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

Optimal. Leaf size=67 \[ \frac{\sqrt{x^2+2} \sqrt{3 x^2-1} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}} x}{\sqrt{3 x^2-1}}\right )|\frac{6}{7}\right )}{\sqrt{7} \sqrt{3 x^4+5 x^2-2}} \]

[Out]

(Sqrt[2 + x^2]*Sqrt[-1 + 3*x^2]*EllipticF[ArcSin[(Sqrt[7/2]*x)/Sqrt[-1 + 3*x^2]]
, 6/7])/(Sqrt[7]*Sqrt[-2 + 5*x^2 + 3*x^4])

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Rubi [A]  time = 0.028874, antiderivative size = 67, 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 \[ \frac{\sqrt{x^2+2} \sqrt{3 x^2-1} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{7}{2}} x}{\sqrt{3 x^2-1}}\right )|\frac{6}{7}\right )}{\sqrt{7} \sqrt{3 x^4+5 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + x^2]*Sqrt[-1 + 3*x^2]*EllipticF[ArcSin[(Sqrt[7/2]*x)/Sqrt[-1 + 3*x^2]]
, 6/7])/(Sqrt[7]*Sqrt[-2 + 5*x^2 + 3*x^4])

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Rubi in Sympy [A]  time = 3.84405, size = 70, normalized size = 1.04 \[ \frac{\sqrt{2} \sqrt{\frac{12 x^{2}}{7} - \frac{4}{7}} \sqrt{2 x^{2} + 4} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{12 x^{2}}{7} - \frac{4}{7}}} \right )}\middle | \frac{6}{7}\right )}{4 \sqrt{3 x^{4} + 5 x^{2} - 2}} \]

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(2)*sqrt(12*x**2/7 - 4/7)*sqrt(2*x**2 + 4)*elliptic_f(asin(sqrt(2)*x/sqrt(12
*x**2/7 - 4/7)), 6/7)/(4*sqrt(3*x**4 + 5*x**2 - 2))

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Mathematica [A]  time = 0.0411476, size = 54, normalized size = 0.81 \[ \frac{\sqrt{1-3 x^2} \sqrt{x^2+2} F\left (\sin ^{-1}\left (\sqrt{3} x\right )|-\frac{1}{6}\right )}{\sqrt{6} \sqrt{3 x^4+5 x^2-2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.013, size = 53, normalized size = 0.8 \[{-{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,i\sqrt{6} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\frac{1}{\sqrt{3\,{x}^{4}+5\,{x}^{2}-2}}}} \]

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