Optimal. Leaf size=147 \[ \frac{13}{16 x^{5/2} \left (x^2+1\right )}+\frac{1}{4 x^{5/2} \left (x^2+1\right )^2}-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{117 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
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Rubi [A] time = 0.0745039, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{13}{16 x^{5/2} \left (x^2+1\right )}+\frac{1}{4 x^{5/2} \left (x^2+1\right )^2}-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{117 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{117 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 290
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 )^3} \, dx &=\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{8} \int \frac{1}{x^{7/2} \left (1+x^2\right )^2} \, dx\\ &=\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}+\frac{117}{32} \int \frac{1}{x^{7/2} \left (1+x^2\right )} \, dx\\ &=-\frac{117}{80 x^{5/2}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}-\frac{117}{32} \int \frac{1}{x^{3/2} \left (1+x^2\right )} \, dx\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}+\frac{117}{32} \int \frac{\sqrt{x}}{1+x^2} \, dx\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}+\frac{117}{16} \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}-\frac{117}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{117}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}+\frac{117}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{117}{64} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{117 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}+\frac{117 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}+\frac{117 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{117 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{117 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{117 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=-\frac{117}{80 x^{5/2}}+\frac{117}{16 \sqrt{x}}+\frac{1}{4 x^{5/2} \left (1+x^2\right )^2}+\frac{13}{16 x^{5/2} \left (1+x^2\right )}-\frac{117 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{117 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}-\frac{117 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0051212, size = 22, normalized size = 0.15 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},3;-\frac{1}{4};-x^2\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 92, normalized size = 0.6 \begin{align*} -{\frac{2}{5}{x}^{-{\frac{5}{2}}}}+6\,{\frac{1}{\sqrt{x}}}+2\,{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ({\frac{21\,{x}^{7/2}}{32}}+{\frac{25\,{x}^{3/2}}{32}} \right ) }+{\frac{117\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{117\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{117\,\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 = 1.93831, size = 144, normalized size = 0.98 \begin{align*} \frac{117}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{117}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{117}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{117}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32}{80 \,{\left (x^{\frac{13}{2}} + 2 \, x^{\frac{9}{2}} + x^{\frac{5}{2}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66257, size = 575, normalized size = 3.91 \begin{align*} -\frac{2340 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 2340 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) + 585 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 585 \, \sqrt{2}{\left (x^{7} + 2 \, x^{5} + x^{3}\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (585 \, x^{6} + 1053 \, x^{4} + 416 \, x^{2} - 32\right )} \sqrt{x}}{640 \,{\left (x^{7} + 2 \, x^{5} + x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 97.5842, size = 678, normalized size = 4.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.15828, size = 143, normalized size = 0.97 \begin{align*} \frac{117}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{117}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) - \frac{117}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{117}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{21 \, x^{\frac{7}{2}} + 25 \, x^{\frac{3}{2}}}{16 \,{\left (x^{2} + 1\right )}^{2}} + \frac{2 \,{\left (15 \, 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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