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

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

[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)*S
qrt[2 + 3*x^4])

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Rubi [A]  time = 0.0072882, 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, Rules used = {220} \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{2}\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+2}} \]

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)*S
qrt[2 + 3*x^4])

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+3 x^4}} \, dx &=\frac{\left (2+\sqrt{6} x^2\right ) \sqrt{\frac{2+3 x^4}{\left (2+\sqrt{6} x^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{2+3 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.025312, size = 25, normalized size = 0.35 \[ -\sqrt [4]{-\frac{1}{6}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt [4]{-\frac{3}{2}} x\right ),-1\right ) \]

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 = 0.196, size = 66, normalized size = 0.9 \begin{align*}{\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}}}} \end{align*}

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. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^4+2)^(1/2),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. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/sqrt(3*x^4 + 2), x)

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Sympy [C]  time = 0.684426, size = 36, normalized size = 0.5 \begin{align*} \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 )} \end{align*}

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*gamma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(3*x^4 + 2), x)