Optimal. Leaf size=305 \[ -\frac{4}{3} \sqrt{x^4+5 x^2+3} x+\frac{1247 \left (2 x^2+\sqrt{13}+5\right ) x}{210 \sqrt{x^4+5 x^2+3}}+\frac{2 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{x^4+5 x^2+3}}-\frac{1247 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{210 \sqrt{x^4+5 x^2+3}}+\frac{1}{35} \left (15 x^2+29\right ) \sqrt{x^4+5 x^2+3} x^3 \]
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Rubi [A] time = 0.464907, antiderivative size = 305, 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 \[ -\frac{4}{3} \sqrt{x^4+5 x^2+3} x+\frac{1247 \left (2 x^2+\sqrt{13}+5\right ) x}{210 \sqrt{x^4+5 x^2+3}}+\frac{2 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{x^4+5 x^2+3}}-\frac{1247 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{210 \sqrt{x^4+5 x^2+3}}+\frac{1}{35} \left (15 x^2+29\right ) \sqrt{x^4+5 x^2+3} x^3 \]
Antiderivative was successfully verified.
[In] Int[x^2*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]
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Rubi in Sympy [A] time = 37.8938, size = 280, normalized size = 0.92 \[ \frac{x^{3} \left (15 x^{2} + 29\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{35} + \frac{1247 x \left (2 x^{2} + \sqrt{13} + 5\right )}{210 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{4 x \sqrt{x^{4} + 5 x^{2} + 3}}{3} - \frac{1247 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{1260 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{2 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{3 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(3*x**2+2)*(x**4+5*x**2+3)**(1/2),x)
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Mathematica [C] time = 0.523429, size = 234, normalized size = 0.77 \[ \frac{-i \sqrt{2} \left (1247 \sqrt{13}-5395\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+1247 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+4 x \left (45 x^8+312 x^6+430 x^4-439 x^2-420\right )}{420 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^2*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]
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Maple [A] time = 0.019, size = 243, normalized size = 0.8 \[{\frac{29\,{x}^{3}}{35}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{4\,x}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+24\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}-{\frac{14964}{35\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{3\,{x}^{5}}{7}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(3*x^2+2)*(x^4+5*x^2+3)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^2,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (3 \, x^{4} + 2 \, x^{2}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^2,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(3*x**2+2)*(x**4+5*x**2+3)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^2,x, algorithm="giac")
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