3.900 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(11/2)) + (16*(1 - x^2)^(7/4))/(21*e*(e*x)^(11/2
)) - (64*(1 - x^2)^(11/4))/(231*e*(e*x)^(11/2))

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Rubi [A]  time = 0.0845373, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(11/2)) + (16*(1 - x^2)^(7/4))/(21*e*(e*x)^(11/2
)) - (64*(1 - x^2)^(11/4))/(231*e*(e*x)^(11/2))

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Rubi in Sympy [A]  time = 9.96972, size = 60, normalized size = 0.79 \[ - \frac{64 \left (- x^{2} + 1\right )^{\frac{11}{4}}}{231 e \left (e x\right )^{\frac{11}{2}}} + \frac{16 \left (- x^{2} + 1\right )^{\frac{7}{4}}}{21 e \left (e x\right )^{\frac{11}{2}}} - \frac{2 \left (- x^{2} + 1\right )^{\frac{3}{4}}}{3 e \left (e x\right )^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(13/2)/(1+x)**(1/4),x)

[Out]

-64*(-x**2 + 1)**(11/4)/(231*e*(e*x)**(11/2)) + 16*(-x**2 + 1)**(7/4)/(21*e*(e*x
)**(11/2)) - 2*(-x**2 + 1)**(3/4)/(3*e*(e*x)**(11/2))

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Mathematica [A]  time = 0.0258965, size = 40, normalized size = 0.53 \[ -\frac{2 \left (1-x^2\right )^{3/4} \left (32 x^4+24 x^2+21\right ) \sqrt{e x}}{231 e^7 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]

[Out]

(-2*Sqrt[e*x]*(1 - x^2)^(3/4)*(21 + 24*x^2 + 32*x^4))/(231*e^7*x^6)

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Maple [A]  time = 0.006, size = 33, normalized size = 0.4 \[ -{\frac{2\,x \left ( 32\,{x}^{4}+24\,{x}^{2}+21 \right ) }{231} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x)

[Out]

-2/231*x*(1+x)^(3/4)*(32*x^4+24*x^2+21)*(1-x)^(3/4)/(e*x)^(13/2)

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Maxima [A]  time = 1.42837, size = 53, normalized size = 0.7 \[ \frac{2 \,{\left (32 \, x^{7} - 8 \, x^{5} - 3 \, x^{3} - 21 \, x\right )}}{231 \, e^{\frac{13}{2}}{\left (x + 1\right )}^{\frac{1}{4}} x^{\frac{13}{2}}{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

2/231*(32*x^7 - 8*x^5 - 3*x^3 - 21*x)/(e^(13/2)*(x + 1)^(1/4)*x^(13/2)*(-x + 1)^
(1/4))

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Fricas [A]  time = 0.221371, size = 57, normalized size = 0.75 \[ \frac{2 \,{\left (32 \, x^{6} - 8 \, x^{4} - 3 \, x^{2} - 21\right )}}{231 \, \sqrt{e x} e^{6}{\left (x + 1\right )}^{\frac{1}{4}} x^{5}{\left (-x + 1\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

2/231*(32*x^6 - 8*x^4 - 3*x^2 - 21)/(sqrt(e*x)*e^6*(x + 1)^(1/4)*x^5*(-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)**(13/2)/(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)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out