Optimal. Leaf size=83 \[ -\frac {9 x}{8 \sqrt [4]{3 x^2+2}}+\frac {3 \left (3 x^2+2\right )^{3/4}}{8 x}-\frac {\left (3 x^2+2\right )^{3/4}}{6 x^3}+\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
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Rubi [A] time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {325, 227, 196} \[ -\frac {9 x}{8 \sqrt [4]{3 x^2+2}}+\frac {3 \left (3 x^2+2\right )^{3/4}}{8 x}-\frac {\left (3 x^2+2\right )^{3/4}}{6 x^3}+\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 227
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{2+3 x^2}} \, dx &=-\frac {\left (2+3 x^2\right )^{3/4}}{6 x^3}-\frac {3}{4} \int \frac {1}{x^2 \sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac {\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (2+3 x^2\right )^{3/4}}{8 x}-\frac {9}{16} \int \frac {1}{\sqrt [4]{2+3 x^2}} \, dx\\ &=-\frac {9 x}{8 \sqrt [4]{2+3 x^2}}-\frac {\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (2+3 x^2\right )^{3/4}}{8 x}+\frac {9}{8} \int \frac {1}{\left (2+3 x^2\right )^{5/4}} \, dx\\ &=-\frac {9 x}{8 \sqrt [4]{2+3 x^2}}-\frac {\left (2+3 x^2\right )^{3/4}}{6 x^3}+\frac {3 \left (2+3 x^2\right )^{3/4}}{8 x}+\frac {3 \sqrt {3} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{4\ 2^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 29, normalized size = 0.35 \[ -\frac {\, _2F_1\left (-\frac {3}{2},\frac {1}{4};-\frac {1}{2};-\frac {3 x^2}{2}\right )}{3 \sqrt [4]{2} x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}}{3 \, x^{6} + 2 \, x^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.27, size = 45, normalized size = 0.54 \[ -\frac {9 \,2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{32}+\frac {27 x^{4}+6 x^{2}-8}{24 \left (3 x^{2}+2\right )^{\frac {1}{4}} x^{3}} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{4}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (3\,x^2+2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.84, size = 32, normalized size = 0.39 \[ - \frac {2^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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