3.903 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{3/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}} \]

[Out]

(-2*(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(e^2*(1 - x^2)^(1/4)
)

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Rubi [A]  time = 0.0630773, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{e^2 \sqrt [4]{1-x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(1/4)*(e*x)^(3/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(e^2*(1 - x^2)^(1/4)
)

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Rubi in Sympy [A]  time = 7.56291, size = 39, normalized size = 0.93 \[ - \frac{2 \sqrt{e x} \sqrt [4]{1 - \frac{1}{x^{2}}} E\left (\frac{\operatorname{asin}{\left (\frac{1}{x} \right )}}{2}\middle | 2\right )}{e^{2} \sqrt [4]{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(3/2)/(1+x)**(1/4),x)

[Out]

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

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Mathematica [C]  time = 0.0279611, size = 44, normalized size = 1.05 \[ -\frac{2 x \left (2 x^2 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right )+3 \left (1-x^2\right )^{3/4}\right )}{3 (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(1/4)*(e*x)^(3/2)*(1 + x)^(1/4)),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{1-x}}} \left ( ex \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x)

[Out]

int(1/(1-x)^(1/4)/(e*x)^(3/2)/(1+x)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (e x\right )^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed 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(1/((e*x)^(3/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out