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

Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-5 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4-5 x^2+2}} \]

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - 5*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(3/2)^(1/4)*x], (12 + 5*Sqrt[6])/24])/(2*6^(1/4)*Sqrt[2 - 5*x^2 + 3*x^4])

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Rubi [A]  time = 0.0526538, antiderivative size = 92, 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 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4-5 x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12+5 \sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4-5 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

((2 + Sqrt[6]*x^2)*Sqrt[(2 - 5*x^2 + 3*x^4)/(2 + Sqrt[6]*x^2)^2]*EllipticF[2*Arc
Tan[(3/2)^(1/4)*x], (12 + 5*Sqrt[6])/24])/(2*6^(1/4)*Sqrt[2 - 5*x^2 + 3*x^4])

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Rubi in Sympy [A]  time = 4.0521, size = 90, normalized size = 0.98 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{3 x^{4} - 5 x^{2} + 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | \frac{1}{2} + \frac{5 \sqrt{6}}{24}\right )}{12 \sqrt{3 x^{4} - 5 x^{2} + 2}} \]

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]

6**(3/4)*sqrt((3*x**4 - 5*x**2 + 2)/(sqrt(6)*x**2/2 + 1)**2)*(sqrt(6)*x**2/2 + 1
)*elliptic_f(2*atan(2**(3/4)*3**(1/4)*x/2), 1/2 + 5*sqrt(6)/24)/(12*sqrt(3*x**4
- 5*x**2 + 2))

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Mathematica [A]  time = 0.0414906, size = 53, normalized size = 0.58 \[ \frac{\sqrt{2-3 x^2} \sqrt{1-x^2} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|\frac{2}{3}\right )}{\sqrt{9 x^4-15 x^2+6}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 - 3*x^2]*Sqrt[1 - x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2/3])/Sqrt[6 - 15*
x^2 + 9*x^4]

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Maple [A]  time = 0.012, size = 42, normalized size = 0.5 \[{\frac{1}{2}\sqrt{-{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\it EllipticF} \left ( x,{\frac{\sqrt{6}}{2}} \right ){\frac{1}{\sqrt{3\,{x}^{4}-5\,{x}^{2}+2}}}} \]

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

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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)

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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)

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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)

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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)