Optimal. Leaf size=243 \[ x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.359897, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {1668, 1678, 1676, 1169, 634, 618, 204, 628} \[ x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+58 x+\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{7678611 \sqrt{3}-8595619} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 1668
Rule 1678
Rule 1676
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^3} \, dx &=-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{1}{96} \int \frac{2250-2850 x^2-4800 x^4+2400 x^6-672 x^{10}+480 x^{12}}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{-201960+193248 x^2+87552 x^4-78336 x^6+23040 x^8}{3+2 x^2+x^4} \, dx}{4608}\\ &=-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{\int \left (267264-124416 x^2+23040 x^4-\frac{216 \left (4647-148 x^2\right )}{3+2 x^2+x^4}\right ) \, dx}{4608}\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{3}{64} \int \frac{4647-148 x^2}{3+2 x^2+x^4} \, dx\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \sqrt{3 \left (1+\sqrt{3}\right )} \int \frac{4647 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (4647+148 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \sqrt{3 \left (1+\sqrt{3}\right )} \int \frac{4647 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (4647+148 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}-\frac{1}{256} \left (3 \sqrt{7220107-458504 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx-\frac{1}{256} \left (3 \sqrt{7220107-458504 \sqrt{3}}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{512} \left (3 \sqrt{8595619+7678611 \sqrt{3}}\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}{512} \left (3 \sqrt{8595619+7678611 \sqrt{3}}\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\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )+\frac{1}{128} \left (3 \sqrt{7220107-458504 \sqrt{3}}\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}{128} \left (3 \sqrt{7220107-458504 \sqrt{3}}\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 )\\ &=58 x-9 x^3+x^5-\frac{25 x \left (15+7 x^2\right )}{16 \left (3+2 x^2+x^4\right )^2}+\frac{x \left (3305+252 x^2\right )}{64 \left (3+2 x^2+x^4\right )}+\frac{3}{256} \sqrt{-8595619+7678611 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )-\frac{3}{256} \sqrt{-8595619+7678611 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{3}{512} \sqrt{8595619+7678611 \sqrt{3}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.222089, size = 156, normalized size = 0.64 \[ x^5-9 x^3+\frac{\left (252 x^2+3305\right ) x}{64 \left (x^4+2 x^2+3\right )}-\frac{25 \left (7 x^2+15\right ) x}{16 \left (x^4+2 x^2+3\right )^2}+58 x+\frac{3 \left (148 \sqrt{2}+4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{3 \left (148 \sqrt{2}-4795 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 429, normalized size = 1.8 \begin{align*}{x}^{5}-9\,{x}^{3}+58\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ({\frac{63\,{x}^{7}}{16}}+{\frac{3809\,{x}^{5}}{64}}+{\frac{3333\,{x}^{3}}{32}}+{\frac{8415\,x}{64}} \right ) }+{\frac{5091\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-28770+28770\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4647\,\sqrt{3}}{64\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{5091\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{14385\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -10182+10182\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-28770+28770\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{4647\,\sqrt{3}}{64\,\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} - 9 \, x^{3} + 58 \, x + \frac{252 \, x^{7} + 3809 \, x^{5} + 6666 \, x^{3} + 8415 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{3}{64} \, \int \frac{148 \, x^{2} - 4647}{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 = 1.7942, size = 2853, normalized size = 11.74 \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 = 0.591164, size = 82, normalized size = 0.34 \begin{align*} x^{5} - 9 x^{3} + 58 x + \frac{252 x^{7} + 3809 x^{5} + 6666 x^{3} + 8415 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 3 \operatorname{RootSum}{\left (17179869184 t^{4} - 2253289947136 t^{2} + 176883200667963, \left ( t \mapsto t \log{\left (- \frac{56941871104 t^{3}}{55104008440689} - \frac{1957224667904 t}{55104008440689} + 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^{10}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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