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

Optimal. Leaf size=72 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2}} \]

[Out]

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

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Rubi [A]  time = 0.0339236, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 3*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.45892, size = 71, normalized size = 0.99 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{3 x^{4} + 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{3 x^{4} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6**(3/4)*sqrt((3*x**4 + 2)/(sqrt(6)*x**2/2 + 1)**2)*(sqrt(6)*x**2/2 + 1)*ellipti
c_f(2*atan(2**(3/4)*3**(1/4)*x/2), 1/2)/(12*sqrt(3*x**4 + 2))

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Mathematica [C]  time = 0.0411105, size = 25, normalized size = 0.35 \[ -\sqrt [4]{-\frac{1}{6}} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{3}{2}} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + 3*x^4],x]

[Out]

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

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Maple [C]  time = 0.08, size = 66, normalized size = 0.9 \[{\frac{\sqrt{2}}{4\,\sqrt{i\sqrt{6}}}\sqrt{4-2\,i\sqrt{6}{x}^{2}}\sqrt{4+2\,i\sqrt{6}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{2}\sqrt{i\sqrt{6}}}{2}},i \right ){\frac{1}{\sqrt{3\,{x}^{4}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^4+2)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(3*x^4 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(3*x^4 + 2), x)

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Sympy [A]  time = 1.74014, size = 36, normalized size = 0.5 \[ \frac{\sqrt{2} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{3 x^{4} e^{i \pi }}{2}} \right )}}{8 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 3*x**4*exp_polar(I*pi)/2)/(8*gamm
a(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(3*x^4 + 2), x)