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

Optimal. Leaf size=326 \[ -\frac{133 x \left (2 x^2+\sqrt{13}+5\right )}{1053 \sqrt{x^4+5 x^2+3}}+\frac{266 \sqrt{x^4+5 x^2+3}}{1053 x}-\frac{5 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{351 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{133 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{1053 \sqrt{x^4+5 x^2+3}}-\frac{5 \sqrt{x^4+5 x^2+3}}{351 x^3}-\frac{8 x^2+7}{39 x^3 \sqrt{x^4+5 x^2+3}} \]

[Out]

(-133*x*(5 + Sqrt[13] + 2*x^2))/(1053*Sqrt[3 + 5*x^2 + x^4]) - (7 + 8*x^2)/(39*x
^3*Sqrt[3 + 5*x^2 + x^4]) - (5*Sqrt[3 + 5*x^2 + x^4])/(351*x^3) + (266*Sqrt[3 +
5*x^2 + x^4])/(1053*x) + (133*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])/(1053*Sqrt[3 + 5*x^2 + x^4]) - (5*Sqrt[
(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Elli
pticF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(351*Sqrt[6*(5 +
Sqrt[13])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.507465, antiderivative size = 326, 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 \[ -\frac{133 x \left (2 x^2+\sqrt{13}+5\right )}{1053 \sqrt{x^4+5 x^2+3}}+\frac{266 \sqrt{x^4+5 x^2+3}}{1053 x}-\frac{5 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{351 \sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}+\frac{133 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{1053 \sqrt{x^4+5 x^2+3}}-\frac{5 \sqrt{x^4+5 x^2+3}}{351 x^3}-\frac{8 x^2+7}{39 x^3 \sqrt{x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

(-133*x*(5 + Sqrt[13] + 2*x^2))/(1053*Sqrt[3 + 5*x^2 + x^4]) - (7 + 8*x^2)/(39*x
^3*Sqrt[3 + 5*x^2 + x^4]) - (5*Sqrt[3 + 5*x^2 + x^4])/(351*x^3) + (266*Sqrt[3 +
5*x^2 + x^4])/(1053*x) + (133*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])/(1053*Sqrt[3 + 5*x^2 + x^4]) - (5*Sqrt[
(6 + (5 - Sqrt[13])*x^2)/(6 + (5 + Sqrt[13])*x^2)]*(6 + (5 + Sqrt[13])*x^2)*Elli
pticF[ArcTan[Sqrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(351*Sqrt[6*(5 +
Sqrt[13])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 40.2134, size = 301, normalized size = 0.92 \[ - \frac{133 x \left (2 x^{2} + \sqrt{13} + 5\right )}{1053 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{133 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{6318 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{5 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{2106 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{266 \sqrt{x^{4} + 5 x^{2} + 3}}{1053 x} - \frac{8 x^{2} + 7}{39 x^{3} \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{5 \sqrt{x^{4} + 5 x^{2} + 3}}{351 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-133*x*(2*x**2 + sqrt(13) + 5)/(1053*sqrt(x**4 + 5*x**2 + 3)) + 133*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*s
qrt(13)/6)/(6318*sqrt(x**4 + 5*x**2 + 3)) - 5*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)/(2106*sqrt(sqrt(13) + 5)*sqrt(
x**4 + 5*x**2 + 3)) + 266*sqrt(x**4 + 5*x**2 + 3)/(1053*x) - (8*x**2 + 7)/(39*x*
*3*sqrt(x**4 + 5*x**2 + 3)) - 5*sqrt(x**4 + 5*x**2 + 3)/(351*x**3)

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Mathematica [C]  time = 0.551696, size = 234, normalized size = 0.72 \[ \frac{532 x^6+2630 x^4+1014 x^2+i \sqrt{2} \left (133 \sqrt{13}-650\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-133 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^3 E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-468}{2106 x^3 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-468 + 1014*x^2 + 2630*x^4 + 532*x^6 - (133*I)*Sqrt[2]*(-5 + Sqrt[13])*x^3*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]*(-650 + 1
33*Sqrt[13])*x^3*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])
/(2106*x^3*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.031, size = 274, normalized size = 0.8 \[ -{\frac{2}{27\,{x}^{3}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{23}{81\,x}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-4\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ( -{\frac{40\,{x}^{3}}{351}}-{\frac{343\,x}{702}} \right ) }-{\frac{10}{117\,\sqrt{-30+6\,\sqrt{13}}}\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+{\frac{1064}{117\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}-6\,{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}} \left ({\frac{19\,{x}^{3}}{234}}+{\frac{40\,x}{117}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(x^4+5*x^2+3)^(1/2)/x^3+23/81*(x^4+5*x^2+3)^(1/2)/x-4*(-40/351*x^3-343/702
*x)/(x^4+5*x^2+3)^(1/2)-10/117/(-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))+1064/117/(-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*(19/23
4*x^3+40/117*x)/(x^4+5*x^2+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((3*x^2 + 2)/((x^8 + 5*x^6 + 3*x^4)*sqrt(x^4 + 5*x^2 + 3)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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