Optimal. Leaf size=182 \[ -\frac{160\ 2^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{567 \sqrt{3}}+\frac{2}{63} \sqrt [4]{2-3 x^2} x^3+\frac{80}{567} \sqrt [4]{2-3 x^2} x+\frac{8\ 2^{3/4} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{8\ 2^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}} \]
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Rubi [A] time = 0.118129, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {443, 232, 321, 400, 441} \[ \frac{2}{63} \sqrt [4]{2-3 x^2} x^3+\frac{80}{567} \sqrt [4]{2-3 x^2} x+\frac{8\ 2^{3/4} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{8\ 2^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-3 x^2}+2^{3/4}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{160\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{567 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 443
Rule 232
Rule 321
Rule 400
Rule 441
Rubi steps
\begin{align*} \int \frac{x^6}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac{16}{27 \left (2-3 x^2\right )^{3/4}}-\frac{4 x^2}{9 \left (2-3 x^2\right )^{3/4}}-\frac{x^4}{3 \left (2-3 x^2\right )^{3/4}}+\frac{64}{27 \left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} \int \frac{x^4}{\left (2-3 x^2\right )^{3/4}} \, dx\right )-\frac{4}{9} \int \frac{x^2}{\left (2-3 x^2\right )^{3/4}} \, dx-\frac{16}{27} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac{64}{27} \int \frac{1}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac{8}{81} x \sqrt [4]{2-3 x^2}+\frac{2}{63} x^3 \sqrt [4]{2-3 x^2}-\frac{16\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{4}{21} \int \frac{x^2}{\left (2-3 x^2\right )^{3/4}} \, dx-\frac{16}{81} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac{16}{27} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx+\frac{16}{9} \int \frac{x^2}{\left (2-3 x^2\right )^{3/4} \left (4-3 x^2\right )} \, dx\\ &=\frac{80}{567} x \sqrt [4]{2-3 x^2}+\frac{2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac{8\ 2^{3/4} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{8\ 2^{3/4} \tanh ^{-1}\left (\frac{2^{3/4}+\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{16\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{81 \sqrt{3}}-\frac{16}{189} \int \frac{1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=\frac{80}{567} x \sqrt [4]{2-3 x^2}+\frac{2}{63} x^3 \sqrt [4]{2-3 x^2}+\frac{8\ 2^{3/4} \tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{8\ 2^{3/4} \tanh ^{-1}\left (\frac{2^{3/4}+\sqrt [4]{2} \sqrt{2-3 x^2}}{\sqrt{3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt{3}}-\frac{160\ 2^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{567 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.229756, size = 190, normalized size = 1.04 \[ \frac{2}{567} x \left (31 \sqrt [4]{2} x^2 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+\frac{\frac{1280 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )}{\left (3 x^2-4\right ) \left (x^2 \left (2 F_1\left (\frac{3}{2};\frac{3}{4},2;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )+3 F_1\left (\frac{3}{2};\frac{7}{4},1;\frac{5}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )+4 F_1\left (\frac{1}{2};\frac{3}{4},1;\frac{3}{2};\frac{3 x^2}{2},\frac{3 x^2}{4}\right )\right )}-27 x^4-102 x^2+80}{\left (2-3 x^2\right )^{3/4}}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{-3\,{x}^{2}+4} \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} - 4\right )}{\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}}{9 \, x^{4} - 18 \, x^{2} + 8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{6}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac{3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac{3}{4}}}\, dx \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} - 4\right )}{\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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