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

Optimal. Leaf size=52 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]

[Out]

((1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt
[2 + 5*x^2 + 3*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.0237831, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{3 x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(Sqrt[2]*Sqrt
[2 + 5*x^2 + 3*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.65851, size = 46, normalized size = 0.88 \[ \frac{\sqrt{\frac{6 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{4} + 5 x^{2} + 2}} \]

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]

sqrt((6*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), -1/2)/(8*sqrt(3*x
**4 + 5*x**2 + 2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.034497, size = 58, normalized size = 1.12 \[ -\frac{i \sqrt{x^2+1} \sqrt{3 x^2+2} F\left (i \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{2}{3}\right )}{\sqrt{9 x^4+15 x^2+6}} \]

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])/Sqrt
[6 + 15*x^2 + 9*x^4]

_______________________________________________________________________________________

Maple [A]  time = 0., size = 44, normalized size = 0.9 \[{-{\frac{i}{2}}\sqrt{{x}^{2}+1}\sqrt{6\,{x}^{2}+4}{\it EllipticF} \left ( ix,{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{3\,{x}^{4}+5\,{x}^{2}+2}}}} \]

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