Optimal. Leaf size=141 \[ \frac {27 \sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{64 \sqrt [4]{2} x}+\frac {27 \sqrt [4]{-3 x^2-2}}{32 x}+\frac {\sqrt [4]{-3 x^2-2}}{10 x^5}-\frac {9 \sqrt [4]{-3 x^2-2}}{40 x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 141, 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 = {325, 234, 220} \[ \frac {27 \sqrt [4]{-3 x^2-2}}{32 x}-\frac {9 \sqrt [4]{-3 x^2-2}}{40 x^3}+\frac {\sqrt [4]{-3 x^2-2}}{10 x^5}+\frac {27 \sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{64 \sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 234
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 {\left (27 \sqrt {\frac {3}{2}} \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 )}{32 x}\\ &=\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} \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 )}{64 \sqrt [4]{2} x}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 48, normalized size = 0.34 \[ -\frac {\left (\frac {3 x^2}{2}+1\right )^{3/4} \, _2F_1\left (-\frac {5}{2},\frac {3}{4};-\frac {3}{2};-\frac {3 x^2}{2}\right )}{5 x^5 \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ \frac {160 \, x^{5} {\rm integral}\left (\frac {81 \, {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}{64 \, {\left (3 \, x^{2} + 2\right )}}, x\right ) + {\left (135 \, x^{4} - 36 \, x^{2} + 16\right )} {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}{160 \, x^{5}} \]
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 [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-3 x^{2}-2\right )^{\frac {3}{4}} x^{6}}\, dx \]
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 \[ \int \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{6}}\,{d x} \]
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 = 1.12, size = 37, normalized size = 0.26 \[ \frac {\sqrt [4]{2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {3}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{10 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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