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

Optimal. Leaf size=257 \[ \frac{\sqrt [4]{3} \sqrt [4]{2-e x} (e x+2)^{3/4}}{e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

[Out]

(3^(1/4)*(2 - e*x)^(1/4)*(2 + e*x)^(3/4))/e + (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 = 0.41337, antiderivative size = 257, 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 \[ \frac{\sqrt [4]{3} \sqrt [4]{2-e x} (e x+2)^{3/4}}{e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

(3^(1/4)*(2 - e*x)^(1/4)*(2 + e*x)^(3/4))/e + (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 = 42.4648, size = 246, normalized size = 0.96 \[ \frac{\sqrt [4]{- 3 e x + 6} \left (e x + 2\right )^{\frac{3}{4}}}{e} + \frac{\sqrt{2} \sqrt [4]{3} \log{\left (- \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{- 3 e x + 6}}{\sqrt [4]{e x + 2}} + \frac{\sqrt{- 3 e x + 6}}{\sqrt{e x + 2}} + \sqrt{3} \right )}}{2 e} - \frac{\sqrt{2} \sqrt [4]{3} \log{\left (\frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{- 3 e x + 6}}{\sqrt [4]{e x + 2}} + \frac{\sqrt{- 3 e x + 6}}{\sqrt{e x + 2}} + \sqrt{3} \right )}}{2 e} - \frac{\sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{- 3 e x + 6}}{3 \sqrt [4]{e x + 2}} - 1 \right )}}{e} - \frac{\sqrt{2} \sqrt [4]{3} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{- 3 e x + 6}}{3 \sqrt [4]{e x + 2}} + 1 \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.053825, size = 81, normalized size = 0.32 \[ \frac{\sqrt{e x+2} \sqrt [4]{4-e^2 x^2} \left (-\sqrt{2} (2-e x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{4} (e x+2)\right )+3 e x-6\right )}{3^{3/4} e (e x-2)} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + e*x]*(4 - e^2*x^2)^(1/4)*(-6 + 3*e*x - Sqrt[2]*(2 - e*x)^(3/4)*Hyperge
ometric2F1[3/4, 3/4, 7/4, (2 + e*x)/4]))/(3^(3/4)*e*(-2 + e*x))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{1\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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 = 0.249761, size = 725, normalized size = 2.82 \[ \frac{4 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \arctan \left (\frac{3^{\frac{1}{4}} \sqrt{2}{\left (e^{2} x + 2 \, e\right )} \frac{1}{e^{4}}^{\frac{1}{4}}}{3^{\frac{1}{4}} \sqrt{2}{\left (e^{2} x + 2 \, e\right )} \frac{1}{e^{4}}^{\frac{1}{4}} + 2 \,{\left (e x + 2\right )} \sqrt{\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} + \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} + \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}} + 2 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}\right ) + 4 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \arctan \left (-\frac{3^{\frac{1}{4}} \sqrt{2}{\left (e^{2} x + 2 \, e\right )} \frac{1}{e^{4}}^{\frac{1}{4}}}{3^{\frac{1}{4}} \sqrt{2}{\left (e^{2} x + 2 \, e\right )} \frac{1}{e^{4}}^{\frac{1}{4}} - 2 \,{\left (e x + 2\right )} \sqrt{-\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} - \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} - \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}} - 2 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}\right ) - 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \log \left (\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} + \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} + \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \log \left (-\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} - \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} - \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 2 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{2 \, e} \]

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]

1/2*(4*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))) + 4*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^(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^(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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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="giac")

[Out]

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