Optimal. Leaf size=72 \[ \frac{x \left (25 x^2+24\right )}{4 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (130 x^2+211\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0655951, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1668, 1678, 1166, 203} \[ \frac{x \left (25 x^2+24\right )}{4 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (130 x^2+211\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac{x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{48-154 x^2-40 x^4}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{748-520 x^2}{2+3 x^2+x^4} \, dx\\ &=\frac{x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{317}{8} \int \frac{1}{1+x^2} \, dx-\frac{447}{8} \int \frac{1}{2+x^2} \, dx\\ &=\frac{x \left (24+25 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{x \left (211+130 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0627458, size = 56, normalized size = 0.78 \[ \frac{1}{16} \left (-\frac{2 x \left (130 x^6+601 x^4+843 x^2+374\right )}{\left (x^4+3 x^2+2\right )^2}+634 \tan ^{-1}(x)-447 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 53, normalized size = 0.7 \begin{align*} -{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{103\,{x}^{3}}{8}}+{\frac{129\,x}{4}} \right ) }-{\frac{447\,\sqrt{2}}{16}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{27\,{x}^{3}}{8}}-{\frac{29\,x}{8}} \right ) }+{\frac{317\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49742, size = 81, normalized size = 1.12 \begin{align*} -\frac{447}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{317}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52068, size = 279, normalized size = 3.88 \begin{align*} -\frac{260 \, x^{7} + 1202 \, x^{5} + 1686 \, x^{3} + 447 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 634 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 748 \, x}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.236459, size = 65, normalized size = 0.9 \begin{align*} - \frac{130 x^{7} + 601 x^{5} + 843 x^{3} + 374 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{317 \operatorname{atan}{\left (x \right )}}{8} - \frac{447 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10467, size = 68, normalized size = 0.94 \begin{align*} -\frac{447}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{317}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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