Optimal. Leaf size=68 \[ \frac{5 x^8}{8}-\frac{9 x^6}{2}+\frac{49 x^4}{2}-\frac{293 x^2}{2}+\frac{415 x^2+414}{2 \left (x^4+3 x^2+2\right )}+2 \log \left (x^2+1\right )+392 \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.126049, antiderivative size = 68, 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} \[ \frac{5 x^8}{8}-\frac{9 x^6}{2}+\frac{49 x^4}{2}-\frac{293 x^2}{2}+\frac{415 x^2+414}{2 \left (x^4+3 x^2+2\right )}+2 \log \left (x^2+1\right )+392 \log \left (x^2+2\right ) \]
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^9 \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^4 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{414+415 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-206-105 x+53 x^2-27 x^3+12 x^4-5 x^5}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac{414+415 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (293-98 x+27 x^2-5 x^3-\frac{4 (198+197 x)}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{293 x^2}{2}+\frac{49 x^4}{2}-\frac{9 x^6}{2}+\frac{5 x^8}{8}+\frac{414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \operatorname{Subst}\left (\int \frac{198+197 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{293 x^2}{2}+\frac{49 x^4}{2}-\frac{9 x^6}{2}+\frac{5 x^8}{8}+\frac{414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )+392 \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^2\right )\\ &=-\frac{293 x^2}{2}+\frac{49 x^4}{2}-\frac{9 x^6}{2}+\frac{5 x^8}{8}+\frac{414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \log \left (1+x^2\right )+392 \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.035152, size = 62, normalized size = 0.91 \[ \frac{1}{8} \left (5 x^8-36 x^6+196 x^4-1172 x^2+\frac{4 \left (415 x^2+414\right )}{x^4+3 x^2+2}+16 \log \left (x^2+1\right )+3136 \log \left (x^2+2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 56, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{8}}{8}}-{\frac{9\,{x}^{6}}{2}}+{\frac{49\,{x}^{4}}{2}}-{\frac{293\,{x}^{2}}{2}}+392\,\ln \left ({x}^{2}+2 \right ) +208\, \left ({x}^{2}+2 \right ) ^{-1}+2\,\ln \left ({x}^{2}+1 \right ) -{\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 = 0.972301, size = 78, normalized size = 1.15 \begin{align*} \frac{5}{8} \, x^{8} - \frac{9}{2} \, x^{6} + \frac{49}{2} \, x^{4} - \frac{293}{2} \, x^{2} + \frac{415 \, x^{2} + 414}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \, \log \left (x^{2} + 2\right ) + 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 = 1.9619, size = 220, normalized size = 3.24 \begin{align*} \frac{5 \, x^{12} - 21 \, x^{10} + 98 \, x^{8} - 656 \, x^{6} - 3124 \, x^{4} - 684 \, x^{2} + 3136 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 16 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 1656}{8 \,{\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 = 0.162926, size = 61, normalized size = 0.9 \begin{align*} \frac{5 x^{8}}{8} - \frac{9 x^{6}}{2} + \frac{49 x^{4}}{2} - \frac{293 x^{2}}{2} + \frac{415 x^{2} + 414}{2 x^{4} + 6 x^{2} + 4} + 2 \log{\left (x^{2} + 1 \right )} + 392 \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 = 1.11876, size = 85, normalized size = 1.25 \begin{align*} \frac{5}{8} \, x^{8} - \frac{9}{2} \, x^{6} + \frac{49}{2} \, x^{4} - \frac{293}{2} \, x^{2} - \frac{394 \, x^{4} + 767 \, x^{2} + 374}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \, \log \left (x^{2} + 2\right ) + 2 \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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