Optimal. Leaf size=83 \[ -\frac{2}{39} \left (2-3 x^2\right )^{3/4} x^5-\frac{40 \left (2-3 x^2\right )^{3/4} x^3}{1053}-\frac{32 \left (2-3 x^2\right )^{3/4} x}{1053}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]
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Rubi [A] time = 0.0224705, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {321, 228} \[ -\frac{2}{39} \left (2-3 x^2\right )^{3/4} x^5-\frac{40 \left (2-3 x^2\right )^{3/4} x^3}{1053}-\frac{32 \left (2-3 x^2\right )^{3/4} x}{1053}+\frac{128 \sqrt [4]{2} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 228
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{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} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{1053 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0244062, size = 54, normalized size = 0.65 \[ -\frac{2 x \left (\left (2-3 x^2\right )^{3/4} \left (27 x^4+20 x^2+16\right )-16\ 2^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{3 x^2}{2}\right )\right )}{1053} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.031, size = 50, normalized size = 0.6 \begin{align*}{\frac{2\,x \left ( 27\,{x}^{4}+20\,{x}^{2}+16 \right ) \left ( 3\,{x}^{2}-2 \right ) }{1053}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}+{\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.
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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.
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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^{6}}{3 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.803028, size = 29, normalized size = 0.35 \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^{2 i \pi }}{2}} \right )}}{14} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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