Optimal. Leaf size=43 \[ \frac {2 x}{\sqrt [4]{3 x^2+2}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {227, 196} \[ \frac {2 x}{\sqrt [4]{3 x^2+2}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{2+3 x^2}} \, dx &=\frac {2 x}{\sqrt [4]{2+3 x^2}}-2 \int \frac {1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=\frac {2 x}{\sqrt [4]{2+3 x^2}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 24, normalized size = 0.56 \[ \frac {x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {3 x^2}{2}\right )}{\sqrt [4]{2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 18, normalized size = 0.42 \[ \frac {2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{2} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 16, normalized size = 0.37 \[ \frac {8^{1/4}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ -\frac {3\,x^2}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.67, size = 26, normalized size = 0.60 \[ \frac {2^{\frac {3}{4}} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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