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

Optimal. Leaf size=90 \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]

[Out]

((1 + Sqrt[2]*x^2)*Sqrt[(2 + 5*x^2 + 4*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*Arc
Tan[2^(1/4)*x], (8 - 5*Sqrt[2])/16])/(2*2^(3/4)*Sqrt[2 + 5*x^2 + 4*x^4])

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Rubi [A]  time = 0.0527348, antiderivative size = 90, 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 \[ \frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{4 x^4+5 x^2+2}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{16} \left (8-5 \sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{4 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + Sqrt[2]*x^2)*Sqrt[(2 + 5*x^2 + 4*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*Arc
Tan[2^(1/4)*x], (8 - 5*Sqrt[2])/16])/(2*2^(3/4)*Sqrt[2 + 5*x^2 + 4*x^4])

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Rubi in Sympy [A]  time = 3.74721, size = 80, normalized size = 0.89 \[ \frac{\sqrt [4]{2} \sqrt{\frac{4 x^{4} + 5 x^{2} + 2}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{5 \sqrt{2}}{16} + \frac{1}{2}\right )}{4 \sqrt{4 x^{4} + 5 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/4)*sqrt((4*x**4 + 5*x**2 + 2)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)*el
liptic_f(2*atan(2**(1/4)*x), -5*sqrt(2)/16 + 1/2)/(4*sqrt(4*x**4 + 5*x**2 + 2))

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Mathematica [C]  time = 0.154818, size = 147, normalized size = 1.63 \[ -\frac{i \sqrt{1-\frac{8 x^2}{-5-i \sqrt{7}}} \sqrt{1-\frac{8 x^2}{-5+i \sqrt{7}}} F\left (i \sinh ^{-1}\left (2 \sqrt{-\frac{2}{-5-i \sqrt{7}}} x\right )|\frac{-5-i \sqrt{7}}{-5+i \sqrt{7}}\right )}{2 \sqrt{2} \sqrt{-\frac{1}{-5-i \sqrt{7}}} \sqrt{4 x^4+5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I/2)*Sqrt[1 - (8*x^2)/(-5 - I*Sqrt[7])]*Sqrt[1 - (8*x^2)/(-5 + I*Sqrt[7])]*El
lipticF[I*ArcSinh[2*Sqrt[-2/(-5 - I*Sqrt[7])]*x], (-5 - I*Sqrt[7])/(-5 + I*Sqrt[
7])])/(Sqrt[2]*Sqrt[-(-5 - I*Sqrt[7])^(-1)]*Sqrt[2 + 5*x^2 + 4*x^4])

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Maple [C]  time = 0.161, size = 87, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( -5/4+i/4\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -5/4-i/4\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-5+i\sqrt{7}},1/4\,\sqrt{9+5\,i\sqrt{7}} \right ) }{\sqrt{-5+i\sqrt{7}}\sqrt{4\,{x}^{4}+5\,{x}^{2}+2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^4+5*x^2+2)^(1/2),x)

[Out]

2/(-5+I*7^(1/2))^(1/2)*(1-(-5/4+1/4*I*7^(1/2))*x^2)^(1/2)*(1-(-5/4-1/4*I*7^(1/2)
)*x^2)^(1/2)/(4*x^4+5*x^2+2)^(1/2)*EllipticF(1/2*x*(-5+I*7^(1/2))^(1/2),1/4*(9+5
*I*7^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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