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

Optimal. Leaf size=322 \[ \frac{13}{3} \sqrt{x^4+5 x^2+3} x-\frac{1924 \left (2 x^2+\sqrt{13}+5\right ) x}{105 \sqrt{x^4+5 x^2+3}}-\frac{13 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{962 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{105 \sqrt{x^4+5 x^2+3}}+\frac{1}{21} \left (7 x^2+11\right ) \sqrt{x^4+5 x^2+3} x^5-\frac{26}{35} \sqrt{x^4+5 x^2+3} x^3 \]

[Out]

(-1924*x*(5 + Sqrt[13] + 2*x^2))/(105*Sqrt[3 + 5*x^2 + x^4]) + (13*x*Sqrt[3 + 5*
x^2 + x^4])/3 - (26*x^3*Sqrt[3 + 5*x^2 + x^4])/35 + (x^5*(11 + 7*x^2)*Sqrt[3 + 5
*x^2 + x^4])/21 + (962*Sqrt[(2*(5 + Sqrt[13]))/3]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/
(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sq
rt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(105*Sqrt[3 + 5*x^2 + x^4]) - (13*Sqrt[(6
+ (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Ellipti
cF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(Sqrt[6*(5 + Sqrt[13
])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.602963, 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 \[ \frac{13}{3} \sqrt{x^4+5 x^2+3} x-\frac{1924 \left (2 x^2+\sqrt{13}+5\right ) x}{105 \sqrt{x^4+5 x^2+3}}-\frac{13 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{962 \sqrt{\frac{2}{3} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{105 \sqrt{x^4+5 x^2+3}}+\frac{1}{21} \left (7 x^2+11\right ) \sqrt{x^4+5 x^2+3} x^5-\frac{26}{35} \sqrt{x^4+5 x^2+3} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^4*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]

[Out]

(-1924*x*(5 + Sqrt[13] + 2*x^2))/(105*Sqrt[3 + 5*x^2 + x^4]) + (13*x*Sqrt[3 + 5*
x^2 + x^4])/3 - (26*x^3*Sqrt[3 + 5*x^2 + x^4])/35 + (x^5*(11 + 7*x^2)*Sqrt[3 + 5
*x^2 + x^4])/21 + (962*Sqrt[(2*(5 + Sqrt[13]))/3]*Sqrt[(6 + (5 - Sqrt[13])*x^2)/
(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*EllipticE[ArcTan[Sqrt[(5 + Sq
rt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(105*Sqrt[3 + 5*x^2 + x^4]) - (13*Sqrt[(6
+ (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Ellipti
cF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(Sqrt[6*(5 + Sqrt[13
])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 42.9635, size = 301, normalized size = 0.93 \[ \frac{x^{5} \left (21 x^{2} + 33\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{63} - \frac{26 x^{3} \sqrt{x^{4} + 5 x^{2} + 3}}{35} - \frac{1924 x \left (2 x^{2} + \sqrt{13} + 5\right )}{105 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{13 x \sqrt{x^{4} + 5 x^{2} + 3}}{3} + \frac{962 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{315 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{13 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{6 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(3*x**2+2)*(x**4+5*x**2+3)**(1/2),x)

[Out]

x**5*(21*x**2 + 33)*sqrt(x**4 + 5*x**2 + 3)/63 - 26*x**3*sqrt(x**4 + 5*x**2 + 3)
/35 - 1924*x*(2*x**2 + sqrt(13) + 5)/(105*sqrt(x**4 + 5*x**2 + 3)) + 13*x*sqrt(x
**4 + 5*x**2 + 3)/3 + 962*sqrt(6)*sqrt((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)/(315*sqrt(x**4 + 5*x**2 + 3)) -
13*sqrt(6)*sqrt((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(13) + 5) + 6))*(x**2*(sq
rt(13) + 5) + 6)*elliptic_f(atan(sqrt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt
(13)/6)/(6*sqrt(sqrt(13) + 5)*sqrt(x**4 + 5*x**2 + 3))

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Mathematica [C]  time = 0.551457, size = 237, normalized size = 0.74 \[ \frac{70 x^{11}+460 x^9+604 x^7+460 x^5+4082 x^3+13 i \sqrt{2} \left (148 \sqrt{13}-635\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-1924 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+2730 x}{210 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4*(2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4],x]

[Out]

(2730*x + 4082*x^3 + 460*x^5 + 604*x^7 + 460*x^9 + 70*x^11 - (1924*I)*Sqrt[2]*(-
5 + Sqrt[13])*Sqrt[(-5 + Sqrt[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] +
2*x^2]*EllipticE[I*ArcSinh[Sqrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6] + (
13*I)*Sqrt[2]*(-635 + 148*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])/(210*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.3, size = 260, normalized size = 0.8 \[{\frac{11\,{x}^{5}}{21}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{26\,{x}^{3}}{35}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13\,x}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-78\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}+{\frac{46176}{35\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{{x}^{7}}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(3*x^2+2)*(x^4+5*x^2+3)^(1/2),x)

[Out]

11/21*x^5*(x^4+5*x^2+3)^(1/2)-26/35*x^3*(x^4+5*x^2+3)^(1/2)+13/3*x*(x^4+5*x^2+3)
^(1/2)-78/(-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))+46176/35/(-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)))+1/3*x^7*(x^4+5*x^2+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (3 \, x^{6} + 2 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^4,x, algorithm="fricas")

[Out]

integral((3*x^6 + 2*x^4)*sqrt(x^4 + 5*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(3*x**2+2)*(x**4+5*x**2+3)**(1/2),x)

[Out]

Integral(x**4*(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. \[ \int \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)*x^4,x, algorithm="giac")

[Out]

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