Optimal. Leaf size=83 \[ -\frac {320\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{2079 \sqrt {3}}+\frac {160 \sqrt [4]{3 x^2+2} x}{2079}+\frac {2}{33} \sqrt [4]{3 x^2+2} x^5-\frac {40}{693} \sqrt [4]{3 x^2+2} x^3 \]
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Rubi [A] time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, 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, 231} \[ \frac {2}{33} \sqrt [4]{3 x^2+2} x^5-\frac {40}{693} \sqrt [4]{3 x^2+2} x^3+\frac {160 \sqrt [4]{3 x^2+2} x}{2079}-\frac {320\ 2^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2079 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 321
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} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2079 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.65 \[ \frac {2 x \left (\sqrt [4]{3 x^2+2} \left (63 x^4-60 x^2+80\right )-80 \sqrt [4]{2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};-\frac {3 x^2}{2}\right )\right )}{2079} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 43, normalized size = 0.52 \[ -\frac {160 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{2079}+\frac {2 \left (63 x^{4}-60 x^{2}+80\right ) \left (3 x^{2}+2\right )^{\frac {1}{4}} x}{2079} \]
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 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (3\,x^2+2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.82, size = 27, normalized size = 0.33 \[ \frac {\sqrt [4]{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{14} \]
Verification of antiderivative is not currently implemented for this CAS.
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