Optimal. Leaf size=260 \[ -\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{160\ 2^{3/4} x}-\frac {189 \sqrt [4]{3 x^2-2} x}{160 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {63 \left (3 x^2-2\right )^{3/4}}{160 x}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x}+\frac {\left (3 x^2-2\right )^{3/4}}{10 x^5}+\frac {7 \left (3 x^2-2\right )^{3/4}}{40 x^3} \]
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Rubi [A] time = 0.13, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {325, 230, 305, 220, 1196} \[ -\frac {189 \sqrt [4]{3 x^2-2} x}{160 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {63 \left (3 x^2-2\right )^{3/4}}{160 x}+\frac {7 \left (3 x^2-2\right )^{3/4}}{40 x^3}+\frac {\left (3 x^2-2\right )^{3/4}}{10 x^5}-\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{160\ 2^{3/4} x}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 230
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [4]{-2+3 x^2}} \, dx &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {21}{20} \int \frac {1}{x^4 \sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63}{80} \int \frac {1}{x^2 \sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {189}{320} \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {\left (63 \sqrt {\frac {3}{2}} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {\left (63 \sqrt {3} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x}+\frac {\left (63 \sqrt {3} \sqrt {x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {2}}}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2+3 x^2}\right )}{160 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{10 x^5}+\frac {7 \left (-2+3 x^2\right )^{3/4}}{40 x^3}+\frac {63 \left (-2+3 x^2\right )^{3/4}}{160 x}-\frac {189 x \sqrt [4]{-2+3 x^2}}{160 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{80\ 2^{3/4} x}-\frac {63 \sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {2}+\sqrt {-2+3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{160\ 2^{3/4} x}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.18 \[ -\frac {\sqrt [4]{1-\frac {3 x^2}{2}} \, _2F_1\left (-\frac {5}{2},\frac {1}{4};-\frac {3}{2};\frac {3 x^2}{2}\right )}{5 x^5 \sqrt [4]{3 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{3 \, x^{8} - 2 \, x^{6}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 72, normalized size = 0.28 \[ -\frac {189 \,2^{\frac {3}{4}} \left (-\mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{640 \mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )^{\frac {1}{4}}}+\frac {189 x^{6}-42 x^{4}-8 x^{2}-32}{160 \left (3 x^{2}-2\right )^{\frac {1}{4}} x^{5}} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^6\,{\left (3\,x^2-2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.98, size = 34, normalized size = 0.13 \[ \frac {2^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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