Optimal. Leaf size=138 \[ \frac{160\ 2^{3/4} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{2079 \sqrt{3} x}+\frac{2}{33} \sqrt [4]{3 x^2-2} x^5+\frac{40}{693} \sqrt [4]{3 x^2-2} x^3+\frac{160 \sqrt [4]{3 x^2-2} x}{2079} \]
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Rubi [A] time = 0.0568948, antiderivative size = 138, 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, 234, 220} \[ \frac{2}{33} \sqrt [4]{3 x^2-2} x^5+\frac{40}{693} \sqrt [4]{3 x^2-2} x^3+\frac{160 \sqrt [4]{3 x^2-2} x}{2079}+\frac{160\ 2^{3/4} \sqrt{\frac{x^2}{\left (\sqrt{3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{3 x^2-2}+\sqrt{2}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac{1}{2}\right )}{2079 \sqrt{3} x} \]
Antiderivative was successfully verified.
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Rule 321
Rule 234
Rule 220
Rubi steps
\begin{align*} \int \frac{x^6}{\left (-2+3 x^2\right )^{3/4}} \, dx &=\frac{2}{33} x^5 \sqrt [4]{-2+3 x^2}+\frac{20}{33} \int \frac{x^4}{\left (-2+3 x^2\right )^{3/4}} \, dx\\ &=\frac{40}{693} x^3 \sqrt [4]{-2+3 x^2}+\frac{2}{33} x^5 \sqrt [4]{-2+3 x^2}+\frac{80}{231} \int \frac{x^2}{\left (-2+3 x^2\right )^{3/4}} \, dx\\ &=\frac{160 x \sqrt [4]{-2+3 x^2}}{2079}+\frac{40}{693} x^3 \sqrt [4]{-2+3 x^2}+\frac{2}{33} x^5 \sqrt [4]{-2+3 x^2}+\frac{320 \int \frac{1}{\left (-2+3 x^2\right )^{3/4}} \, dx}{2079}\\ &=\frac{160 x \sqrt [4]{-2+3 x^2}}{2079}+\frac{40}{693} x^3 \sqrt [4]{-2+3 x^2}+\frac{2}{33} x^5 \sqrt [4]{-2+3 x^2}+\frac{\left (320 \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 )}{2079 x}\\ &=\frac{160 x \sqrt [4]{-2+3 x^2}}{2079}+\frac{40}{693} x^3 \sqrt [4]{-2+3 x^2}+\frac{2}{33} x^5 \sqrt [4]{-2+3 x^2}+\frac{160\ 2^{3/4} \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 )}{2079 \sqrt{3} x}\\ \end{align*}
Mathematica [C] time = 0.0186276, size = 68, normalized size = 0.49 \[ \frac{2 x \left (80 \sqrt [4]{2} \left (2-3 x^2\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{3 x^2}{2}\right )+189 x^6+54 x^4+120 x^2-160\right )}{2079 \left (3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.042, size = 65, normalized size = 0.5 \begin{align*}{\frac{2\,x \left ( 63\,{x}^{4}+60\,{x}^{2}+80 \right ) }{2079}\sqrt [4]{3\,{x}^{2}-2}}+{\frac{160\,\sqrt [4]{2}x}{2079} \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.
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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{3}{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{x^{6}}{{\left (3 \, x^{2} - 2\right )}^{\frac{3}{4}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.787967, size = 31, normalized size = 0.22 \begin{align*} \frac{\sqrt [4]{2} x^{7} e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{3 x^{2}}{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{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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