Optimal. Leaf size=182 \[ \frac {80}{567} \sqrt [4]{2-3 x^2} x+\frac {8\ 2^{3/4} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}-\frac {8\ 2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{27 \sqrt {3}}+\frac {2}{63} \sqrt [4]{2-3 x^2} x^3-\frac {160\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{567 \sqrt {3}} \]
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Rubi [A] time = 0.12, antiderivative size = 182, normalized size of antiderivative = 1.00, 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 232
Rule 321
Rule 400
Rule 441
Rule 443
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*}
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Mathematica [C] time = 0.25, 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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fricas [F] time = 4.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{6}}{9 \, x^{4} - 18 \, x^{2} + 8}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 10.35, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}} \left (-3 x^{2}+4\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x^{6}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x^6}{{\left (2-3\,x^2\right )}^{3/4}\,\left (3\,x^2-4\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x^{6}}{3 x^{2} \left (2 - 3 x^{2}\right )^{\frac {3}{4}} - 4 \left (2 - 3 x^{2}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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