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

Optimal. Leaf size=356 \[ -\frac{4210}{429} \sqrt{x^4+5 x^2+3} x+\frac{176723 \left (2 x^2+\sqrt{13}+5\right ) x}{4290 \sqrt{x^4+5 x^2+3}}+\frac{2105 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{143 \sqrt{x^4+5 x^2+3}}-\frac{176723 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{4290 \sqrt{x^4+5 x^2+3}}+\frac{1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac{1}{429} \left (272 x^2+283\right ) \sqrt{x^4+5 x^2+3} x^5+\frac{1251}{715} \sqrt{x^4+5 x^2+3} x^3 \]

[Out]

(176723*x*(5 + Sqrt[13] + 2*x^2))/(4290*Sqrt[3 + 5*x^2 + x^4]) - (4210*x*Sqrt[3
+ 5*x^2 + x^4])/429 + (1251*x^3*Sqrt[3 + 5*x^2 + x^4])/715 - (x^5*(283 + 272*x^2
)*Sqrt[3 + 5*x^2 + x^4])/429 + (x^5*(71 + 33*x^2)*(3 + 5*x^2 + x^4)^(3/2))/143 -
(176723*Sqrt[(5 + Sqrt[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], (-1
3 + 5*Sqrt[13])/6])/(4290*Sqrt[3 + 5*x^2 + x^4]) + (2105*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])/(143*Sqr
t[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.642794, 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 \[ -\frac{4210}{429} \sqrt{x^4+5 x^2+3} x+\frac{176723 \left (2 x^2+\sqrt{13}+5\right ) x}{4290 \sqrt{x^4+5 x^2+3}}+\frac{2105 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{143 \sqrt{x^4+5 x^2+3}}-\frac{176723 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{4290 \sqrt{x^4+5 x^2+3}}+\frac{1}{143} \left (33 x^2+71\right ) \left (x^4+5 x^2+3\right )^{3/2} x^5-\frac{1}{429} \left (272 x^2+283\right ) \sqrt{x^4+5 x^2+3} x^5+\frac{1251}{715} \sqrt{x^4+5 x^2+3} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(176723*x*(5 + Sqrt[13] + 2*x^2))/(4290*Sqrt[3 + 5*x^2 + x^4]) - (4210*x*Sqrt[3
+ 5*x^2 + x^4])/429 + (1251*x^3*Sqrt[3 + 5*x^2 + x^4])/715 - (x^5*(283 + 272*x^2
)*Sqrt[3 + 5*x^2 + x^4])/429 + (x^5*(71 + 33*x^2)*(3 + 5*x^2 + x^4)^(3/2))/143 -
(176723*Sqrt[(5 + Sqrt[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], (-1
3 + 5*Sqrt[13])/6])/(4290*Sqrt[3 + 5*x^2 + x^4]) + (2105*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])/(143*Sqr
t[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 50.5822, size = 326, normalized size = 0.92 \[ \frac{x^{5} \left (33 x^{2} + 71\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{143} - \frac{x^{5} \left (1904 x^{2} + 1981\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{3003} + \frac{1251 x^{3} \sqrt{x^{4} + 5 x^{2} + 3}}{715} + \frac{176723 x \left (2 x^{2} + \sqrt{13} + 5\right )}{4290 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{4210 x \sqrt{x^{4} + 5 x^{2} + 3}}{429} - \frac{176723 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{25740 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{2105 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{429 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} \]

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)**(3/2),x)

[Out]

x**5*(33*x**2 + 71)*(x**4 + 5*x**2 + 3)**(3/2)/143 - x**5*(1904*x**2 + 1981)*sqr
t(x**4 + 5*x**2 + 3)/3003 + 1251*x**3*sqrt(x**4 + 5*x**2 + 3)/715 + 176723*x*(2*
x**2 + sqrt(13) + 5)/(4290*sqrt(x**4 + 5*x**2 + 3)) - 4210*x*sqrt(x**4 + 5*x**2
+ 3)/429 - 176723*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)/(25740*sqrt(x**4 + 5*x**2 + 3)) + 2105*
sqrt(6)*sqrt((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(13) + 5) + 6))*(x**2*(sqrt(1
3) + 5) + 6)*elliptic_f(atan(sqrt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt(13)
/6)/(429*sqrt(sqrt(13) + 5)*sqrt(x**4 + 5*x**2 + 3))

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Mathematica [C]  time = 0.654941, size = 249, normalized size = 0.7 \[ \frac{-i \sqrt{2} \left (176723 \sqrt{13}-757315\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+176723 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+4 x \left (495 x^{14}+6015 x^{12}+24635 x^{10}+39650 x^8+29003 x^6+3055 x^4-93991 x^2-63150\right )}{8580 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*x*(-63150 - 93991*x^2 + 3055*x^4 + 29003*x^6 + 39650*x^8 + 24635*x^10 + 6015*
x^12 + 495*x^14) + (176723*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 + S
qrt[13])]*x], 19/6 + (5*Sqrt[13])/6] - I*Sqrt[2]*(-757315 + 176723*Sqrt[13])*Sqr
t[(-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])/(8580*Sqrt[3 + 5*x^
2 + x^4])

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Maple [A]  time = 0.027, size = 294, normalized size = 0.8 \[{\frac{236\,{x}^{9}}{143}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1090\,{x}^{7}}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{356\,{x}^{5}}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1251\,{x}^{3}}{715}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{4210\,x}{429}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{25260}{143\,\sqrt{-30+6\,\sqrt{13}}}\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-{\frac{2120676}{715\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{3\,{x}^{11}}{13}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

236/143*x^9*(x^4+5*x^2+3)^(1/2)+1090/429*x^7*(x^4+5*x^2+3)^(1/2)+356/429*x^5*(x^
4+5*x^2+3)^(1/2)+1251/715*x^3*(x^4+5*x^2+3)^(1/2)-4210/429*x*(x^4+5*x^2+3)^(1/2)
+25260/143/(-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))-2120676/715/(-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)))+3/13*x^11*(x^4+5*x^2+3)^(
1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (3 \, x^{10} + 17 \, x^{8} + 19 \, x^{6} + 6 \, x^{4}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x^10 + 17*x^8 + 19*x^6 + 6*x^4)*sqrt(x^4 + 5*x^2 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*(3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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