Integrand size = 24, antiderivative size = 244 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}-\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}} \]
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Time = 0.14 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {126, 327, 335, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt {2}}-\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x} \]
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Rule 126
Rule 210
Rule 217
Rule 246
Rule 327
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x^2}} \, dx \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}+\frac {1}{4} e^2 \int \frac {1}{\sqrt {e x} \sqrt [4]{1-x^2}} \, dx \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {x^4}{e^2}}} \, dx,x,\sqrt {e x}\right ) \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{1+\frac {x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right ) \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}+\frac {1}{4} \text {Subst}\left (\int \frac {e-x^2}{1+\frac {x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )+\frac {1}{4} \text {Subst}\left (\int \frac {e+x^2}{1+\frac {x^4}{e^2}} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right ) \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}-\frac {e^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {1}{8} e^2 \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right )+\frac {1}{8} e^2 \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\frac {\sqrt {e x}}{\sqrt [4]{1-x^2}}\right ) \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}-\frac {e^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}} \\ & = -\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}-\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {(e x)^{3/2} \left (-4 \sqrt {x} \left (1-x^2\right )^{3/4}+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{-x+\sqrt {1-x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{x+\sqrt {1-x^2}}\right )\right )}{8 x^{3/2}} \]
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\[\int \frac {\left (e x \right )^{\frac {3}{2}}}{\left (1-x \right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} \, \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \frac {1}{8} \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) + \frac {1}{8} \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \left (-e^{6}\right )^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac {1}{8} i \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (i \, x^{2} - i\right )}}{x^{2} - 1}\right ) + \frac {1}{8} i \, \left (-e^{6}\right )^{\frac {1}{4}} \log \left (\frac {\sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \left (-e^{6}\right )^{\frac {1}{4}} {\left (-i \, x^{2} + i\right )}}{x^{2} - 1}\right ) \]
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Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]
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