3.899 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{9/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=51 \[ \frac{8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}} \]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(7/2)) + (8*(1 - x^2)^(7/4))/(21*e*(e*x)^(7/2))

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Rubi [A]  time = 0.0644011, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{8 \left (1-x^2\right )^{7/4}}{21 e (e x)^{7/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(1/4)*(e*x)^(9/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(7/2)) + (8*(1 - x^2)^(7/4))/(21*e*(e*x)^(7/2))

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Rubi in Sympy [A]  time = 7.4369, size = 39, normalized size = 0.76 \[ \frac{8 \left (- x^{2} + 1\right )^{\frac{7}{4}}}{21 e \left (e x\right )^{\frac{7}{2}}} - \frac{2 \left (- x^{2} + 1\right )^{\frac{3}{4}}}{3 e \left (e x\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(9/2)/(1+x)**(1/4),x)

[Out]

8*(-x**2 + 1)**(7/4)/(21*e*(e*x)**(7/2)) - 2*(-x**2 + 1)**(3/4)/(3*e*(e*x)**(7/2
))

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Mathematica [A]  time = 0.0233888, size = 35, normalized size = 0.69 \[ -\frac{2 \left (1-x^2\right )^{3/4} \left (4 x^2+3\right ) \sqrt{e x}}{21 e^5 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(1/4)*(e*x)^(9/2)*(1 + x)^(1/4)),x]

[Out]

(-2*Sqrt[e*x]*(1 - x^2)^(3/4)*(3 + 4*x^2))/(21*e^5*x^4)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[ -{\frac{2\,x \left ( 4\,{x}^{2}+3 \right ) }{21} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(9/2)/(1+x)^(1/4),x)

[Out]

-2/21*x*(1+x)^(3/4)*(4*x^2+3)*(1-x)^(3/4)/(e*x)^(9/2)

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Maxima [A]  time = 1.42587, size = 46, normalized size = 0.9 \[ \frac{2 \,{\left (4 \, x^{5} - x^{3} - 3 \, x\right )}}{21 \, e^{\frac{9}{2}}{\left (x + 1\right )}^{\frac{1}{4}} x^{\frac{9}{2}}{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(9/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

2/21*(4*x^5 - x^3 - 3*x)/(e^(9/2)*(x + 1)^(1/4)*x^(9/2)*(-x + 1)^(1/4))

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Fricas [A]  time = 0.212544, size = 50, normalized size = 0.98 \[ \frac{2 \,{\left (4 \, x^{4} - x^{2} - 3\right )}}{21 \, \sqrt{e x} e^{4}{\left (x + 1\right )}^{\frac{1}{4}} x^{3}{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(9/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

2/21*(4*x^4 - x^2 - 3)/(sqrt(e*x)*e^4*(x + 1)^(1/4)*x^3*(-x + 1)^(1/4))

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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)**(9/2)/(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)^(9/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out