, size = 423, normalized size = 3.92 \begin{align*} -\frac{20 \, \sqrt{2} x^{3} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 20 \, \sqrt{2} x^{3} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) + 5 \, \sqrt{2} x^{3} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 5 \, \sqrt{2} x^{3} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (5 \, x^{2} - 1\right )} \sqrt{x}}{20 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.1108, size = 105, normalized size = 0.97 \begin{align*} \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} + \frac{2}{\sqrt{x}} - \frac{2}{5 x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.69134, size = 116, normalized size = 1.07 \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 ) + \frac{2 \,{\left (5 \, x^{2} - 1\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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