3.119 \(\int \frac{1}{\sqrt{-2-5 x^2-3 x^4}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{\sqrt{-3 x^2-2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]

[Out]

-((Sqrt[-2 - 3*x^2]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 +
3*x^2)/(1 + x^2)]))

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Rubi [A]  time = 0.0547113, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\sqrt{-3 x^2-2} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{3 x^2+2}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 - 5*x^2 - 3*x^4],x]

[Out]

-((Sqrt[-2 - 3*x^2]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt[1 + x^2]*Sqrt[(2 +
3*x^2)/(1 + x^2)]))

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Rubi in Sympy [A]  time = 8.2786, size = 56, normalized size = 1.08 \[ - \frac{3 \sqrt{2} \sqrt{- 6 x^{2} - 4} F\left (\operatorname{atan}{\left (x \right )}\middle | - \frac{1}{2}\right )}{\sqrt{- \frac{- 36 x^{2} - 24}{x^{2} + 1}} \sqrt{6 x^{2} + 6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(2)*sqrt(-6*x**2 - 4)*elliptic_f(atan(x), -1/2)/(sqrt(-(-36*x**2 - 24)/(x
**2 + 1))*sqrt(6*x**2 + 6))

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Mathematica [C]  time = 0.0429724, size = 63, normalized size = 1.21 \[ -\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{2}{3}\right )}{\sqrt{3} \sqrt{-3 x^4-5 x^2-2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-2 - 5*x^2 - 3*x^4],x]

[Out]

((-I)*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3])/(Sqr
t[3]*Sqrt[-2 - 5*x^2 - 3*x^4])

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Maple [A]  time = 0.051, size = 50, normalized size = 1. \[{-{\frac{i}{6}}\sqrt{6}\sqrt{6\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{6},{\frac{\sqrt{6}}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{4}-5\,{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4-5*x^2-2)^(1/2),x)

[Out]

-1/6*I*6^(1/2)*(6*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-3*x^4-5*x^2-2)^(1/2)*EllipticF(1/
2*I*x*6^(1/2),1/3*6^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, x^{4} - 5 \, x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 - 5*x^2 - 2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-3*x^4 - 5*x^2 - 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 - 5*x^2 - 2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 3 x^{4} - 5 x^{2} - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(-3*x**4 - 5*x**2 - 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-3 \, x^{4} - 5 \, x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^4 - 5*x^2 - 2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-3*x^4 - 5*x^2 - 2), x)