Optimal. Leaf size=61 \[ \frac{5 x^6}{6}-\frac{27 x^4}{4}+49 x^2-\frac{207 x^2+206}{2 \left (x^4+3 x^2+2\right )}-\frac{5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.117539, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {1663, 1660, 1657, 632, 31} \[ \frac{5 x^6}{6}-\frac{27 x^4}{4}+49 x^2-\frac{207 x^2+206}{2 \left (x^4+3 x^2+2\right )}-\frac{5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1660
Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^7 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 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 (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{102+53 x-27 x^2+12 x^3-5 x^4}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-98+27 x-5 x^2+\frac{298+293 x}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=49 x^2-\frac{27 x^4}{4}+\frac{5 x^6}{6}-\frac{206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{298+293 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=49 x^2-\frac{27 x^4}{4}+\frac{5 x^6}{6}-\frac{206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )-144 \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^2\right )\\ &=49 x^2-\frac{27 x^4}{4}+\frac{5 x^6}{6}-\frac{206+207 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{5}{2} \log \left (1+x^2\right )-144 \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0272733, size = 61, normalized size = 1. \[ \frac{5 x^6}{6}-\frac{27 x^4}{4}+49 x^2+\frac{-207 x^2-206}{2 \left (x^4+3 x^2+2\right )}-\frac{5}{2} \log \left (x^2+1\right )-144 \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 51, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{6}}{6}}-{\frac{27\,{x}^{4}}{4}}+49\,{x}^{2}-144\,\ln \left ({x}^{2}+2 \right ) -104\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{5\,\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{1}{2\,{x}^{2}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01567, size = 72, normalized size = 1.18 \begin{align*} \frac{5}{6} \, x^{6} - \frac{27}{4} \, x^{4} + 49 \, x^{2} - \frac{207 \, x^{2} + 206}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac{5}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94918, size = 208, normalized size = 3.41 \begin{align*} \frac{10 \, x^{10} - 51 \, x^{8} + 365 \, x^{6} + 1602 \, x^{4} - 66 \, x^{2} - 1728 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 30 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 1236}{12 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155932, size = 54, normalized size = 0.89 \begin{align*} \frac{5 x^{6}}{6} - \frac{27 x^{4}}{4} + 49 x^{2} - \frac{207 x^{2} + 206}{2 x^{4} + 6 x^{2} + 4} - \frac{5 \log{\left (x^{2} + 1 \right )}}{2} - 144 \log{\left (x^{2} + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11072, size = 78, normalized size = 1.28 \begin{align*} \frac{5}{6} \, x^{6} - \frac{27}{4} \, x^{4} + 49 \, x^{2} + \frac{293 \, x^{4} + 465 \, x^{2} + 174}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - 144 \, \log \left (x^{2} + 2\right ) - \frac{5}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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