Optimal. Leaf size=81 \[ x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.112492, antiderivative size = 81, 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 = {1668, 1678, 1676, 1166, 203} \[ x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1676
Rule 1166
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{828-2478 x^2-840 x^4+424 x^6-216 x^8+96 x^{10}-40 x^{12}}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{-4952-2700 x^2+3136 x^4-864 x^6+160 x^8}{2+3 x^2+x^4} \, dx\\ &=\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (6848-1344 x^2+160 x^4-\frac{36 \left (518+571 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{9}{8} \int \frac{518+571 x^2}{2+3 x^2+x^4} \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{477}{8} \int \frac{1}{1+x^2} \, dx-702 \int \frac{1}{2+x^2} \, dx\\ &=214 x-14 x^3+x^5+\frac{x \left (414+415 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (824+1669 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0603729, size = 71, normalized size = 0.88 \[ \frac{x \left (8 x^{12}-64 x^{10}+1144 x^8+10581 x^6+26775 x^4+26736 x^2+9324\right )}{8 \left (x^4+3 x^2+2\right )^2}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 64, normalized size = 0.8 \begin{align*}{x}^{5}-14\,{x}^{3}+214\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{105\,{x}^{3}}{8}}-{\frac{79\,x}{4}} \right ) }-351\,\arctan \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2}+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{11\,{x}^{3}}{8}}-{\frac{13\,x}{8}} \right ) }+{\frac{477\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49385, size = 96, normalized size = 1.19 \begin{align*} x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{477}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58518, size = 325, normalized size = 4.01 \begin{align*} \frac{8 \, x^{13} - 64 \, x^{11} + 1144 \, x^{9} + 10581 \, x^{7} + 26775 \, x^{5} + 26736 \, x^{3} - 2808 \, \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 ) + 477 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 9324 \, x}{8 \,{\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 = 0.240526, size = 75, normalized size = 0.93 \begin{align*} x^{5} - 14 x^{3} + 214 x + \frac{1669 x^{7} + 5831 x^{5} + 6640 x^{3} + 2476 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{477 \operatorname{atan}{\left (x \right )}}{8} - 351 \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 = 1.09603, size = 82, normalized size = 1.01 \begin{align*} x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{477}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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