Integrand size = 24, antiderivative size = 148 \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {\sqrt [4]{2} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {451, 234, 406} \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \arcsin \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \arctan \left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {2-3 x^2}+2^{3/4}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}} \]
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Rule 234
Rule 406
Rule 451
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{3 \sqrt [4]{2-3 x^2}}+\frac {4}{3 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{\sqrt [4]{2-3 x^2}} \, dx\right )+\frac {4}{3} \int \frac {1}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx \\ & = \frac {\sqrt [4]{2} \tan ^{-1}\left (\frac {2^{3/4}-\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}+\frac {\sqrt [4]{2} \tanh ^{-1}\left (\frac {2^{3/4}+\sqrt [4]{2} \sqrt {2-3 x^2}}{\sqrt {3} x \sqrt [4]{2-3 x^2}}\right )}{3 \sqrt {3}}-\frac {2 \sqrt [4]{2} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 5.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.25 \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\frac {x^3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},\frac {3 x^2}{2},\frac {3 x^2}{4}\right )}{12 \sqrt [4]{2}} \]
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\[\int \frac {x^{2}}{\left (-3 x^{2}+2\right )^{\frac {1}{4}} \left (-3 x^{2}+4\right )}d x\]
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\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=- \int \frac {x^{2}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \]
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\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=\int { -\frac {x^{2}}{{\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx=-\int \frac {x^2}{{\left (2-3\,x^2\right )}^{1/4}\,\left (3\,x^2-4\right )} \,d x \]
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