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

Optimal. Leaf size=53 \[ \frac{\sqrt{2 x^2+3} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+3}{x^2+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0547891, antiderivative size = 53, 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 \[ \frac{\sqrt{2 x^2+3} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+3}{x^2+1}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.39057, size = 54, normalized size = 1.02 \[ \frac{4 \sqrt{3} \sqrt{4 x^{2} + 6} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{3}\right )}{3 \sqrt{- \frac{- 16 x^{2} - 24}{x^{2} + 1}} \sqrt{- 4 x^{2} - 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(3)*sqrt(4*x**2 + 6)*elliptic_f(atan(x), 1/3)/(3*sqrt(-(-16*x**2 - 24)/(x*
*2 + 1))*sqrt(-4*x**2 - 4))

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Mathematica [C]  time = 0.0417511, size = 63, normalized size = 1.19 \[ -\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|\frac{3}{2}\right )}{\sqrt{2} \sqrt{-2 x^4-5 x^2-3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.049, size = 44, normalized size = 0.8 \[{-{\frac{i}{3}}\sqrt{{x}^{2}+1}\sqrt{6\,{x}^{2}+9}{\it EllipticF} \left ( ix,{\frac{\sqrt{6}}{3}} \right ){\frac{1}{\sqrt{-2\,{x}^{4}-5\,{x}^{2}-3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4-5*x^2-3)^(1/2),x)

[Out]

-1/3*I*(x^2+1)^(1/2)*(6*x^2+9)^(1/2)/(-2*x^4-5*x^2-3)^(1/2)*EllipticF(I*x,1/3*6^
(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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