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

Optimal. Leaf size=286 \[ -\frac{11 x \left (2 x^2+\sqrt{13}+5\right )}{26 \sqrt{x^4+5 x^2+3}}+\frac{x \left (11 x^2+8\right )}{13 \sqrt{x^4+5 x^2+3}}-\frac{4 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{26 \sqrt{x^4+5 x^2+3}} \]

[Out]

(-11*x*(5 + Sqrt[13] + 2*x^2))/(26*Sqrt[3 + 5*x^2 + x^4]) + (x*(8 + 11*x^2))/(13
*Sqrt[3 + 5*x^2 + x^4]) + (11*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], (-13 + 5*Sqrt[13])/6])/(26*Sqrt[3 + 5*x^2 + x^4]) - (4*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[1
3])/6])/(13*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.318153, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{11 x \left (2 x^2+\sqrt{13}+5\right )}{26 \sqrt{x^4+5 x^2+3}}+\frac{x \left (11 x^2+8\right )}{13 \sqrt{x^4+5 x^2+3}}-\frac{4 \sqrt{\frac{2}{3 \left (5+\sqrt{13}\right )}} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{13 \sqrt{x^4+5 x^2+3}}+\frac{11 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{26 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-11*x*(5 + Sqrt[13] + 2*x^2))/(26*Sqrt[3 + 5*x^2 + x^4]) + (x*(8 + 11*x^2))/(13
*Sqrt[3 + 5*x^2 + x^4]) + (11*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], (-13 + 5*Sqrt[13])/6])/(26*Sqrt[3 + 5*x^2 + x^4]) - (4*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[1
3])/6])/(13*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 27.73, size = 260, normalized size = 0.91 \[ \frac{x \left (11 x^{2} + 8\right )}{13 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{11 x \left (2 x^{2} + \sqrt{13} + 5\right )}{26 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{11 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{156 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{4 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{39 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} \]

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)**(3/2),x)

[Out]

x*(11*x**2 + 8)/(13*sqrt(x**4 + 5*x**2 + 3)) - 11*x*(2*x**2 + sqrt(13) + 5)/(26*
sqrt(x**4 + 5*x**2 + 3)) + 11*sqrt(6)*sqrt((x**2*(-sqrt(13) + 5) + 6)/(x**2*(sqr
t(13) + 5) + 6))*sqrt(sqrt(13) + 5)*(x**2*(sqrt(13) + 5) + 6)*elliptic_e(atan(sq
rt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt(13)/6)/(156*sqrt(x**4 + 5*x**2 + 3
)) - 4*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*s
qrt(13)/6)/(39*sqrt(sqrt(13) + 5)*sqrt(x**4 + 5*x**2 + 3))

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Mathematica [C]  time = 0.495049, size = 219, normalized size = 0.77 \[ \frac{4 x \left (11 x^2+8\right )+i \sqrt{2} \left (11 \sqrt{13}-39\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-11 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )}{52 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(4*x*(8 + 11*x^2) - (11*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] + I*Sqrt[2]*(-39 + 11*Sqrt[13])*Sqrt[(-5 + Sqr
t[13] - 2*x^2)/(-5 + Sqrt[13])]*Sqrt[5 + Sqrt[13] + 2*x^2]*EllipticF[I*ArcSinh[S
qrt[2/(5 + Sqrt[13])]*x], 19/6 + (5*Sqrt[13])/6])/(52*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.022, size = 240, normalized size = 0.8 \[ -4\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ( 1/13\,{x}^{3}+{\frac{5\,x}{26}} \right ) }-{\frac{48}{13\,\sqrt{-30+6\,\sqrt{13}}}\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{396}{13\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-6\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ( -{\frac{5\,{x}^{3}}{26}}-3/13\,x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(1/13*x^3+5/26*x)/(x^4+5*x^2+3)^(1/2)-48/13/(-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)*El
lipticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))+396/13/(-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)))-6*(-5/26*x^3-3/13*x)/(x^4+5*x^2+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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