3.152 \(\int x^2 \left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx\)

Optimal. Leaf size=305 \[ -\frac{4}{3} \sqrt{x^4+5 x^2+3} x+\frac{1247 \left (2 x^2+\sqrt{13}+5\right ) x}{210 \sqrt{x^4+5 x^2+3}}+\frac{2 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{x^4+5 x^2+3}}-\frac{1247 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{210 \sqrt{x^4+5 x^2+3}}+\frac{1}{35} \left (15 x^2+29\right ) \sqrt{x^4+5 x^2+3} x^3 \]

[Out]

(1247*x*(5 + Sqrt[13] + 2*x^2))/(210*Sqrt[3 + 5*x^2 + x^4]) - (4*x*Sqrt[3 + 5*x^
2 + x^4])/3 + (x^3*(29 + 15*x^2)*Sqrt[3 + 5*x^2 + x^4])/35 - (1247*Sqrt[(5 + Sqr
t[13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt
[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2
10*Sqrt[3 + 5*x^2 + x^4]) + (2*Sqrt[2/(3*(5 + Sqrt[13]))]*Sqrt[(6 + (5 - Sqrt[13
])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[ArcTan[Sqrt
[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4]

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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]

[Out]

(1247*x*(5 + Sqrt[13] + 2*x^2))/(210*Sqrt[3 + 5*x^2 + x^4]) - (4*x*Sqrt[3 + 5*x^
2 + x^4])/3 + (x^3*(29 + 15*x^2)*Sqrt[3 + 5*x^2 + x^4])/35 - (1247*Sqrt[(5 + Sqr
t[13])/6]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt
[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(2
10*Sqrt[3 + 5*x^2 + x^4]) + (2*Sqrt[2/(3*(5 + Sqrt[13]))]*Sqrt[(6 + (5 - Sqrt[13
])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticF[ArcTan[Sqrt
[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/Sqrt[3 + 5*x^2 + x^4]

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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)

[Out]

x**3*(15*x**2 + 29)*sqrt(x**4 + 5*x**2 + 3)/35 + 1247*x*(2*x**2 + sqrt(13) + 5)/
(210*sqrt(x**4 + 5*x**2 + 3)) - 4*x*sqrt(x**4 + 5*x**2 + 3)/3 - 1247*sqrt(6)*sqr
t((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(13) + 5) + 6))*sqrt(sqrt(13) + 5)*(x**2
*(sqrt(13) + 5) + 6)*elliptic_e(atan(sqrt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*
sqrt(13)/6)/(1260*sqrt(x**4 + 5*x**2 + 3)) + 2*sqrt(6)*sqrt((x**2*(-sqrt(13) + 5
) + 6)/(x**2*(sqrt(13) + 5) + 6))*(x**2*(sqrt(13) + 5) + 6)*elliptic_f(atan(sqrt
(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt(13)/6)/(3*sqrt(sqrt(13) + 5)*sqrt(x*
*4 + 5*x**2 + 3))

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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]

[Out]

(4*x*(-420 - 439*x^2 + 430*x^4 + 312*x^6 + 45*x^8) + (1247*I)*Sqrt[2]*(-5 + Sqrt
[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*E
llipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]
*(-5395 + 1247*Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 +
Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[
13])/6])/(420*Sqrt[3 + 5*x^2 + x^4])

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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)

[Out]

29/35*x^3*(x^4+5*x^2+3)^(1/2)-4/3*x*(x^4+5*x^2+3)^(1/2)+24/(-30+6*13^(1/2))^(1/2
)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/2))*x^2)^(1/2)/(x^4+5*x^2
+3)^(1/2)*EllipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-14964
/35/(-30+6*13^(1/2))^(1/2)*(1-(-5/6+1/6*13^(1/2))*x^2)^(1/2)*(1-(-5/6-1/6*13^(1/
2))*x^2)^(1/2)/(x^4+5*x^2+3)^(1/2)/(5+13^(1/2))*(EllipticF(1/6*x*(-30+6*13^(1/2)
)^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-EllipticE(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(
1/2)+1/6*39^(1/2)))+3/7*x^5*(x^4+5*x^2+3)^(1/2)

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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")

[Out]

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^2, x)

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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")

[Out]

integral((3*x^4 + 2*x^2)*sqrt(x^4 + 5*x^2 + 3), x)

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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)

[Out]

Integral(x**2*(3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3), 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")

[Out]

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^2, x)