3.896 \(\int \frac{(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=244 \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]

[Out]

-(e*Sqrt[e*x]*(1 - x^2)^(3/4))/2 - (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt
[e]*(1 - x^2)^(1/4))])/(4*Sqrt[2]) + (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sq
rt[e]*(1 - x^2)^(1/4))])/(4*Sqrt[2]) - (e^(3/2)*Log[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1
- x^2] - (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)])/(8*Sqrt[2]) + (e^(3/2)*Log[Sqrt[
e] + (Sqrt[e]*x)/Sqrt[1 - x^2] + (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)])/(8*Sqrt[2
])

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Rubi [A]  time = 0.414125, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(3/2)/((1 - x)^(1/4)*(1 + x)^(1/4)),x]

[Out]

-(e*Sqrt[e*x]*(1 - x^2)^(3/4))/2 - (e^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt
[e]*(1 - x^2)^(1/4))])/(4*Sqrt[2]) + (e^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sq
rt[e]*(1 - x^2)^(1/4))])/(4*Sqrt[2]) - (e^(3/2)*Log[Sqrt[e] + (Sqrt[e]*x)/Sqrt[1
- x^2] - (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)])/(8*Sqrt[2]) + (e^(3/2)*Log[Sqrt[
e] + (Sqrt[e]*x)/Sqrt[1 - x^2] + (Sqrt[2]*Sqrt[e*x])/(1 - x^2)^(1/4)])/(8*Sqrt[2
])

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Rubi in Sympy [A]  time = 36.4365, size = 204, normalized size = 0.84 \[ - \frac{\sqrt{2} e^{\frac{3}{2}} \log{\left (- \frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{16} + \frac{\sqrt{2} e^{\frac{3}{2}} \log{\left (\frac{\sqrt{2} \sqrt{e} \sqrt{e x}}{\sqrt [4]{- x^{2} + 1}} + \frac{e x}{\sqrt{- x^{2} + 1}} + e \right )}}{16} - \frac{\sqrt{2} e^{\frac{3}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{8} + \frac{\sqrt{2} e^{\frac{3}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{- x^{2} + 1}} \right )}}{8} - \frac{e \sqrt{e x} \left (- x^{2} + 1\right )^{\frac{3}{4}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)/(1-x)**(1/4)/(1+x)**(1/4),x)

[Out]

-sqrt(2)*e**(3/2)*log(-sqrt(2)*sqrt(e)*sqrt(e*x)/(-x**2 + 1)**(1/4) + e*x/sqrt(-
x**2 + 1) + e)/16 + sqrt(2)*e**(3/2)*log(sqrt(2)*sqrt(e)*sqrt(e*x)/(-x**2 + 1)**
(1/4) + e*x/sqrt(-x**2 + 1) + e)/16 - sqrt(2)*e**(3/2)*atan(1 - sqrt(2)*sqrt(e*x
)/(sqrt(e)*(-x**2 + 1)**(1/4)))/8 + sqrt(2)*e**(3/2)*atan(1 + sqrt(2)*sqrt(e*x)/
(sqrt(e)*(-x**2 + 1)**(1/4)))/8 - e*sqrt(e*x)*(-x**2 + 1)**(3/4)/2

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Mathematica [C]  time = 0.0326395, size = 39, normalized size = 0.16 \[ \frac{1}{2} e \sqrt{e x} \left (\, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};x^2\right )-\left (1-x^2\right )^{3/4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^(3/2)/((1 - x)^(1/4)*(1 + x)^(1/4)),x]

[Out]

(e*Sqrt[e*x]*(-(1 - x^2)^(3/4) + Hypergeometric2F1[1/4, 1/4, 5/4, x^2]))/2

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{1 \left ( ex \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(3/2)/(1-x)^(1/4)/(1+x)^(1/4),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e x\right )^{\frac{3}{2}}}{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.248055, size = 610, normalized size = 2.5 \[ -\frac{1}{2} \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )}}{2 \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )} + 2 \,{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}\right ) + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )}}{2 \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}}{\left (x^{2} - 1\right )} + 2 \,{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}\right ) + \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="fricas")

[Out]

-1/2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) + 1/4*sqrt(2)*(e^6)^(1/4)*arctan(s
qrt(2)*(e^6)^(1/4)*(x^2 - 1)/(2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) + sqrt(
2)*(e^6)^(1/4)*(x^2 - 1) + 2*(x^2 - 1)*sqrt(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) - s
qrt(2)*(e^6)^(1/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1
))/(x^2 - 1)))) + 1/4*sqrt(2)*(e^6)^(1/4)*arctan(sqrt(2)*(e^6)^(1/4)*(x^2 - 1)/(
2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(2)*(e^6)^(1/4)*(x^2 - 1) + 2*(
x^2 - 1)*sqrt(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) + sqrt(2)*(e^6)^(1/4)*sqrt(e*x)*e
*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1)))) + 1/16*sqrt(2)
*(e^6)^(1/4)*log(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) + sqrt(2)*(e^6)^(1/4)*sqrt(e*x
)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1)) - 1/16*sqrt(2
)*(e^6)^(1/4)*log(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) - sqrt(2)*(e^6)^(1/4)*sqrt(e*
x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1))

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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)**(3/2)/(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)^(3/2)/((x + 1)^(1/4)*(-x + 1)^(1/4)),x, algorithm="giac")

[Out]

Timed out