Optimal. Leaf size=27 \[ \frac {2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{\sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {232} \[ \frac {2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 232
Rubi steps
\begin {align*} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx &=\frac {2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 27, normalized size = 1.00 \[ \frac {2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x^{2} - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.29, size = 18, normalized size = 0.67 \[ \frac {2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.76, size = 16, normalized size = 0.59 \[ \frac {2^{1/4}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {3}{2};\ \frac {3\,x^2}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.69, size = 27, normalized size = 1.00 \[ \frac {\sqrt [4]{2} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________