Optimal. Leaf size=139 \[ \frac{160\ 2^{3/4} \sqrt{-\frac{x^2}{\left (\sqrt{-3 x^2-2}+\sqrt{2}\right )^2}} \left (\sqrt{-3 x^2-2}+\sqrt{2}\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac{1}{2}\right )}{2079 \sqrt{3} x}-\frac{2}{33} \sqrt [4]{-3 x^2-2} x^5+\frac{40}{693} \sqrt [4]{-3 x^2-2} x^3-\frac{160 \sqrt [4]{-3 x^2-2} x}{2079} \]
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Rubi [A] time = 0.0575328, antiderivative size = 139, 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 = 0.0195166, size = 68, normalized size = 0.49 \[ \frac{2 x \left (-80 \sqrt [4]{2} \left (3 x^2+2\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{3 x^2}{2}\right )+189 x^6-54 x^4+120 x^2+160\right )}{2079 \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [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*} -\frac{2}{2079} \,{\left (63 \, x^{5} - 60 \, x^{3} + 80 \, x\right )}{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}} +{\rm integral}\left (\frac{320 \,{\left (-3 \, x^{2} - 2\right )}^{\frac{1}{4}}}{2079 \,{\left (3 \, x^{2} + 2\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.802413, size = 36, normalized size = 0.26 \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} e^{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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