Optimal. Leaf size=244 \[ -\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}-\frac {e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x} \]
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Rubi [A] time = 0.20, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {125, 321, 329, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}+\frac {e^{3/2} \log \left (\frac {\sqrt {e} x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2}}+\sqrt {e}\right )}{8 \sqrt {2}}-\frac {e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} e \left (1-x^2\right )^{3/4} \sqrt {e x} \]
Antiderivative was successfully verified.
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Rule 125
Rule 204
Rule 211
Rule 240
Rule 321
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx &=\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 \operatorname {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 \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 \operatorname {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 \operatorname {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} \operatorname {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} \operatorname {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*}
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Mathematica [A] time = 0.10, size = 190, normalized size = 0.78 \[ -\frac {(e x)^{3/2} \left (8 \sqrt {x} \left (1-x^2\right )^{3/4}+\sqrt {2} \log \left (\frac {x}{\sqrt {1-x^2}}-\frac {\sqrt {2} \sqrt {x}}{\sqrt [4]{1-x^2}}+1\right )-\sqrt {2} \log \left (\frac {x}{\sqrt {1-x^2}}+\frac {\sqrt {2} \sqrt {x}}{\sqrt [4]{1-x^2}}+1\right )+2 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {x}}{\sqrt [4]{1-x^2}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [4]{1-x^2}}+1\right )\right )}{16 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 478, normalized size = 1.96 \[ -\frac {1}{2} \, \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \frac {1}{4} \, \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \arctan \left (-\frac {e^{6} x^{2} - e^{6} + \sqrt {2} {\left (e^{6}\right )}^{\frac {3}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \sqrt {2} {\left (e^{6}\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )} \sqrt {-\frac {e^{3} \sqrt {x + 1} x \sqrt {-x + 1} - \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \sqrt {e^{6}} {\left (x^{2} - 1\right )}}{x^{2} - 1}}}{e^{6} x^{2} - e^{6}}\right ) + \frac {1}{4} \, \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \arctan \left (\frac {e^{6} x^{2} - e^{6} - \sqrt {2} {\left (e^{6}\right )}^{\frac {3}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \sqrt {2} {\left (e^{6}\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )} \sqrt {-\frac {e^{3} \sqrt {x + 1} x \sqrt {-x + 1} + \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \sqrt {e^{6}} {\left (x^{2} - 1\right )}}{x^{2} - 1}}}{e^{6} x^{2} - e^{6}}\right ) + \frac {1}{16} \, \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \log \left (-\frac {e^{3} \sqrt {x + 1} x \sqrt {-x + 1} + \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \sqrt {e^{6}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac {1}{16} \, \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \log \left (-\frac {e^{3} \sqrt {x + 1} x \sqrt {-x + 1} - \sqrt {2} {\left (e^{6}\right )}^{\frac {1}{4}} \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \sqrt {e^{6}} {\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{\frac {3}{2}}}{\left (-x +1\right )^{\frac {1}{4}} \left (x +1\right )^{\frac {1}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3/2}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.18, size = 114, normalized size = 0.47 \[ - \frac {i e^{\frac {3}{2}} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{8}, - \frac {1}{8} & - \frac {1}{2}, - \frac {1}{4}, 0, 1 \\-1, - \frac {5}{8}, - \frac {1}{2}, - \frac {1}{8}, 0, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \Gamma \left (\frac {1}{4}\right )} - \frac {e^{\frac {3}{2}} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {5}{4}, - \frac {9}{8}, - \frac {3}{4}, - \frac {5}{8}, - \frac {1}{4}, 1 & \\- \frac {9}{8}, - \frac {5}{8} & - \frac {5}{4}, -1, - \frac {3}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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