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

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

[Out]

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

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Rubi [A]  time = 0.0222564, 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 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{2 x^2+3}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{3}\right )}{\sqrt{3} \sqrt{2 x^4+5 x^2+3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.63815, size = 49, normalized size = 0.94 \[ \frac{\sqrt{6} \sqrt{\frac{4 x^{2} + 6}{x^{2} + 1}} \left (6 x^{2} + 6\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{3}\right )}{36 \sqrt{2 x^{4} + 5 x^{2} + 3}} \]

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]

sqrt(6)*sqrt((4*x**2 + 6)/(x**2 + 1))*(6*x**2 + 6)*elliptic_f(atan(x), 1/3)/(36*
sqrt(2*x**4 + 5*x**2 + 3))

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Mathematica [C]  time = 0.0421946, size = 58, normalized size = 1.12 \[ -\frac{i \sqrt{x^2+1} \sqrt{2 x^2+3} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{3}} x\right )|\frac{3}{2}\right )}{\sqrt{4 x^4+10 x^2+6}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.049, size = 50, normalized size = 1. \[{-{\frac{i}{6}}\sqrt{6}\sqrt{6\,{x}^{2}+9}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{i}{3}}x\sqrt{6},{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+5\,{x}^{2}+3}}}} \]

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*I*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
*I*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)