Integrand size = 24, antiderivative size = 60 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}+\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {126, 321, 323, 342, 234} \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}} \]
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Rule 126
Rule 234
Rule 321
Rule 323
Rule 342
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {e x}}{\sqrt [4]{1-x^2}} \, dx \\ & = -\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}-\frac {1}{2} e^2 \int \frac {1}{(e x)^{3/2} \sqrt [4]{1-x^2}} \, dx \\ & = -\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}-\frac {\left (\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \int \frac {1}{\sqrt [4]{1-\frac {1}{x^2}} x^2} \, dx}{2 \sqrt [4]{1-x^2}} \\ & = -\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}+\frac {\left (\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^2}} \, dx,x,\frac {1}{x}\right )}{2 \sqrt [4]{1-x^2}} \\ & = -\frac {e \left (1-x^2\right )^{3/4}}{\sqrt {e x}}+\frac {\sqrt [4]{1-\frac {1}{x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {2}{3} x \sqrt {e x} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},x^2\right ) \]
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\[\int \frac {\sqrt {e x}}{\left (1-x \right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}d x\]
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\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 2.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {i \sqrt {e} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{8}, \frac {3}{8} & 0, \frac {1}{4}, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{8}, 0, \frac {3}{8}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \Gamma \left (\frac {1}{4}\right )} - \frac {\sqrt {e} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {3}{4}, - \frac {5}{8}, - \frac {1}{4}, - \frac {1}{8}, \frac {1}{4}, 1 & \\- \frac {5}{8}, - \frac {1}{8} & - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\sqrt {e x}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int \frac {\sqrt {e\,x}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]
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