Optimal. Leaf size=81 \[ \frac{5 x^6}{6}-\frac{17 x^4}{4}+\frac{19 x^2}{2}+\frac{25 \left (5 x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{19}{2} \log \left (x^4+2 x^2+3\right )-\frac{455 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.127361, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1663, 1660, 1657, 634, 618, 204, 628} \[ \frac{5 x^6}{6}-\frac{17 x^4}{4}+\frac{19 x^2}{2}+\frac{25 \left (5 x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{19}{2} \log \left (x^4+2 x^2+3\right )-\frac{455 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 \left (4+x+3 x^2+5 x^3\right )}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-150+200 x-56 x^3+40 x^4}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (152-136 x+40 x^2-\frac{2 (303-152 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac{19 x^2}{2}-\frac{17 x^4}{4}+\frac{5 x^6}{6}+\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{303-152 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{19 x^2}{2}-\frac{17 x^4}{4}+\frac{5 x^6}{6}+\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{19}{2} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{455}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{19 x^2}{2}-\frac{17 x^4}{4}+\frac{5 x^6}{6}+\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{19}{2} \log \left (3+2 x^2+x^4\right )+\frac{455}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac{19 x^2}{2}-\frac{17 x^4}{4}+\frac{5 x^6}{6}+\frac{25 \left (3+5 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{455 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{19}{2} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0313759, size = 73, normalized size = 0.9 \[ \frac{1}{48} \left (40 x^6-204 x^4+456 x^2+\frac{150 \left (5 x^2+3\right )}{x^4+2 x^2+3}+456 \log \left (x^4+2 x^2+3\right )-1365 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 69, normalized size = 0.9 \begin{align*}{\frac{5\,{x}^{6}}{6}}-{\frac{17\,{x}^{4}}{4}}+{\frac{19\,{x}^{2}}{2}}+{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{125\,{x}^{2}}{4}}+{\frac{75}{4}} \right ) }+{\frac{19\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{2}}-{\frac{455\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49928, size = 89, normalized size = 1.1 \begin{align*} \frac{5}{6} \, x^{6} - \frac{17}{4} \, x^{4} + \frac{19}{2} \, x^{2} - \frac{455}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (5 \, x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{19}{2} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54505, size = 255, normalized size = 3.15 \begin{align*} \frac{40 \, x^{10} - 124 \, x^{8} + 168 \, x^{6} + 300 \, x^{4} - 1365 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 2118 \, x^{2} + 456 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 450}{48 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.168, size = 80, normalized size = 0.99 \begin{align*} \frac{5 x^{6}}{6} - \frac{17 x^{4}}{4} + \frac{19 x^{2}}{2} + \frac{125 x^{2} + 75}{8 x^{4} + 16 x^{2} + 24} + \frac{19 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{2} - \frac{455 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12135, size = 96, normalized size = 1.19 \begin{align*} \frac{5}{6} \, x^{6} - \frac{17}{4} \, x^{4} + \frac{19}{2} \, x^{2} - \frac{455}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{76 \, x^{4} + 27 \, x^{2} + 153}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{19}{2} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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