3.921 \(\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.411857, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 45.9498, size = 248, normalized size = 0.96 \[ - \frac{4 \sqrt [4]{- 3 e x + 6}}{e \sqrt [4]{e x + 2}} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{2 e} + \frac{\sqrt{2} \sqrt [4]{3} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{2 e} + \frac{\sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{e} - \frac{\sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(3/2),x)

[Out]

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

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Mathematica [C]  time = 0.0746053, size = 85, normalized size = 0.33 \[ \frac{\sqrt [4]{4-e^2 x^2} \left (\sqrt{2} (2-e x)^{3/4} (e x+2) \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{4} (e x+2)\right )-12 e x+24\right )}{3^{3/4} e (e x-2) \sqrt{e x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(3/2),x]

[Out]

((4 - e^2*x^2)^(1/4)*(24 - 12*e*x + Sqrt[2]*(2 - e*x)^(3/4)*(2 + e*x)*Hypergeome
tric2F1[3/4, 3/4, 7/4, (2 + e*x)/4]))/(3^(3/4)*e*(-2 + e*x)*Sqrt[2 + e*x])

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)

[Out]

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(3/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(3/2), x)

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Fricas [A]  time = 0.264906, size = 779, normalized size = 3.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(4*3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)*arctan(3^(1/4)*sqrt(2)*(e^2
*x + 2*e)*(e^(-4))^(1/4)/(3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4) + 2*(e*x
+ 2)*sqrt((3^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4
) + sqrt(3)*(e^3*x + 2*e^2)*sqrt(e^(-4)) + sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) + 2
*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))) + 4*3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-
4))^(1/4)*arctan(-3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)/(3^(1/4)*sqrt(2)*
(e^2*x + 2*e)*(e^(-4))^(1/4) - 2*(e*x + 2)*sqrt(-(3^(1/4)*sqrt(2)*(-3*e^2*x^2 +
12)^(1/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) - sqrt(3)*(e^3*x + 2*e^2)*sqrt(e^(-4))
- sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) - 2*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)))
- 3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)*log((3^(1/4)*sqrt(2)*(-3*e^2*x^2
+ 12)^(1/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) + sqrt(3)*(e^3*x + 2*e^2)*sqrt(e^(-4)
) + sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) + 3^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(
1/4)*log(-(3^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4
) - sqrt(3)*(e^3*x + 2*e^2)*sqrt(e^(-4)) - sqrt(-3*e^2*x^2 + 12))/(e*x + 2)) + 8
*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2))/(e^2*x + 2*e)

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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((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.42914, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(3/2),x, algorithm="giac")

[Out]

Done