Optimal. Leaf size=221 \[ -\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{2\ 2^{3/4} x}-\frac {3 \sqrt [4]{3 x^2-2} x}{2 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {\left (3 x^2-2\right )^{3/4}}{2 x}+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2^{3/4} x} \]
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Rubi [A] time = 0.09, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {3 \sqrt [4]{3 x^2-2} x}{2 \left (\sqrt {3 x^2-2}+\sqrt {2}\right )}+\frac {\left (3 x^2-2\right )^{3/4}}{2 x}-\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2\ 2^{3/4} x}+\frac {\sqrt {3} \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2^{3/4} x} \]
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^2 \sqrt [4]{-2+3 x^2}} \, dx &=\frac {\left (-2+3 x^2\right )^{3/4}}{2 x}-\frac {3}{4} \int \frac {1}{\sqrt [4]{-2+3 x^2}} \, dx\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{2 x}-\frac {\left (\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 )}{2 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{2 x}-\frac {\left (\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 )}{2 x}+\frac {\left (\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 )}{2 x}\\ &=\frac {\left (-2+3 x^2\right )^{3/4}}{2 x}-\frac {3 x \sqrt [4]{-2+3 x^2}}{2 \left (\sqrt {2}+\sqrt {-2+3 x^2}\right )}+\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 ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2+3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{2^{3/4} 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 )}{2\ 2^{3/4} x}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 46, normalized size = 0.21 \[ -\frac {\sqrt [4]{1-\frac {3 x^2}{2}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {3 x^2}{2}\right )}{x \sqrt [4]{3 x^2-2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{3 \, x^{4} - 2 \, x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.30, size = 55, normalized size = 0.25 \[ -\frac {3 \,2^{\frac {3}{4}} \left (-\mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {3 x^{2}}{2}\right )}{8 \mathrm {signum}\left (\frac {3 x^{2}}{2}-1\right )^{\frac {1}{4}}}+\frac {\left (3 x^{2}-2\right )^{\frac {3}{4}}}{2 x} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 2\right )}^{\frac {1}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.89, size = 36, normalized size = 0.16 \[ -\frac {2\,3^{3/4}\,{\left (3-\frac {2}{x^2}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {2}{3\,x^2}\right )}{9\,x\,{\left (3\,x^2-2\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.76, size = 31, normalized size = 0.14 \[ \frac {2^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2}}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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