3.902 \(\int \frac{\sqrt{e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=60 \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]

[Out]

-((e*(1 - x^2)^(3/4))/Sqrt[e*x]) + ((1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCs
c[x]/2, 2])/(1 - x^2)^(1/4)

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Rubi [A]  time = 0.0859836, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[e*x]/((1 - x)^(1/4)*(1 + x)^(1/4)),x]

[Out]

-((e*(1 - x^2)^(3/4))/Sqrt[e*x]) + ((1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCs
c[x]/2, 2])/(1 - x^2)^(1/4)

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Rubi in Sympy [A]  time = 9.53554, size = 49, normalized size = 0.82 \[ - \frac{e \left (- x^{2} + 1\right )^{\frac{3}{4}}}{\sqrt{e x}} + \frac{\sqrt{e x} \sqrt [4]{1 - \frac{1}{x^{2}}} E\left (\frac{\operatorname{asin}{\left (\frac{1}{x} \right )}}{2}\middle | 2\right )}{\sqrt [4]{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(1/2)/(1-x)**(1/4)/(1+x)**(1/4),x)

[Out]

-e*(-x**2 + 1)**(3/4)/sqrt(e*x) + sqrt(e*x)*(1 - 1/x**2)**(1/4)*elliptic_e(asin(
1/x)/2, 2)/(-x**2 + 1)**(1/4)

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Mathematica [C]  time = 0.0135497, size = 25, normalized size = 0.42 \[ \frac{2}{3} x \sqrt{e x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[e*x]/((1 - x)^(1/4)*(1 + x)^(1/4)),x]

[Out]

(2*x*Sqrt[e*x]*Hypergeometric2F1[1/4, 3/4, 7/4, x^2])/3

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1\sqrt{ex}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)

[Out]

int((e*x)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

integral(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Sympy [A]  time = 28.8983, size = 105, normalized size = 1.75 \[ \frac{i \sqrt{e}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{8}, \frac{3}{8} & 0, \frac{1}{4}, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{8}, 0, \frac{3}{8}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \Gamma \left (\frac{1}{4}\right )} - \frac{\sqrt{e}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{4}, - \frac{5}{8}, - \frac{1}{4}, - \frac{1}{8}, \frac{1}{4}, 1 & \\- \frac{5}{8}, - \frac{1}{8} & - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(1/2)/(1-x)**(1/4)/(1+x)**(1/4),x)

[Out]

I*sqrt(e)*meijerg(((-1/8, 3/8), (0, 1/4, 1/2, 1)), ((-1/2, -1/8, 0, 3/8, 1/2, 0)
, ()), exp_polar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*gamma(1/4)) - sqrt(e)*meijerg(
((-3/4, -5/8, -1/4, -1/8, 1/4, 1), ()), ((-5/8, -1/8), (-3/4, -1/2, -1/4, 0)), x
**(-2))/(4*pi*gamma(1/4))

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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(sqrt(e*x)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out