Optimal. Leaf size=105 \[ \frac {\sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{2} x}+\frac {\sqrt [4]{-3 x^2-2}}{2 x} \]
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Rubi [A] time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {325, 234, 220} \[ \frac {\sqrt [4]{-3 x^2-2}}{2 x}+\frac {\sqrt {3} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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Rule 220
Rule 234
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (-2-3 x^2\right )^{3/4}} \, dx &=\frac {\sqrt [4]{-2-3 x^2}}{2 x}-\frac {3}{4} \int \frac {1}{\left (-2-3 x^2\right )^{3/4}} \, dx\\ &=\frac {\sqrt [4]{-2-3 x^2}}{2 x}+\frac {\left (\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 )}{2 x}\\ &=\frac {\sqrt [4]{-2-3 x^2}}{2 x}+\frac {\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 )}{4 \sqrt [4]{2} x}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 0.44 \[ -\frac {\left (\frac {3 x^2}{2}+1\right )^{3/4} \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {1}{2};-\frac {3 x^2}{2}\right )}{x \left (-3 x^2-2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ \frac {2 \, x {\rm integral}\left (\frac {3 \, {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}{4 \, {\left (3 \, x^{2} + 2\right )}}, x\right ) + {\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}{2 \, x} \]
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 \[ \int \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
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 \[ \int \frac {1}{\left (-3 x^{2}-2\right )^{\frac {3}{4}} x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.73, size = 36, normalized size = 0.34 \[ -\frac {2\,3^{1/4}\,{\left (\frac {2}{x^2}+3\right )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{4},\frac {5}{4};\ \frac {9}{4};\ -\frac {2}{3\,x^2}\right )}{15\,x\,{\left (-3\,x^2-2\right )}^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.84, size = 34, normalized size = 0.32 \[ \frac {\sqrt [4]{2} e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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