Optimal. Leaf size=82 \[ \frac {\sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{\sqrt [4]{2} \sqrt {3} x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {234, 220} \[ \frac {\sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\sqrt [4]{2} \sqrt {3} x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 234
Rubi steps
\begin {align*} \int \frac {1}{\left (-2+3 x^2\right )^{3/4}} \, dx &=\frac {\left (\sqrt {\frac {2}{3}} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{x}\\ &=\frac {\sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{\sqrt [4]{2} \sqrt {3} x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 43, normalized size = 0.52 \[ \frac {x \left (1-\frac {3 x^2}{2}\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\frac {3 x^2}{2}\right )}{\left (3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}, x\right ) \]
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 = 40, normalized size = 0.49 \[ \frac {2^{\frac {1}{4}} \left (-\mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )\right )^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{2 \mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )^{\frac {3}{4}}} \]
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 = 4.87, size = 34, normalized size = 0.41 \[ \frac {2^{1/4}\,x\,{\left (2-3\,x^2\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {3}{2};\ \frac {3\,x^2}{2}\right )}{2\,{\left (3\,x^2-2\right )}^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.71, size = 29, normalized size = 0.35 \[ \frac {\sqrt [4]{2} x e^{- \frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________