Optimal. Leaf size=83 \[ \frac{320\ 2^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{2079 \sqrt{3}}-\frac{2}{33} \sqrt [4]{2-3 x^2} x^5-\frac{40}{693} \sqrt [4]{2-3 x^2} x^3-\frac{160 \sqrt [4]{2-3 x^2} x}{2079} \]
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Rubi [A] time = 0.0226623, 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, 232} \[ -\frac{2}{33} \sqrt [4]{2-3 x^2} x^5-\frac{40}{693} \sqrt [4]{2-3 x^2} x^3-\frac{160 \sqrt [4]{2-3 x^2} x}{2079}+\frac{320\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2079 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 232
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{320\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{2079 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0353716, size = 59, normalized size = 0.71 \[ \frac{320\ 2^{3/4} \sqrt{3} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )-6 x \sqrt [4]{2-3 x^2} \left (63 x^4+60 x^2+80\right )}{6237} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{{x}^{6} \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \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{{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{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 = 0.80729, size = 29, normalized size = 0.35 \begin{align*} \frac{\sqrt [4]{2} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{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{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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