Optimal. Leaf size=237 \[ x^5-\frac{17 x^3}{3}+\frac{25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+19 x+\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.293025, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1668, 1676, 1169, 634, 618, 204, 628} \[ x^5-\frac{17 x^3}{3}+\frac{25 \left (3-x^2\right ) x}{8 \left (x^4+2 x^2+3\right )}+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+19 x+\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{16} \sqrt{\frac{3}{2} \left (5011 \sqrt{3}-8669\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1676
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^6 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{-450+1050 x^2-336 x^6+240 x^8}{3+2 x^2+x^4} \, dx\\ &=\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \left (912-816 x^2+240 x^4-\frac{54 \left (59-31 x^2\right )}{3+2 x^2+x^4}\right ) \, dx\\ &=19 x-\frac{17 x^3}{3}+x^5+\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{9}{8} \int \frac{59-31 x^2}{3+2 x^2+x^4} \, dx\\ &=19 x-\frac{17 x^3}{3}+x^5+\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{32} \left (3 \sqrt{3 \left (1+\sqrt{3}\right )}\right ) \int \frac{59 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (59+31 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{32} \left (3 \sqrt{3 \left (1+\sqrt{3}\right )}\right ) \int \frac{59 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (59+31 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=19 x-\frac{17 x^3}{3}+x^5+\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{1}{16} \left (3 \sqrt{\frac{3}{2} \left (3182-1829 \sqrt{3}\right )}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{16} \left (3 \sqrt{\frac{3}{2} \left (3182-1829 \sqrt{3}\right )}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{32} \left (3 \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )}\right ) \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{32} \left (3 \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )}\right ) \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=19 x-\frac{17 x^3}{3}+x^5+\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{8} \left (3 \sqrt{\frac{3}{2} \left (3182-1829 \sqrt{3}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )+\frac{1}{8} \left (3 \sqrt{\frac{3}{2} \left (3182-1829 \sqrt{3}\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )\\ &=19 x-\frac{17 x^3}{3}+x^5+\frac{25 x \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{3}{16} \sqrt{\frac{3}{2} \left (-8669+5011 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{16} \sqrt{\frac{3}{2} \left (-8669+5011 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{32} \sqrt{\frac{3}{2} \left (8669+5011 \sqrt{3}\right )} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.164887, size = 132, normalized size = 0.56 \[ x^5-\frac{17 x^3}{3}-\frac{25 \left (x^2-3\right ) x}{8 \left (x^4+2 x^2+3\right )}+19 x+\frac{9 \left (31 \sqrt{2}+90 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{16 \sqrt{2-2 i \sqrt{2}}}+\frac{9 \left (31 \sqrt{2}-90 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{16 \sqrt{2+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 419, normalized size = 1.8 \begin{align*}{x}^{5}-{\frac{17\,{x}^{3}}{3}}+19\,x+{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{25\,{x}^{3}}{8}}+{\frac{75\,x}{8}} \right ) }+{\frac{57\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{16}}+{\frac{405\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{64}}+{\frac{ \left ( -114+114\,\sqrt{3} \right ) \sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-810+810\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{177\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{57\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{16}}-{\frac{405\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{64}}+{\frac{ \left ( -114+114\,\sqrt{3} \right ) \sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-810+810\,\sqrt{3}}{32\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{177\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} x^{5} - \frac{17}{3} \, x^{3} + 19 \, x - \frac{25 \,{\left (x^{3} - 3 \, x\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{9}{8} \, \int \frac{31 \, x^{2} - 59}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66491, size = 2055, normalized size = 8.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.541024, size = 63, normalized size = 0.27 \begin{align*} x^{5} - \frac{17 x^{3}}{3} + 19 x - \frac{25 x^{3} - 75 x}{8 x^{4} + 16 x^{2} + 24} + 3 \operatorname{RootSum}{\left (1048576 t^{4} - 53262336 t^{2} + 677973267, \left ( t \mapsto t \log{\left (- \frac{2490368 t^{3}}{13484601} + \frac{20518496 t}{4494867} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{6}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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