Optimal. Leaf size=54 \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.108365, antiderivative size = 54, 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} \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right ) \]
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^5 \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^2 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-50-27 x+12 x^2-5 x^3}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac{102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (27-5 x-\frac{2 (52+49 x)}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{27 x^2}{2}+\frac{5 x^4}{4}+\frac{102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+\operatorname{Subst}\left (\int \frac{52+49 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{27 x^2}{2}+\frac{5 x^4}{4}+\frac{102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )+46 \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^2\right )\\ &=-\frac{27 x^2}{2}+\frac{5 x^4}{4}+\frac{102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \log \left (1+x^2\right )+46 \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0244659, size = 54, normalized size = 1. \[ \frac{5 x^4}{4}-\frac{27 x^2}{2}+\frac{103 x^2+102}{2 \left (x^4+3 x^2+2\right )}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 46, normalized size = 0.9 \begin{align*}{\frac{5\,{x}^{4}}{4}}-{\frac{27\,{x}^{2}}{2}}+46\,\ln \left ({x}^{2}+2 \right ) +52\, \left ({x}^{2}+2 \right ) ^{-1}+3\,\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 = 0.996348, size = 65, normalized size = 1.2 \begin{align*} \frac{5}{4} \, x^{4} - \frac{27}{2} \, x^{2} + \frac{103 \, x^{2} + 102}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \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 = 1.96442, size = 186, normalized size = 3.44 \begin{align*} \frac{5 \, x^{8} - 39 \, x^{6} - 152 \, x^{4} + 98 \, x^{2} + 184 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 12 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 204}{4 \,{\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 = 0.154507, size = 48, normalized size = 0.89 \begin{align*} \frac{5 x^{4}}{4} - \frac{27 x^{2}}{2} + \frac{103 x^{2} + 102}{2 x^{4} + 6 x^{2} + 4} + 3 \log{\left (x^{2} + 1 \right )} + 46 \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 = 1.13079, size = 72, normalized size = 1.33 \begin{align*} \frac{5}{4} \, x^{4} - \frac{27}{2} \, x^{2} - \frac{49 \, x^{4} + 44 \, x^{2} - 4}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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