Optimal. Leaf size=99 \[ \frac{2}{39} \left (3 x^2+2\right )^{3/4} x^5-\frac{40 \left (3 x^2+2\right )^{3/4} x^3}{1053}+\frac{32 \left (3 x^2+2\right )^{3/4} x}{1053}-\frac{128 x}{1053 \sqrt [4]{3 x^2+2}}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0282477, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {321, 227, 196} \[ \frac{2}{39} \left (3 x^2+2\right )^{3/4} x^5-\frac{40 \left (3 x^2+2\right )^{3/4} x^3}{1053}+\frac{32 \left (3 x^2+2\right )^{3/4} x}{1053}-\frac{128 x}{1053 \sqrt [4]{3 x^2+2}}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 321
Rule 227
Rule 196
Rubi steps
\begin{align*} \int \frac{x^6}{\sqrt [4]{2+3 x^2}} \, dx &=\frac{2}{39} x^5 \left (2+3 x^2\right )^{3/4}-\frac{20}{39} \int \frac{x^4}{\sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac{40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac{2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac{80}{351} \int \frac{x^2}{\sqrt [4]{2+3 x^2}} \, dx\\ &=\frac{32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac{40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac{2}{39} x^5 \left (2+3 x^2\right )^{3/4}-\frac{64 \int \frac{1}{\sqrt [4]{2+3 x^2}} \, dx}{1053}\\ &=-\frac{128 x}{1053 \sqrt [4]{2+3 x^2}}+\frac{32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac{40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac{2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac{128 \int \frac{1}{\left (2+3 x^2\right )^{5/4}} \, dx}{1053}\\ &=-\frac{128 x}{1053 \sqrt [4]{2+3 x^2}}+\frac{32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac{40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac{2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.026938, size = 54, normalized size = 0.55 \[ \frac{2 x \left (\left (3 x^2+2\right )^{3/4} \left (27 x^4-20 x^2+16\right )-16\ 2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{3 x^2}{2}\right )\right )}{1053} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.028, size = 43, normalized size = 0.4 \begin{align*}{\frac{2\,x \left ( 27\,{x}^{4}-20\,{x}^{2}+16 \right ) }{1053} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{4}}}}-{\frac{32\,{2}^{3/4}x}{1053}{\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^{6}}{{\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{x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac{1}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.786577, size = 27, normalized size = 0.27 \begin{align*} \frac{2^{\frac{3}{4}} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{14} \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^{6}}{{\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]