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

Optimal. Leaf size=14 \[ -F\left (\left .\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right )\right |3\right ) \]

[Out]

-EllipticF[ArcCos[Sqrt[2/3]*x], 3]

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Rubi [A]  time = 0.0401121, antiderivative size = 14, 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 \[ -F\left (\left .\cos ^{-1}\left (\sqrt{\frac{2}{3}} x\right )\right |3\right ) \]

Antiderivative was successfully verified.

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

[Out]

-EllipticF[ArcCos[Sqrt[2/3]*x], 3]

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Rubi in Sympy [A]  time = 9.43334, size = 12, normalized size = 0.86 \[ - F\left (\operatorname{acos}{\left (\frac{\sqrt{6} x}{3} \right )}\middle | 3\right ) \]

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]

-elliptic_f(acos(sqrt(6)*x/3), 3)

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Mathematica [B]  time = 0.0395096, size = 53, normalized size = 3.79 \[ \frac{\sqrt{3-2 x^2} \sqrt{1-x^2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|\frac{3}{2}\right )}{\sqrt{-4 x^4+10 x^2-6}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 - 2*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[2/3]*x], 3/2])/Sqrt[-6 + 10
*x^2 - 4*x^4]

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Maple [A]  time = 0.048, size = 50, normalized size = 3.6 \[{\frac{\sqrt{6}}{6}\sqrt{-6\,{x}^{2}+9}\sqrt{-{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{6}}{3}},{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{-2\,{x}^{4}+5\,{x}^{2}-3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-2*x^4+5*x^2-3)^(1/2),x)

[Out]

1/6*6^(1/2)*(-6*x^2+9)^(1/2)*(-x^2+1)^(1/2)/(-2*x^4+5*x^2-3)^(1/2)*EllipticF(1/3
*x*6^(1/2),1/2*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. \[{\rm integral}\left (\frac{1}{\sqrt{-2 \, x^{4} + 5 \, x^{2} - 3}}, x\right ) \]

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(1/sqrt(-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)