Integrand size = 24, antiderivative size = 76 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}} \]
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Time = 0.02 (sec) , antiderivative size = 76, 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 = {126, 279, 270} \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]
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Rule 126
Rule 270
Rule 279
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(e x)^{13/2} \sqrt [4]{1-x^2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}-\frac {8}{3} \int \frac {\left (1-x^2\right )^{3/4}}{(e x)^{13/2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}+\frac {32}{21} \int \frac {\left (1-x^2\right )^{7/4}}{(e x)^{13/2}} \, dx \\ & = -\frac {2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac {16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac {64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 x \left (1-x^2\right )^{3/4} \left (21+24 x^2+32 x^4\right )}{231 (e x)^{13/2}} \]
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Time = 1.52 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 x \left (1-x \right )^{\frac {3}{4}} \left (1+x \right )^{\frac {3}{4}} \left (32 x^{4}+24 x^{2}+21\right )}{231 \left (e x \right )^{\frac {13}{2}}}\) | \(33\) |
risch | \(\frac {2 \left (e^{2} x^{2} \left (1-x \right ) \left (1+x \right )\right )^{\frac {1}{4}} \left (1+x \right )^{\frac {3}{4}} \left (-1+x \right ) \left (32 x^{4}+24 x^{2}+21\right )}{231 \sqrt {e x}\, \left (1-x \right )^{\frac {1}{4}} e^{6} x^{5} \left (-e^{2} x^{2} \left (-1+x \right ) \left (1+x \right )\right )^{\frac {1}{4}}}\) | \(74\) |
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Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {2 \, {\left (32 \, x^{4} + 24 \, x^{2} + 21\right )} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{231 \, e^{7} x^{6}} \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=\int { \frac {1}{\left (e x\right )^{\frac {13}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Time = 1.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {e\,x}\,\left (\frac {2}{11\,e^7}+\frac {2\,x^2}{77\,e^7}+\frac {16\,x^4}{231\,e^7}-\frac {64\,x^6}{231\,e^7}\right )}{x^6\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \]
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