3.168 \(\int \frac{\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=331 \[ -\frac{722 \sqrt{x^4+5 x^2+3}}{15 x}+\frac{361 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}+\frac{103 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{361 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}}-\frac{\left (2-5 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{5 x^5}-\frac{\left (40-87 x^2\right ) \sqrt{x^4+5 x^2+3}}{5 x^3} \]

[Out]

(361*x*(5 + Sqrt[13] + 2*x^2))/(15*Sqrt[3 + 5*x^2 + x^4]) - (722*Sqrt[3 + 5*x^2
+ x^4])/(15*x) - ((40 - 87*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5*x^3) - ((2 - 5*x^2)*(3
+ 5*x^2 + x^4)^(3/2))/(5*x^5) - (361*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[S
qrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(15*Sqrt[3 + 5*x^2 + x^4]) + (1
03*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[6*(
5 + Sqrt[13])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi [A]  time = 0.493636, antiderivative size = 331, 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 \[ -\frac{722 \sqrt{x^4+5 x^2+3}}{15 x}+\frac{361 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}+\frac{103 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{361 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}}-\frac{\left (2-5 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{5 x^5}-\frac{\left (40-87 x^2\right ) \sqrt{x^4+5 x^2+3}}{5 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^6,x]

[Out]

(361*x*(5 + Sqrt[13] + 2*x^2))/(15*Sqrt[3 + 5*x^2 + x^4]) - (722*Sqrt[3 + 5*x^2
+ x^4])/(15*x) - ((40 - 87*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5*x^3) - ((2 - 5*x^2)*(3
+ 5*x^2 + x^4)^(3/2))/(5*x^5) - (361*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[S
qrt[(5 + Sqrt[13])/6]*x], (-13 + 5*Sqrt[13])/6])/(15*Sqrt[3 + 5*x^2 + x^4]) + (1
03*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[6*(
5 + Sqrt[13])]*Sqrt[3 + 5*x^2 + x^4])

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Rubi in Sympy [A]  time = 39.8939, size = 306, normalized size = 0.92 \[ \frac{361 x \left (2 x^{2} + \sqrt{13} + 5\right )}{15 \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{361 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \sqrt{\sqrt{13} + 5} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{90 \sqrt{x^{4} + 5 x^{2} + 3}} + \frac{103 \sqrt{6} \sqrt{\frac{x^{2} \left (- \sqrt{13} + 5\right ) + 6}{x^{2} \left (\sqrt{13} + 5\right ) + 6}} \left (x^{2} \left (\sqrt{13} + 5\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{6} x \sqrt{\sqrt{13} + 5}}{6} \right )}\middle | - \frac{13}{6} + \frac{5 \sqrt{13}}{6}\right )}{6 \sqrt{\sqrt{13} + 5} \sqrt{x^{4} + 5 x^{2} + 3}} - \frac{722 \sqrt{x^{4} + 5 x^{2} + 3}}{15 x} - \frac{\left (- 261 x^{2} + 120\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{15 x^{3}} - \frac{\left (- 15 x^{2} + 6\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)*(x**4+5*x**2+3)**(3/2)/x**6,x)

[Out]

361*x*(2*x**2 + sqrt(13) + 5)/(15*sqrt(x**4 + 5*x**2 + 3)) - 361*sqrt(6)*sqrt((x
**2*(-sqrt(13) + 5) + 6)/(x**2*(sqrt(13) + 5) + 6))*sqrt(sqrt(13) + 5)*(x**2*(sq
rt(13) + 5) + 6)*elliptic_e(atan(sqrt(6)*x*sqrt(sqrt(13) + 5)/6), -13/6 + 5*sqrt
(13)/6)/(90*sqrt(x**4 + 5*x**2 + 3)) + 103*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)/(6*sqrt(sqrt(13) + 5)*sqrt(x**4 +
5*x**2 + 3)) - 722*sqrt(x**4 + 5*x**2 + 3)/(15*x) - (-261*x**2 + 120)*sqrt(x**4
+ 5*x**2 + 3)/(15*x**3) - (-15*x**2 + 6)*(x**4 + 5*x**2 + 3)**(3/2)/(15*x**5)

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Mathematica [C]  time = 0.604923, size = 244, normalized size = 0.74 \[ \frac{30 x^{10}-634 x^8-4040 x^6-3438 x^4-810 x^2-i \sqrt{2} \left (361 \sqrt{13}-260\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+361 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-108}{30 x^5 \sqrt{x^4+5 x^2+3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((2 + 3*x^2)*(3 + 5*x^2 + x^4)^(3/2))/x^6,x]

[Out]

(-108 - 810*x^2 - 3438*x^4 - 4040*x^6 - 634*x^8 + 30*x^10 + (361*I)*Sqrt[2]*(-5
+ Sqrt[13])*x^5*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]*(-260 + 361*Sqrt[13])*x^5*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])/(30*x^5*Sqrt[3 + 5*x^2 + x^4])

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Maple [A]  time = 0.028, size = 259, normalized size = 0.8 \[ -{\frac{6}{5\,{x}^{5}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-7\,{\frac{\sqrt{{x}^{4}+5\,{x}^{2}+3}}{{x}^{3}}}-{\frac{392}{15\,x}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+618\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}-{\frac{8664}{5\,\sqrt{-30+6\,\sqrt{13}} \left ( 5+\sqrt{13} \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+x\sqrt{{x}^{4}+5\,{x}^{2}+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)*(x^4+5*x^2+3)^(3/2)/x^6,x)

[Out]

-6/5/x^5*(x^4+5*x^2+3)^(1/2)-7*(x^4+5*x^2+3)^(1/2)/x^3-392/15*(x^4+5*x^2+3)^(1/2
)/x+618/(-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))-8664/5/(-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))*(Elli
pticF(1/6*x*(-30+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2))-EllipticE(1/6*x*(-3
0+6*13^(1/2))^(1/2),5/6*3^(1/2)+1/6*39^(1/2)))+x*(x^4+5*x^2+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x^{6}}\,{d x} \]

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^6,x, algorithm="maxima")

[Out]

integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^6, x)

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

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^6,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

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**6,x)

[Out]

Integral((3*x**2 + 2)*(x**4 + 5*x**2 + 3)**(3/2)/x**6, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x^{6}}\,{d x} \]

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^6,x, algorithm="giac")

[Out]

integrate((x^4 + 5*x^2 + 3)^(3/2)*(3*x^2 + 2)/x^6, x)