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

Optimal. Leaf size=10 \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-6\right ) \]

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -6]

_______________________________________________________________________________________

Rubi [A]  time = 0.0408244, antiderivative size = 10, 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 \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-6\right ) \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -6]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.77856, size = 12, normalized size = 1.2 \[ F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -6\right ) \]

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]

elliptic_f(asin(sqrt(2)*x/2), -6)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0401972, size = 65, normalized size = 6.5 \[ -\frac{i \sqrt{1-\frac{x^2}{2}} \sqrt{3 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{3} x\right )|-\frac{1}{6}\right )}{\sqrt{3} \sqrt{-3 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*Sqrt[1 - x^2/2]*Sqrt[1 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3]*x], -1/6])/(Sq
rt[3]*Sqrt[2 + 5*x^2 - 3*x^4])

_______________________________________________________________________________________

Maple [B]  time = 0., size = 51, normalized size = 5.1 \[{\frac{\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{3\,{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{6} \right ){\frac{1}{\sqrt{-3\,{x}^{4}+5\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^4+5*x^2+2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-3 \, x^{4} + 5 \, x^{2} + 2}}, x\right ) \]

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(1/sqrt(-3*x^4 + 5*x^2 + 2), x)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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)