Optimal. Leaf size=356 \[ \frac{2105 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{143 \sqrt{x^4+5 x^2+3}}+\frac{1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac{1}{429} \left (272 x^2+283\right ) \sqrt{x^4+5 x^2+3} x^5+\frac{1251}{715} \sqrt{x^4+5 x^2+3} x^3-\frac{4210}{429} \sqrt{x^4+5 x^2+3} x+\frac{176723 \left (2 x^2+\sqrt{13}+5\right ) x}{4290 \sqrt{x^4+5 x^2+3}}-\frac{176723 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{4290 \sqrt{x^4+5 x^2+3}} \]
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Rubi [A] time = 0.256476, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac{1}{429} \left (272 x^2+283\right ) \sqrt{x^4+5 x^2+3} x^5+\frac{1251}{715} \sqrt{x^4+5 x^2+3} x^3-\frac{4210}{429} \sqrt{x^4+5 x^2+3} x+\frac{176723 \left (2 x^2+\sqrt{13}+5\right ) x}{4290 \sqrt{x^4+5 x^2+3}}+\frac{2105 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{143 \sqrt{x^4+5 x^2+3}}-\frac{176723 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{4290 \sqrt{x^4+5 x^2+3}} \]
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 ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{3}{143} \int x^4 \left (-69-272 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx\\ &=-\frac{1}{429} x^5 \left (283+272 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{\int \frac{x^4 \left (16674+26271 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx}{3003}\\ &=\frac{1251}{715} x^3 \sqrt{3+5 x^2+x^4}-\frac{1}{429} x^5 \left (283+272 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{\int \frac{x^2 \left (236439+442050 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx}{15015}\\ &=-\frac{4210}{429} x \sqrt{3+5 x^2+x^4}+\frac{1251}{715} x^3 \sqrt{3+5 x^2+x^4}-\frac{1}{429} x^5 \left (283+272 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{\int \frac{1326150+3711183 x^2}{\sqrt{3+5 x^2+x^4}} \, dx}{45045}\\ &=-\frac{4210}{429} x \sqrt{3+5 x^2+x^4}+\frac{1251}{715} x^3 \sqrt{3+5 x^2+x^4}-\frac{1}{429} x^5 \left (283+272 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{4210}{143} \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx+\frac{176723 \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx}{2145}\\ &=\frac{176723 x \left (5+\sqrt{13}+2 x^2\right )}{4290 \sqrt{3+5 x^2+x^4}}-\frac{4210}{429} x \sqrt{3+5 x^2+x^4}+\frac{1251}{715} x^3 \sqrt{3+5 x^2+x^4}-\frac{1}{429} x^5 \left (283+272 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{143} x^5 \left (71+33 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{176723 \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 )}{4290 \sqrt{3+5 x^2+x^4}}+\frac{2105 \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 ) 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 )}{143 \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.317198, size = 249, normalized size = 0.7 \[ \frac{-i \sqrt{2} \left (176723 \sqrt{13}-757315\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+4 x \left (495 x^{14}+6015 x^{12}+24635 x^{10}+39650 x^8+29003 x^6+3055 x^4-93991 x^2-63150\right )+176723 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )}{8580 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.02, size = 294, normalized size = 0.8 \begin{align*}{\frac{3\,{x}^{11}}{13}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{236\,{x}^{9}}{143}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1090\,{x}^{7}}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{356\,{x}^{5}}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1251\,{x}^{3}}{715}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{4210\,x}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{25260}{143\,\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{2120676}{715\,\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{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\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^{10} + 17 \, x^{8} + 19 \, x^{6} + 6 \, 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 ) \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{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\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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