Optimal. Leaf size=99 \[ \frac {32 \left (3 x^2+2\right )^{3/4} x}{1053}-\frac {128 x}{1053 \sqrt [4]{3 x^2+2}}+\frac {2}{39} \left (3 x^2+2\right )^{3/4} x^5-\frac {40 \left (3 x^2+2\right )^{3/4} x^3}{1053}+\frac {128 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{1053 \sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 227, 196} \[ \frac {2}{39} \left (3 x^2+2\right )^{3/4} x^5-\frac {40 \left (3 x^2+2\right )^{3/4} x^3}{1053}+\frac {32 \left (3 x^2+2\right )^{3/4} x}{1053}-\frac {128 x}{1053 \sqrt [4]{3 x^2+2}}+\frac {128 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{1053 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 321
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt [4]{2+3 x^2}} \, dx &=\frac {2}{39} x^5 \left (2+3 x^2\right )^{3/4}-\frac {20}{39} \int \frac {x^4}{\sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac {40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac {2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac {80}{351} \int \frac {x^2}{\sqrt [4]{2+3 x^2}} \, dx\\ &=\frac {32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac {40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac {2}{39} x^5 \left (2+3 x^2\right )^{3/4}-\frac {64 \int \frac {1}{\sqrt [4]{2+3 x^2}} \, dx}{1053}\\ &=-\frac {128 x}{1053 \sqrt [4]{2+3 x^2}}+\frac {32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac {40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac {2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac {128 \int \frac {1}{\left (2+3 x^2\right )^{5/4}} \, dx}{1053}\\ &=-\frac {128 x}{1053 \sqrt [4]{2+3 x^2}}+\frac {32 x \left (2+3 x^2\right )^{3/4}}{1053}-\frac {40 x^3 \left (2+3 x^2\right )^{3/4}}{1053}+\frac {2}{39} x^5 \left (2+3 x^2\right )^{3/4}+\frac {128 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{1053 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 54, normalized size = 0.55 \[ \frac {2 x \left (\left (3 x^2+2\right )^{3/4} \left (27 x^4-20 x^2+16\right )-16\ 2^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {3 x^2}{2}\right )\right )}{1053} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}, x\right ) \]
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 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 43, normalized size = 0.43 \[ -\frac {32 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{1053}+\frac {2 \left (27 x^{4}-20 x^{2}+16\right ) \left (3 x^{2}+2\right )^{\frac {3}{4}} x}{1053} \]
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 \[ \int \frac {x^{6}}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (3\,x^2+2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.83, size = 27, normalized size = 0.27 \[ \frac {2^{\frac {3}{4}} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{14} \]
Verification of antiderivative is not currently implemented for this CAS.
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