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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]