Optimal. Leaf size=47 \[ -\frac{\left (2-3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0337604, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (2-3 x^2\right )^{3/4}}{2 x}-\frac{\sqrt{3} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(2 - 3*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 3.44161, size = 39, normalized size = 0.83 \[ - \frac{\sqrt [4]{2} \sqrt{3} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{6} x}{2} \right )}}{2}\middle | 2\right )}{2} - \frac{\left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(-3*x**2+2)**(1/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0283924, size = 46, normalized size = 0.98 \[ -\frac{3 x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )}{4 \sqrt [4]{2}}-\frac{\left (2-3 x^2\right )^{3/4}}{2 x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(2 - 3*x^2)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.041, size = 40, normalized size = 0.9 \[{\frac{3\,{x}^{2}-2}{2\,x}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}-{\frac{3\,{2}^{3/4}x}{8}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(-3*x^2+2)^(1/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-3*x^2 + 2)^(1/4)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-3*x^2 + 2)^(1/4)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.38581, size = 31, normalized size = 0.66 \[ - \frac{2^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(-3*x**2+2)**(1/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-3*x^2 + 2)^(1/4)*x^2),x, algorithm="giac")
[Out]