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

Optimal. Leaf size=93 \[ -\frac{e^3 \left (1-x^2\right )^{3/4}}{2 \sqrt{e x}}+\frac{e^2 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{2 \sqrt [4]{1-x^2}}-\frac{1}{3} e \left (1-x^2\right )^{3/4} (e x)^{3/2} \]

[Out]

-(e^3*(1 - x^2)^(3/4))/(2*Sqrt[e*x]) - (e*(e*x)^(3/2)*(1 - x^2)^(3/4))/3 + (e^2*
(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(2*(1 - x^2)^(1/4))

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Rubi [A]  time = 0.108526, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{e^3 \left (1-x^2\right )^{3/4}}{2 \sqrt{e x}}+\frac{e^2 \sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{2 \sqrt [4]{1-x^2}}-\frac{1}{3} e \left (1-x^2\right )^{3/4} (e x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

-(e^3*(1 - x^2)^(3/4))/(2*Sqrt[e*x]) - (e*(e*x)^(3/2)*(1 - x^2)^(3/4))/3 + (e^2*
(1 - x^(-2))^(1/4)*Sqrt[e*x]*EllipticE[ArcCsc[x]/2, 2])/(2*(1 - x^2)^(1/4))

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Rubi in Sympy [A]  time = 12.297, size = 76, normalized size = 0.82 \[ - \frac{e^{3} \left (- x^{2} + 1\right )^{\frac{3}{4}}}{2 \sqrt{e x}} + \frac{e^{2} \sqrt{e x} \sqrt [4]{1 - \frac{1}{x^{2}}} E\left (\frac{\operatorname{asin}{\left (\frac{1}{x} \right )}}{2}\middle | 2\right )}{2 \sqrt [4]{- x^{2} + 1}} - \frac{e \left (e x\right )^{\frac{3}{2}} \left (- x^{2} + 1\right )^{\frac{3}{4}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**3*(-x**2 + 1)**(3/4)/(2*sqrt(e*x)) + e**2*sqrt(e*x)*(1 - 1/x**2)**(1/4)*elli
ptic_e(asin(1/x)/2, 2)/(2*(-x**2 + 1)**(1/4)) - e*(e*x)**(3/2)*(-x**2 + 1)**(3/4
)/3

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Mathematica [C]  time = 0.0307382, size = 39, normalized size = 0.42 \[ -\frac{1}{3} e (e x)^{3/2} \left (\left (1-x^2\right )^{3/4}-\, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( ex \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{5}{2}}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{e x} e^{2} x^{2}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out