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

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

[Out]

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

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Rubi [A]  time = 0.430861, 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 \[ -\frac{(2-e x)^{3/4} (e x+2)^{5/4}}{2 \sqrt [4]{3} e}-\frac{5 (2-e x)^{3/4} \sqrt [4]{e x+2}}{2 \sqrt [4]{3} e}-\frac{5 \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} \sqrt [4]{3} e}+\frac{5 \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} \sqrt [4]{3} e}+\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} \sqrt [4]{3} e}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} \sqrt [4]{3} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.416, size = 282, normalized size = 0.95 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{4}} \left (e x + 2\right )^{\frac{5}{4}}}{6 e} - \frac{5 \left (- 3 e x + 6\right )^{\frac{3}{4}} \sqrt [4]{e x + 2}}{6 e} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{12 e} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{12 e} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{6 e} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{6 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-3*e*x + 6)**(3/4)*(e*x + 2)**(5/4)/(6*e) - 5*(-3*e*x + 6)**(3/4)*(e*x + 2)**(
1/4)/(6*e) - 5*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))/(12*e) + 5*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))/(12*e) - 5*sqrt(2)*3**(3/4)*atan(1 - sqrt(2)*3**(1/4
)*(e*x + 2)**(1/4)/(-3*e*x + 6)**(1/4))/(6*e) + 5*sqrt(2)*3**(3/4)*atan(1 + sqrt
(2)*3**(1/4)*(e*x + 2)**(1/4)/(-3*e*x + 6)**(1/4))/(6*e)

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Mathematica [C]  time = 0.0713482, size = 79, normalized size = 0.27 \[ \frac{\sqrt{e x+2} \left (e^2 x^2+10 \sqrt{2} \sqrt [4]{2-e x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{4} (e x+2)\right )+5 e x-14\right )}{2 e \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + e*x]*(-14 + 5*e*x + e^2*x^2 + 10*Sqrt[2]*(2 - e*x)^(1/4)*Hypergeometri
c2F1[1/4, 1/4, 5/4, (2 + e*x)/4]))/(2*e*(12 - 3*e^2*x^2)^(1/4))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{1 \left ( ex+2 \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.260304, size = 887, normalized size = 2.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(60*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))) + 60*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)*sqr
t(-(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))) - 15*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)) + 15*sqrt(2)*(1/3)^(1/4)*(e^2*x + 2*e)*(e^(-4))^(1/4)*lo
g(-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)*(e*x + 7)*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((e*x+2)**(3/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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