Optimal. Leaf size=49 \[ \frac{\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0663581, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1668, 1676, 1166, 203} \[ \frac{\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1676
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 )^2} \, dx &=\frac{x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{48-2 x^2-20 x^4}{2+3 x^2+x^4} \, dx\\ &=\frac{x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (-20+\frac{2 \left (44+29 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=5 x+\frac{x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \int \frac{44+29 x^2}{2+3 x^2+x^4} \, dx\\ &=5 x+\frac{x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-7 \int \frac{1}{2+x^2} \, dx-\frac{15}{2} \int \frac{1}{1+x^2} \, dx\\ &=5 x+\frac{x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.038853, size = 50, normalized size = 1.02 \[ \frac{25 x^3+24 x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 41, normalized size = 0.8 \begin{align*} 5\,x+13\,{\frac{x}{{x}^{2}+2}}-{\frac{7\,\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{15\,\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48706, size = 58, normalized size = 1.18 \begin{align*} -\frac{7}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x + \frac{25 \, x^{3} + 24 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{15}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13975, size = 180, normalized size = 3.67 \begin{align*} \frac{10 \, x^{5} + 55 \, x^{3} - 7 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 15 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) + 44 \, x}{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 = 0.185744, size = 48, normalized size = 0.98 \begin{align*} 5 x + \frac{25 x^{3} + 24 x}{2 x^{4} + 6 x^{2} + 4} - \frac{15 \operatorname{atan}{\left (x \right )}}{2} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10117, size = 58, normalized size = 1.18 \begin{align*} -\frac{7}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x + \frac{25 \, x^{3} + 24 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{15}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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