Optimal. Leaf size=244 \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]
[Out]
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Rubi [A] time = 0.414125, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^(3/2)/((1 - x)^(1/4)*(1 + x)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 36.4365, size = 204, normalized size = 0.84 \[ - \frac{\sqrt{2} e^{\frac{3}{2}} \log{\left (- \frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{16} + \frac{\sqrt{2} e^{\frac{3}{2}} \log{\left (\frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{16} - \frac{\sqrt{2} e^{\frac{3}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{8} + \frac{\sqrt{2} e^{\frac{3}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{8} - \frac{e \sqrt{e x} \left (- x^{2} + 1\right )^{\frac{3}{4}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)/(1-x)**(1/4)/(1+x)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0326395, size = 39, normalized size = 0.16 \[ \frac{1}{2} e \sqrt{e x} \left (\, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};x^2\right )-\left (1-x^2\right )^{3/4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^(3/2)/((1 - x)^(1/4)*(1 + x)^(1/4)),x]
[Out]
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Maple [F] time = 0.079, size = 0, normalized size = 0. \[ \int{1 \left ( ex \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)/(1-x)^(1/4)/(1+x)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{3}{2}}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248055, size = 610, normalized size = 2.5 \[ -\frac{1}{2} \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )}}{2 \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )} + 2 \,{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}\right ) + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )}}{2 \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )} + 2 \,{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}\right ) + \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(3/2)/(1-x)**(1/4)/(1+x)**(1/4),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")
[Out]