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

Optimal. Leaf size=297 \[ -\frac{\sqrt [4]{3} (e x+2)^{3/4} (2-e x)^{5/4}}{2 e}+\frac{3 \sqrt [4]{3} (e x+2)^{3/4} \sqrt [4]{2-e x}}{2 e}+\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}-\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{3 \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]

[Out]

(3*3^(1/4)*(2 - e*x)^(1/4)*(2 + e*x)^(3/4))/(2*e) - (3^(1/4)*(2 - e*x)^(5/4)*(2
+ e*x)^(3/4))/(2*e) + (3*3^(1/4)*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^
(1/4)])/(Sqrt[2]*e) - (3*3^(1/4)*ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^
(1/4)])/(Sqrt[2]*e) + (3*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]])/(2*Sqrt[2]*e) - (3*3^(1/4)*Log[(S
qrt[2 - e*x] + Sqrt[2]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) + Sqrt[2 + e*x])/Sqrt[2 +
e*x]])/(2*Sqrt[2]*e)

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Rubi [A]  time = 0.514895, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{\sqrt [4]{3} (e x+2)^{3/4} (2-e x)^{5/4}}{2 e}+\frac{3 \sqrt [4]{3} (e x+2)^{3/4} \sqrt [4]{2-e x}}{2 e}+\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}-\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{3 \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]

Antiderivative was successfully verified.

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

[Out]

(3*3^(1/4)*(2 - e*x)^(1/4)*(2 + e*x)^(3/4))/(2*e) - (3^(1/4)*(2 - e*x)^(5/4)*(2
+ e*x)^(3/4))/(2*e) + (3*3^(1/4)*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^
(1/4)])/(Sqrt[2]*e) - (3*3^(1/4)*ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^
(1/4)])/(Sqrt[2]*e) + (3*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]])/(2*Sqrt[2]*e) - (3*3^(1/4)*Log[(S
qrt[2 - e*x] + Sqrt[2]*(2 - e*x)^(1/4)*(2 + e*x)^(1/4) + Sqrt[2 + e*x])/Sqrt[2 +
e*x]])/(2*Sqrt[2]*e)

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Rubi in Sympy [A]  time = 48.0118, size = 280, normalized size = 0.94 \[ \frac{\sqrt [4]{- 3 e x + 6} \left (e x + 2\right )^{\frac{7}{4}}}{2 e} - \frac{\sqrt [4]{- 3 e x + 6} \left (e x + 2\right )^{\frac{3}{4}}}{2 e} + \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (- \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{- 3 e x + 6}}{\sqrt [4]{e x + 2}} + \frac{\sqrt{- 3 e x + 6}}{\sqrt{e x + 2}} + \sqrt{3} \right )}}{4 e} - \frac{3 \sqrt{2} \sqrt [4]{3} \log{\left (\frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{- 3 e x + 6}}{\sqrt [4]{e x + 2}} + \frac{\sqrt{- 3 e x + 6}}{\sqrt{e x + 2}} + \sqrt{3} \right )}}{4 e} - \frac{3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{- 3 e x + 6}}{3 \sqrt [4]{e x + 2}} - 1 \right )}}{2 e} - \frac{3 \sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{- 3 e x + 6}}{3 \sqrt [4]{e x + 2}} + 1 \right )}}{2 e} \]

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)**(1/4)*(e*x + 2)**(7/4)/(2*e) - (-3*e*x + 6)**(1/4)*(e*x + 2)**(3/4
)/(2*e) + 3*sqrt(2)*3**(1/4)*log(-sqrt(2)*3**(1/4)*(-3*e*x + 6)**(1/4)/(e*x + 2)
**(1/4) + sqrt(-3*e*x + 6)/sqrt(e*x + 2) + sqrt(3))/(4*e) - 3*sqrt(2)*3**(1/4)*l
og(sqrt(2)*3**(1/4)*(-3*e*x + 6)**(1/4)/(e*x + 2)**(1/4) + sqrt(-3*e*x + 6)/sqrt
(e*x + 2) + sqrt(3))/(4*e) - 3*sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(3/4)*(-3*e*x +
6)**(1/4)/(3*(e*x + 2)**(1/4)) - 1)/(2*e) - 3*sqrt(2)*3**(1/4)*atan(sqrt(2)*3**(
3/4)*(-3*e*x + 6)**(1/4)/(3*(e*x + 2)**(1/4)) + 1)/(2*e)

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Mathematica [C]  time = 0.0754712, size = 86, normalized size = 0.29 \[ \frac{\sqrt{e x+2} \sqrt [4]{12-3 e^2 x^2} \left (e^2 x^2-\sqrt{2} (2-e x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{4} (e x+2)\right )-e x-2\right )}{2 e (e x-2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int \sqrt{ex+2}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}\, dx \]

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. \[ \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246971, size = 733, normalized size = 2.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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GIAC/XCAS [A]  time = 0.449487, 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)*sqrt(e*x + 2),x, algorithm="giac")

[Out]

Done