3.904 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{7/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{4 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{5 e^4 \sqrt [4]{1-x^2}}-\frac{2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}} \]

[Out]

(-2*(1 - x^2)^(3/4))/(5*e*(e*x)^(5/2)) - (4*(1 - x^(-2))^(1/4)*Sqrt[e*x]*Ellipti
cE[ArcCsc[x]/2, 2])/(5*e^4*(1 - x^2)^(1/4))

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Rubi [A]  time = 0.0872142, antiderivative size = 70, 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 \[ -\frac{4 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{5 e^4 \sqrt [4]{1-x^2}}-\frac{2 \left (1-x^2\right )^{3/4}}{5 e (e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - x^2)^(3/4))/(5*e*(e*x)^(5/2)) - (4*(1 - x^(-2))^(1/4)*Sqrt[e*x]*Ellipti
cE[ArcCsc[x]/2, 2])/(5*e^4*(1 - x^2)^(1/4))

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Rubi in Sympy [A]  time = 9.83284, size = 61, normalized size = 0.87 \[ - \frac{2 \left (- x^{2} + 1\right )^{\frac{3}{4}}}{5 e \left (e x\right )^{\frac{5}{2}}} - \frac{4 \sqrt{e x} \sqrt [4]{1 - \frac{1}{x^{2}}} E\left (\frac{\operatorname{asin}{\left (\frac{1}{x} \right )}}{2}\middle | 2\right )}{5 e^{4} \sqrt [4]{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(7/2)/(1+x)**(1/4),x)

[Out]

-2*(-x**2 + 1)**(3/4)/(5*e*(e*x)**(5/2)) - 4*sqrt(e*x)*(1 - 1/x**2)**(1/4)*ellip
tic_e(asin(1/x)/2, 2)/(5*e**4*(-x**2 + 1)**(1/4))

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Mathematica [C]  time = 0.0507474, size = 51, normalized size = 0.73 \[ \frac{x \left (-8 x^4 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right )-6 \left (1-x^2\right )^{3/4} \left (2 x^2+1\right )\right )}{15 (e x)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-6*(1 - x^2)^(3/4)*(1 + 2*x^2) - 8*x^4*Hypergeometric2F1[1/4, 3/4, 7/4, x^2]
))/(15*(e*x)^(7/2))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{1-x}}} \left ( ex \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/(1-x)^(1/4)/(e*x)^(7/2)/(1+x)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (e x\right )^{\frac{7}{2}}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(7/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(7/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{e x} e^{3}{\left (x + 1\right )}^{\frac{1}{4}} x^{3}{\left (-x + 1\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(7/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(e*x)*e^3*(x + 1)^(1/4)*x^3*(-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)**(7/2)/(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)^(7/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out