Optimal. Leaf size=49 \[ \frac{5 x^2}{2}-\frac{51 x^2+50}{2 \left (x^4+3 x^2+2\right )}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right ) \]
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Rubi [A] time = 0.0856682, antiderivative size = 49, 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} \[ \frac{5 x^2}{2}-\frac{51 x^2+50}{2 \left (x^4+3 x^2+2\right )}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right ) \]
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^3 \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 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{24+12 x-5 x^2}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-5+\frac{34+27 x}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{34+27 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{7}{2} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^2\right )-10 \operatorname{Subst}\left (\int \frac{1}{2+x} \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{7}{2} \log \left (1+x^2\right )-10 \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0220387, size = 49, normalized size = 1. \[ \frac{5 x^2}{2}+\frac{-51 x^2-50}{2 \left (x^4+3 x^2+2\right )}-\frac{7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 41, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{2}}{2}}-10\,\ln \left ({x}^{2}+2 \right ) -26\, \left ({x}^{2}+2 \right ) ^{-1}-{\frac{7\,\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 = 1.02707, size = 58, normalized size = 1.18 \begin{align*} \frac{5}{2} \, x^{2} - \frac{51 \, x^{2} + 50}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - 10 \, \log \left (x^{2} + 2\right ) - \frac{7}{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 = 1.97511, size = 169, normalized size = 3.45 \begin{align*} \frac{5 \, x^{6} + 15 \, x^{4} - 41 \, x^{2} - 20 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 7 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 50}{2 \,{\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 = 0.156256, size = 42, normalized size = 0.86 \begin{align*} \frac{5 x^{2}}{2} - \frac{51 x^{2} + 50}{2 x^{4} + 6 x^{2} + 4} - \frac{7 \log{\left (x^{2} + 1 \right )}}{2} - 10 \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 = 1.0934, size = 61, normalized size = 1.24 \begin{align*} \frac{5}{2} \, x^{2} - \frac{51 \, x^{2} + 50}{2 \,{\left (x^{2} + 2\right )}{\left (x^{2} + 1\right )}} - 10 \, \log \left (x^{2} + 2\right ) - \frac{7}{2} \, \log \left (x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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