Optimal. Leaf size=80 \[ \frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.100481, antiderivative size = 80, 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} \[ \frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
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^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}-\frac{1}{8} \int \frac{-412+1230 x^2+424 x^4-216 x^6+96 x^8-40 x^{10}}{\left (2+3 x^2+x^4\right )^2} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \frac{728+1500 x^2-864 x^4+160 x^6}{2+3 x^2+x^4} \, dx\\ &=-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{32} \int \left (-1344+160 x^2+\frac{4 \left (854+1303 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}+\frac{1}{8} \int \frac{854+1303 x^2}{2+3 x^2+x^4} \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{449}{8} \int \frac{1}{1+x^2} \, dx+219 \int \frac{1}{2+x^2} \, dx\\ &=-42 x+\frac{5 x^3}{3}-\frac{x \left (206+207 x^2\right )}{4 \left (2+3 x^2+x^4\right )^2}+\frac{x \left (24-409 x^2\right )}{8 \left (2+3 x^2+x^4\right )}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0552934, size = 66, normalized size = 0.82 \[ \frac{x \left (40 x^{10}-768 x^8-6755 x^6-16233 x^4-15416 x^2-5124\right )}{24 \left (x^4+3 x^2+2\right )^2}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 62, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{3}}-42\,x+16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{53\,{x}^{3}}{16}}-{\frac{27\,x}{8}} \right ) }+{\frac{219\,\sqrt{2}}{2}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{15\,{x}^{3}}{8}}-{\frac{17\,x}{8}} \right ) }-{\frac{449\,\arctan \left ( x \right ) }{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49531, size = 92, normalized size = 1.15 \begin{align*} \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{449}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53826, size = 313, normalized size = 3.91 \begin{align*} \frac{40 \, x^{11} - 768 \, x^{9} - 6755 \, x^{7} - 16233 \, x^{5} - 15416 \, x^{3} + 2628 \, \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 ) - 1347 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 5124 \, x}{24 \,{\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 = 0.238198, size = 75, normalized size = 0.94 \begin{align*} \frac{5 x^{3}}{3} - 42 x - \frac{409 x^{7} + 1203 x^{5} + 1160 x^{3} + 364 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac{449 \operatorname{atan}{\left (x \right )}}{8} + \frac{219 \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 = 1.11295, size = 78, normalized size = 0.98 \begin{align*} \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{449}{8} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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