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

Optimal. Leaf size=58 \[ \frac{\sqrt{\frac{x^2+2}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{3}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]

[Out]

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

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Rubi [A]  time = 0.0245325, antiderivative size = 58, 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 \[ \frac{\sqrt{\frac{x^2+2}{2 x^2+1}} \left (2 x^2+1\right ) F\left (\tan ^{-1}\left (\sqrt{2} x\right )|\frac{3}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.4931, size = 56, normalized size = 0.97 \[ \frac{\sqrt{2} \sqrt{\frac{2 x^{2} + 4}{8 x^{2} + 4}} \left (8 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{2} x \right )}\middle | \frac{3}{4}\right )}{8 \sqrt{2 x^{4} + 5 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.036071, size = 58, normalized size = 1. \[ -\frac{i \sqrt{x^2+2} \sqrt{2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{2} x\right )|\frac{1}{4}\right )}{2 \sqrt{2 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.081, size = 48, normalized size = 0.8 \[{-{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,2 \right ) \sqrt{2\,{x}^{2}+4}\sqrt{2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+5\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4+5*x^2+2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(2*x^4 + 5*x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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