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

Optimal. Leaf size=63 \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]

[Out]

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

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Rubi [A]  time = 0.0258684, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{x^2+1} \sqrt{3 x^2-2} F\left (\sin ^{-1}\left (\frac{\sqrt{5} x}{\sqrt{3 x^2-2}}\right )|\frac{3}{5}\right )}{\sqrt{5} \sqrt{3 x^4+x^2-2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 + x^2 + 3*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.97203, size = 68, normalized size = 1.08 \[ \frac{\sqrt{2} \sqrt{\frac{6 x^{2}}{5} - \frac{4}{5}} \sqrt{4 x^{2} + 4} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{\sqrt{\frac{6 x^{2}}{5} - \frac{4}{5}}} \right )}\middle | \frac{3}{5}\right )}{4 \sqrt{3 x^{4} + x^{2} - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**4+x**2-2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0505583, size = 48, normalized size = 0.76 \[ \frac{\sqrt{\left (\frac{2}{3}-x^2\right ) \left (x^2+1\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3 x^4+x^2-2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [C]  time = 0.012, size = 43, normalized size = 0.7 \[{-{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{6} \right ) \sqrt{{x}^{2}+1}\sqrt{-6\,{x}^{2}+4}{\frac{1}{\sqrt{3\,{x}^{4}+{x}^{2}-2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+x^2-2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + x^2 - 2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} + x^{2} - 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + x^2 - 2),x, algorithm="fricas")

[Out]

integral(1/sqrt(3*x^4 + x^2 - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{4} + x^{2} - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**4+x**2-2)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**4 + x**2 - 2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + x^{2} - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^4 + x^2 - 2),x, algorithm="giac")

[Out]

integrate(1/sqrt(3*x^4 + x^2 - 2), x)