Optimal. Leaf size=82 \[ \frac{\sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{\sqrt [4]{2} \sqrt{3} x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0259494, antiderivative size = 82, normalized size of antiderivative = 1., 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 234
Rule 220
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 = 0.0060872, 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]
________________________________________________________________________________________
Maple [C] time = 0.028, size = 40, normalized size = 0.5 \begin{align*}{\frac{\sqrt [4]{2}x}{2} \left ( -{\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})} \left ({\it signum} \left ( -1+{\frac{3\,{x}^{2}}{2}} \right ) \right ) ^{-{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.610121, size = 29, normalized size = 0.35 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]