Optimal. Leaf size=85 \[ -\frac {27 \sqrt {3} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{32 \sqrt [4]{2}}-\frac {27 \sqrt [4]{3 x^2+2}}{32 x}-\frac {\sqrt [4]{3 x^2+2}}{10 x^5}+\frac {9 \sqrt [4]{3 x^2+2}}{40 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 85, 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 = {325, 231} \[ -\frac {27 \sqrt [4]{3 x^2+2}}{32 x}+\frac {9 \sqrt [4]{3 x^2+2}}{40 x^3}-\frac {\sqrt [4]{3 x^2+2}}{10 x^5}-\frac {27 \sqrt {3} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{32 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (2+3 x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{2+3 x^2}}{10 x^5}-\frac {27}{20} \int \frac {1}{x^4 \left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{2+3 x^2}}{40 x^3}+\frac {27}{16} \int \frac {1}{x^2 \left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{2+3 x^2}}{40 x^3}-\frac {27 \sqrt [4]{2+3 x^2}}{32 x}-\frac {81}{64} \int \frac {1}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{2+3 x^2}}{10 x^5}+\frac {9 \sqrt [4]{2+3 x^2}}{40 x^3}-\frac {27 \sqrt [4]{2+3 x^2}}{32 x}-\frac {27 \sqrt {3} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{32 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.34 \[ -\frac {\, _2F_1\left (-\frac {5}{2},\frac {3}{4};-\frac {3}{2};-\frac {3 x^2}{2}\right )}{5\ 2^{3/4} x^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{3 \, x^{8} + 2 \, x^{6}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 50, normalized size = 0.59 \[ -\frac {81 \,2^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{2}, \frac {3}{4}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{128}-\frac {405 x^{6}+162 x^{4}-24 x^{2}+32}{160 \left (3 x^{2}+2\right )^{\frac {3}{4}} x^{5}} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}} x^{6}}\,{d x} \]
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 \[ \int \frac {1}{x^6\,{\left (3\,x^2+2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.08, size = 32, normalized size = 0.38 \[ - \frac {\sqrt [4]{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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