3.905 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{11/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{8 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}} \]

[Out]

(-2*(1 - x^2)^(3/4))/(9*e*(e*x)^(9/2)) - (4*(1 - x^2)^(3/4))/(15*e^3*(e*x)^(5/2)
) - (8*(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(15*e^6*(1 - x^2)
^(1/4))

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Rubi [A]  time = 0.111424, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{8 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{15 e^6 \sqrt [4]{1-x^2}}-\frac{4 \left (1-x^2\right )^{3/4}}{15 e^3 (e x)^{5/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{9 e (e x)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((1 - x)^(1/4)*(e*x)^(11/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(9*e*(e*x)^(9/2)) - (4*(1 - x^2)^(3/4))/(15*e^3*(e*x)^(5/2)
) - (8*(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(15*e^6*(1 - x^2)
^(1/4))

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Rubi in Sympy [A]  time = 12.7473, size = 83, normalized size = 0.87 \[ - \frac{2 \left (- x^{2} + 1\right )^{\frac{3}{4}}}{9 e \left (e x\right )^{\frac{9}{2}}} - \frac{4 \left (- x^{2} + 1\right )^{\frac{3}{4}}}{15 e^{3} \left (e x\right )^{\frac{5}{2}}} - \frac{8 \sqrt{e x} \sqrt [4]{1 - \frac{1}{x^{2}}} E\left (\frac{\operatorname{asin}{\left (\frac{1}{x} \right )}}{2}\middle | 2\right )}{15 e^{6} \sqrt [4]{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/4)/(e*x)**(11/2)/(1+x)**(1/4),x)

[Out]

-2*(-x**2 + 1)**(3/4)/(9*e*(e*x)**(9/2)) - 4*(-x**2 + 1)**(3/4)/(15*e**3*(e*x)**
(5/2)) - 8*sqrt(e*x)*(1 - 1/x**2)**(1/4)*elliptic_e(asin(1/x)/2, 2)/(15*e**6*(-x
**2 + 1)**(1/4))

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Mathematica [C]  time = 0.0476439, size = 60, normalized size = 0.63 \[ -\frac{2 \sqrt{e x} \left (8 x^6 \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right )+\left (1-x^2\right )^{3/4} \left (12 x^4+6 x^2+5\right )\right )}{45 e^6 x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((1 - x)^(1/4)*(e*x)^(11/2)*(1 + x)^(1/4)),x]

[Out]

(-2*Sqrt[e*x]*((1 - x^2)^(3/4)*(5 + 6*x^2 + 12*x^4) + 8*x^6*Hypergeometric2F1[1/
4, 3/4, 7/4, x^2]))/(45*e^6*x^5)

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [4]{1-x}}} \left ( ex \right ) ^{-{\frac{11}{2}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x)

[Out]

int(1/(1-x)^(1/4)/(e*x)^(11/2)/(1+x)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (e x\right )^{\frac{11}{2}}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{e x} e^{5}{\left (x + 1\right )}^{\frac{1}{4}} x^{5}{\left (-x + 1\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(e*x)*e^5*(x + 1)^(1/4)*x^5*(-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)**(11/2)/(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)^(11/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out