Integrand size = 24, antiderivative size = 30 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=-\frac {2 (1-x)^{3/4} (1+x)^{3/4}}{3 e (e x)^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {97} \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=-\frac {2 (1-x)^{3/4} (x+1)^{3/4}}{3 e (e x)^{3/2}} \]
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Rule 97
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-x)^{3/4} (1+x)^{3/4}}{3 e (e x)^{3/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \left (1-x^2\right )^{3/4}}{3 (e x)^{5/2}} \]
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Time = 0.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(-\frac {2 x \left (1-x \right )^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{4}}}{3 \left (e x \right )^{\frac {5}{2}}}\) | \(21\) |
risch | \(\frac {2 \left (e^{2} x^{2} \left (1-x \right ) \left (1+x \right )\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{4}} \left (-1+x \right )}{3 \sqrt {e x}\, \left (1-x \right )^{\frac {1}{4}} e^{2} x \left (-e^{2} x^{2} \left (-1+x \right ) \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(62\) |
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none
Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \, \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{3 \, e^{3} x^{2}} \]
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Result contains complex when optimal does not.
Time = 37.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {11}{8}, \frac {15}{8}, 1 & \frac {3}{2}, \frac {7}{4}, 2 \\1, \frac {11}{8}, \frac {3}{2}, \frac {15}{8}, 2 & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi e^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{4}, \frac {7}{8}, \frac {5}{4}, \frac {11}{8}, \frac {7}{4}, 1 & \\\frac {7}{8}, \frac {11}{8} & \frac {3}{4}, 1, \frac {5}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi e^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Time = 1.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {e\,x}\,\left (\frac {2}{3\,e^3}-\frac {2\,x^2}{3\,e^3}\right )}{x^2\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \]
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