Optimal. Leaf size=63 \[ \frac {3 x}{2 \sqrt [4]{3 x^2+2}}-\frac {\left (3 x^2+2\right )^{3/4}}{2 x}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, 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, 227, 196} \[ \frac {3 x}{2 \sqrt [4]{3 x^2+2}}-\frac {\left (3 x^2+2\right )^{3/4}}{2 x}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 325
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 {3 x}{2 \sqrt [4]{2+3 x^2}}-\frac {\left (2+3 x^2\right )^{3/4}}{2 x}-\frac {3}{2} \int \frac {1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=\frac {3 x}{2 \sqrt [4]{2+3 x^2}}-\frac {\left (2+3 x^2\right )^{3/4}}{2 x}-\frac {\sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{2^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.43 \[ -\frac {\, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};-\frac {3 x^2}{2}\right )}{\sqrt [4]{2} x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, 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 = 0.27, size = 33, normalized size = 0.52 \[ \frac {3 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{8}-\frac {\left (3 x^{2}+2\right )^{\frac {3}{4}}}{2 x} \]
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 = 5.02, size = 36, normalized size = 0.57 \[ -\frac {2\,3^{3/4}\,{\left (\frac {2}{x^2}+3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ -\frac {2}{3\,x^2}\right )}{9\,x\,{\left (3\,x^2+2\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.74, size = 29, normalized size = 0.46 \[ - \frac {2^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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