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

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

[Out]

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

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Rubi [A]  time = 0.0325061, 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 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{2}\right )}{2 \sqrt [4]{6} \sqrt{2 x^4+3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[3 + 2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.44617, size = 71, normalized size = 0.99 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 3}{\left (\frac{\sqrt{6} x^{2}}{3} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{3} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{2 x^{4} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0425801, size = 25, normalized size = 0.35 \[ -\sqrt [4]{-\frac{1}{6}} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{2}{3}} x\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[3 + 2*x^4],x]

[Out]

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

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Maple [C]  time = 0.063, size = 66, normalized size = 0.9 \[{\frac{\sqrt{3}}{9\,\sqrt{i\sqrt{6}}}\sqrt{9-3\,i\sqrt{6}{x}^{2}}\sqrt{9+3\,i\sqrt{6}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{3}\sqrt{i\sqrt{6}}}{3}},i \right ){\frac{1}{\sqrt{2\,{x}^{4}+3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*x^4+3)^(1/2),x)

[Out]

1/9*3^(1/2)/(I*6^(1/2))^(1/2)*(9-3*I*6^(1/2)*x^2)^(1/2)*(9+3*I*6^(1/2)*x^2)^(1/2
)/(2*x^4+3)^(1/2)*EllipticF(1/3*x*3^(1/2)*(I*6^(1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(2*x^4 + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(2*x^4 + 3), x)

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Sympy [A]  time = 1.76405, size = 36, normalized size = 0.5 \[ \frac{\sqrt{3} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{2 x^{4} e^{i \pi }}{3}} \right )}}{12 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), 2*x**4*exp_polar(I*pi)/3)/(12*gam
ma(5/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(2*x^4 + 3), x)