3.898 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 (1-x)^{3/4} (x+1)^{3/4}}{3 e (e x)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0322629, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{2 (1-x)^{3/4} (x+1)^{3/4}}{3 e (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(1/4)*(e*x)^(5/2)*(1 + x)^(1/4)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.28365, size = 26, normalized size = 0.87 \[ - \frac{2 \left (- x + 1\right )^{\frac{3}{4}} \left (x + 1\right )^{\frac{3}{4}}}{3 e \left (e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(5/2)/(1+x)**(1/4),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.016833, size = 23, normalized size = 0.77 \[ -\frac{2 x \left (1-x^2\right )^{3/4}}{3 (e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(1/4)*(e*x)^(5/2)*(1 + x)^(1/4)),x]

[Out]

(-2*x*(1 - x^2)^(3/4))/(3*(e*x)^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 21, normalized size = 0.7 \[ -{\frac{2\,x}{3} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(5/2)/(1+x)^(1/4),x)

[Out]

-2/3*x*(1+x)^(3/4)*(1-x)^(3/4)/(e*x)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.42236, size = 36, normalized size = 1.2 \[ \frac{2 \,{\left (x^{3} - x\right )}}{3 \, e^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{1}{4}} x^{\frac{5}{2}}{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(5/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

2/3*(x^3 - x)/(e^(5/2)*(x + 1)^(1/4)*x^(5/2)*(-x + 1)^(1/4))

_______________________________________________________________________________________

Fricas [A]  time = 0.217819, size = 41, normalized size = 1.37 \[ \frac{2 \,{\left (x^{2} - 1\right )}}{3 \, \sqrt{e x} e^{2}{\left (x + 1\right )}^{\frac{1}{4}} x{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(5/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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)**(5/2)/(1+x)**(1/4),x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(5/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out