Optimal. Leaf size=47 \[ \frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}}-\frac{2}{15} x \left (2-3 x^2\right )^{3/4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0083717, 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, Rules used = {321, 228} \[ \frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}}-\frac{2}{15} x \left (2-3 x^2\right )^{3/4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 321
Rule 228
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt [4]{2-3 x^2}} \, dx &=-\frac{2}{15} x \left (2-3 x^2\right )^{3/4}+\frac{4}{15} \int \frac{1}{\sqrt [4]{2-3 x^2}} \, dx\\ &=-\frac{2}{15} x \left (2-3 x^2\right )^{3/4}+\frac{8 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{15 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0080976, size = 41, normalized size = 0.87 \[ -\frac{2}{15} x \left (\left (2-3 x^2\right )^{3/4}-2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.026, size = 38, normalized size = 0.8 \begin{align*}{\frac{2\,x \left ( 3\,{x}^{2}-2 \right ) }{15}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}+{\frac{2\,{2}^{3/4}x}{15}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,{\frac{3\,{x}^{2}}{2}})}} \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{x^{2}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{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{{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{2}}{3 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.610562, size = 29, normalized size = 0.62 \begin{align*} \frac{2^{\frac{3}{4}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{2 i \pi }}{2}} \right )}}{6} \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{x^{2}}{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]