Optimal. Leaf size=56 \[ \frac{5 x^3}{3}-\frac{\left (51 x^2+50\right ) x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0727921, antiderivative size = 56, 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} \[ \frac{5 x^3}{3}-\frac{\left (51 x^2+50\right ) x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1676
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^4 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=-\frac{x \left (50+51 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \frac{-100-6 x^2+48 x^4-20 x^6}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (50+51 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{4} \int \left (108-20 x^2-\frac{2 \left (158+145 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=-27 x+\frac{5 x^3}{3}-\frac{x \left (50+51 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{1}{2} \int \frac{158+145 x^2}{2+3 x^2+x^4} \, dx\\ &=-27 x+\frac{5 x^3}{3}-\frac{x \left (50+51 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{13}{2} \int \frac{1}{1+x^2} \, dx+66 \int \frac{1}{2+x^2} \, dx\\ &=-27 x+\frac{5 x^3}{3}-\frac{x \left (50+51 x^2\right )}{2 \left (2+3 x^2+x^4\right )}+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0427317, size = 57, normalized size = 1.02 \[ \frac{5 x^3}{3}+\frac{-51 x^3-50 x}{2 \left (x^4+3 x^2+2\right )}-27 x+\frac{13}{2} \tan ^{-1}(x)+33 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 46, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{3}}-27\,x-26\,{\frac{x}{{x}^{2}+2}}+33\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2}+{\frac{x}{2\,{x}^{2}+2}}+{\frac{13\,\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48765, size = 65, normalized size = 1.16 \begin{align*} \frac{5}{3} \, x^{3} + 33 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 27 \, x - \frac{51 \, x^{3} + 50 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{13}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90295, size = 198, normalized size = 3.54 \begin{align*} \frac{10 \, x^{7} - 132 \, x^{5} - 619 \, x^{3} + 198 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 39 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) - 474 \, x}{6 \,{\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 = 0.189747, size = 53, normalized size = 0.95 \begin{align*} \frac{5 x^{3}}{3} - 27 x - \frac{51 x^{3} + 50 x}{2 x^{4} + 6 x^{2} + 4} + \frac{13 \operatorname{atan}{\left (x \right )}}{2} + 33 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14651, size = 65, normalized size = 1.16 \begin{align*} \frac{5}{3} \, x^{3} + 33 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 27 \, x - \frac{51 \, x^{3} + 50 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + \frac{13}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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