Optimal. Leaf size=322 \[ -\frac{13 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{1}{21} \left (7 x^2+11\right ) \sqrt{x^4+5 x^2+3} x^5-\frac{26}{35} \sqrt{x^4+5 x^2+3} x^3+\frac{13}{3} \sqrt{x^4+5 x^2+3} x-\frac{1924 \left (2 x^2+\sqrt{13}+5\right ) x}{105 \sqrt{x^4+5 x^2+3}}+\frac{962 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{105 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.273842, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1273, 1279, 1189, 1099, 1135} \[ \frac{1}{21} \left (7 x^2+11\right ) \sqrt{x^4+5 x^2+3} x^5-\frac{26}{35} \sqrt{x^4+5 x^2+3} x^3+\frac{13}{3} \sqrt{x^4+5 x^2+3} x-\frac{1924 \left (2 x^2+\sqrt{13}+5\right ) x}{105 \sqrt{x^4+5 x^2+3}}-\frac{13 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{962 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{105 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1273
Rule 1279
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int x^4 \left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx &=\frac{1}{21} x^5 \left (11+7 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{63} \int \frac{x^4 \left (-117-234 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{26}{35} x^3 \sqrt{3+5 x^2+x^4}+\frac{1}{21} x^5 \left (11+7 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1}{315} \int \frac{x^2 \left (-2106-4095 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{13}{3} x \sqrt{3+5 x^2+x^4}-\frac{26}{35} x^3 \sqrt{3+5 x^2+x^4}+\frac{1}{21} x^5 \left (11+7 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{945} \int \frac{-12285-34632 x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{13}{3} x \sqrt{3+5 x^2+x^4}-\frac{26}{35} x^3 \sqrt{3+5 x^2+x^4}+\frac{1}{21} x^5 \left (11+7 x^2\right ) \sqrt{3+5 x^2+x^4}-13 \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx-\frac{3848}{105} \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{1924 x \left (5+\sqrt{13}+2 x^2\right )}{105 \sqrt{3+5 x^2+x^4}}+\frac{13}{3} x \sqrt{3+5 x^2+x^4}-\frac{26}{35} x^3 \sqrt{3+5 x^2+x^4}+\frac{1}{21} x^5 \left (11+7 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{962 \sqrt{\frac{2}{3} \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 )}{105 \sqrt{3+5 x^2+x^4}}-\frac{13 \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 )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.325432, size = 237, normalized size = 0.74 \[ \frac{13 i \sqrt{2} \left (148 \sqrt{13}-635\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+70 x^{11}+460 x^9+604 x^7+460 x^5+4082 x^3-1924 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+2730 x}{210 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.098, size = 260, normalized size = 0.8 \begin{align*}{\frac{{x}^{7}}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{11\,{x}^{5}}{21}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{26\,{x}^{3}}{35}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13\,x}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-78\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}+{\frac{46176}{35\,\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}}}} \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 \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{4}\,{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 ({\left (3 \, x^{6} + 2 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}, 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 x^{4} \left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}\, 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 \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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