Optimal. Leaf size=307 \[ \frac{11 \sqrt{\frac{3}{2 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{13 \sqrt{x^4+5 x^2+3}}+\frac{\left (11 x^2+8\right ) x^3}{13 \sqrt{x^4+5 x^2+3}}-\frac{11}{13} \sqrt{x^4+5 x^2+3} x+\frac{43 \left (2 x^2+\sqrt{13}+5\right ) x}{13 \sqrt{x^4+5 x^2+3}}-\frac{43 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.163282, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1275, 1279, 1189, 1099, 1135} \[ \frac{\left (11 x^2+8\right ) x^3}{13 \sqrt{x^4+5 x^2+3}}-\frac{11}{13} \sqrt{x^4+5 x^2+3} x+\frac{43 \left (2 x^2+\sqrt{13}+5\right ) x}{13 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{3}{2 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{x^4+5 x^2+3}}-\frac{43 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1275
Rule 1279
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \frac{x^4 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac{x^3 \left (8+11 x^2\right )}{13 \sqrt{3+5 x^2+x^4}}+\frac{1}{13} \int \frac{x^2 \left (-24-33 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{x^3 \left (8+11 x^2\right )}{13 \sqrt{3+5 x^2+x^4}}-\frac{11}{13} x \sqrt{3+5 x^2+x^4}-\frac{1}{39} \int \frac{-99-258 x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{x^3 \left (8+11 x^2\right )}{13 \sqrt{3+5 x^2+x^4}}-\frac{11}{13} x \sqrt{3+5 x^2+x^4}+\frac{33}{13} \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx+\frac{86}{13} \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{43 x \left (5+\sqrt{13}+2 x^2\right )}{13 \sqrt{3+5 x^2+x^4}}+\frac{x^3 \left (8+11 x^2\right )}{13 \sqrt{3+5 x^2+x^4}}-\frac{11}{13} x \sqrt{3+5 x^2+x^4}-\frac{43 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{6+\left (5-\sqrt{13}\right ) x^2}{6+\left (5+\sqrt{13}\right ) x^2}} \left (6+\left (5+\sqrt{13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{3+5 x^2+x^4}}+\frac{11 \sqrt{\frac{3}{2 \left (5+\sqrt{13}\right )}} \sqrt{\frac{6+\left (5-\sqrt{13}\right ) x^2}{6+\left (5+\sqrt{13}\right ) x^2}} \left (6+\left (5+\sqrt{13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.300614, size = 219, normalized size = 0.71 \[ \frac{-i \sqrt{2} \left (43 \sqrt{13}-182\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-2 x \left (47 x^2+33\right )+43 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )}{26 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.021, size = 240, normalized size = 0.8 \begin{align*} -6\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ({\frac{19\,{x}^{3}}{26}}+{\frac{15\,x}{26}} \right ) }+{\frac{198}{13\,\sqrt{-30+6\,\sqrt{13}}}\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{3096}{13\,\sqrt{-30+6\,\sqrt{13}} \left ( \sqrt{13}+5 \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-4\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ( -{\frac{5\,{x}^{3}}{26}}-3/13\,x \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*} \int \frac{{\left (3 \, x^{2} + 2\right )} x^{4}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (3 \, x^{6} + 2 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{8} + 10 \, x^{6} + 31 \, x^{4} + 30 \, x^{2} + 9}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}\, dx \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 (3 \, x^{2} + 2\right )} x^{4}}{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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