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

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

[Out]

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

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Rubi [A]  time = 0.349381, 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 \[ -\frac{(2-e x)^{3/4} \sqrt [4]{e x+2}}{\sqrt [4]{3} e}-\frac{\log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} \sqrt [4]{3} e}+\frac{\log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} \sqrt [4]{3} e}+\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 42.6031, size = 252, normalized size = 0.98 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{4}} \sqrt [4]{e x + 2}}{3 e} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{6 e} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{6 e} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{3 e} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0603923, size = 68, normalized size = 0.26 \[ \frac{\sqrt{e x+2} \left (2 \sqrt{2} \sqrt [4]{2-e x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{4} (e x+2)\right )+e x-2\right )}{e \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{1\sqrt{ex+2}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.25397, size = 880, normalized size = 3.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(12*sqrt(2)*(1/3)^(1/4)*(e^2*x + 2*e)*(e^(-4))^(1/4)*arctan(3*sqrt(2)*(1/3)^
(1/4)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4)/(2*sqrt(3)*(e^2*x^2 - 4)*sqrt((sqrt(2)*(1/3
)^(1/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) + 3*sqrt(1/3)*(e^
4*x^2 - 4*e^2)*sqrt(e^(-4)) - sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e^2*x^2 - 4)) +
3*sqrt(2)*(1/3)^(1/4)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4) + 2*(-3*e^2*x^2 + 12)^(3/4)
*sqrt(e*x + 2))) + 12*sqrt(2)*(1/3)^(1/4)*(e^2*x + 2*e)*(e^(-4))^(1/4)*arctan(3*
sqrt(2)*(1/3)^(1/4)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4)/(2*sqrt(3)*(e^2*x^2 - 4)*sqrt
(-(sqrt(2)*(1/3)^(1/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) -
3*sqrt(1/3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) + sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e
^2*x^2 - 4)) - 3*sqrt(2)*(1/3)^(1/4)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4) + 2*(-3*e^2*
x^2 + 12)^(3/4)*sqrt(e*x + 2))) - 3*sqrt(2)*(1/3)^(1/4)*(e^2*x + 2*e)*(e^(-4))^(
1/4)*log(3*(sqrt(2)*(1/3)^(1/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e*(e^(-4))
^(1/4) + 3*sqrt(1/3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) - sqrt(-3*e^2*x^2 + 12)*(e*x
+ 2))/(e^2*x^2 - 4)) + 3*sqrt(2)*(1/3)^(1/4)*(e^2*x + 2*e)*(e^(-4))^(1/4)*log(-
3*(sqrt(2)*(1/3)^(1/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) -
3*sqrt(1/3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) + sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e
^2*x^2 - 4)) - 2*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2*x + 2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3^{\frac{3}{4}} \int \frac{\sqrt{e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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