3.926 \(\int \frac{(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx\)

Optimal. Leaf size=328 \[ -\frac{(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}-\frac{3^{3/4} (2-e x)^{3/4} (e x+2)^{5/4}}{2 e}-\frac{5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{e x+2}}{2 e}-\frac{5\ 3^{3/4} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{5\ 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]

[Out]

(-5*3^(3/4)*(2 - e*x)^(3/4)*(2 + e*x)^(1/4))/(2*e) - (3^(3/4)*(2 - e*x)^(3/4)*(2
+ e*x)^(5/4))/(2*e) - ((2 - e*x)^(3/4)*(2 + e*x)^(9/4))/(3*3^(1/4)*e) + (5*3^(3
/4)*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(Sqrt[2]*e) - (5*3^(3
/4)*ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(Sqrt[2]*e) - (5*3^(3
/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) + (5*3^(3/4)*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)

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Rubi [A]  time = 0.527572, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}-\frac{3^{3/4} (2-e x)^{3/4} (e x+2)^{5/4}}{2 e}-\frac{5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{e x+2}}{2 e}-\frac{5\ 3^{3/4} \log \left (\frac{\sqrt{2-e x}-\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \log \left (\frac{\sqrt{2-e x}+\sqrt{2} \sqrt [4]{e x+2} \sqrt [4]{2-e x}+\sqrt{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{5\ 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{5\ 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]

Antiderivative was successfully verified.

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

[Out]

(-5*3^(3/4)*(2 - e*x)^(3/4)*(2 + e*x)^(1/4))/(2*e) - (3^(3/4)*(2 - e*x)^(3/4)*(2
+ e*x)^(5/4))/(2*e) - ((2 - e*x)^(3/4)*(2 + e*x)^(9/4))/(3*3^(1/4)*e) + (5*3^(3
/4)*ArcTan[1 - (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(Sqrt[2]*e) - (5*3^(3
/4)*ArcTan[1 + (Sqrt[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)])/(Sqrt[2]*e) - (5*3^(3
/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) + (5*3^(3/4)*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)

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Rubi in Sympy [A]  time = 53.6514, size = 304, normalized size = 0.93 \[ - \frac{\left (- 3 e x + 6\right )^{\frac{3}{4}} \left (e x + 2\right )^{\frac{9}{4}}}{9 e} - \frac{\left (- 3 e x + 6\right )^{\frac{3}{4}} \left (e x + 2\right )^{\frac{5}{4}}}{2 e} - \frac{5 \left (- 3 e x + 6\right )^{\frac{3}{4}} \sqrt [4]{e x + 2}}{2 e} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{4 e} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (\sqrt{3} + \frac{3 \sqrt{e x + 2}}{\sqrt{- 3 e x + 6}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{4 e} - \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{2 e} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{3} \sqrt [4]{e x + 2}}{\sqrt [4]{- 3 e x + 6}} \right )}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+2)**(5/2)/(-3*e**2*x**2+12)**(1/4),x)

[Out]

-(-3*e*x + 6)**(3/4)*(e*x + 2)**(9/4)/(9*e) - (-3*e*x + 6)**(3/4)*(e*x + 2)**(5/
4)/(2*e) - 5*(-3*e*x + 6)**(3/4)*(e*x + 2)**(1/4)/(2*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))/(4*e) + 5*sqrt(2)*3**(3/4)*log(sqrt(3) + 3*sqrt(e*x + 2)/sq
rt(-3*e*x + 6) + sqrt(2)*3**(3/4)*(e*x + 2)**(1/4)/(-3*e*x + 6)**(1/4))/(4*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
))/(2*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))/(2*e)

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Mathematica [C]  time = 0.0837946, size = 88, normalized size = 0.27 \[ \frac{\sqrt{e x+2} \left (2 e^3 x^3+13 e^2 x^2+90 \sqrt{2} \sqrt [4]{2-e x} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};\frac{1}{4} (e x+2)\right )+37 e x-142\right )}{6 e \sqrt [4]{12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2 + e*x]*(-142 + 37*e*x + 13*e^2*x^2 + 2*e^3*x^3 + 90*Sqrt[2]*(2 - e*x)^(1
/4)*Hypergeometric2F1[1/4, 1/4, 5/4, (2 + e*x)/4]))/(6*e*(12 - 3*e^2*x^2)^(1/4))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1 \left ( ex+2 \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x+2)^(5/2)/(-3*e^2*x^2+12)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + 2\right )}^{\frac{5}{2}}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(1/4),x, algorithm="maxima")

[Out]

integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(1/4), x)

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Fricas [A]  time = 0.261429, size = 888, normalized size = 2.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(1/4),x, algorithm="fricas")

[Out]

1/36*(180*27^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)*arctan(27^(1/4)*sqrt(2)*
(e^3*x^2 - 4*e)*(e^(-4))^(1/4)/(27^(1/4)*sqrt(2)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4)
+ 2*(e^2*x^2 - 4)*sqrt((27^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e
*(e^(-4))^(1/4) + 3*sqrt(3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) - 3*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))) + 180
*27^(1/4)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)*arctan(-27^(1/4)*sqrt(2)*(e^3*x^2
- 4*e)*(e^(-4))^(1/4)/(27^(1/4)*sqrt(2)*(e^3*x^2 - 4*e)*(e^(-4))^(1/4) - 2*(e^2
*x^2 - 4)*sqrt(-(27^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)*e*(e^(-4
))^(1/4) - 3*sqrt(3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) + 3*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))) - 45*27^(1/4
)*sqrt(2)*(e^2*x + 2*e)*(e^(-4))^(1/4)*log((27^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(
3/4)*sqrt(e*x + 2)*e*(e^(-4))^(1/4) + 3*sqrt(3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) -
3*sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e^2*x^2 - 4)) + 45*27^(1/4)*sqrt(2)*(e^2*x
+ 2*e)*(e^(-4))^(1/4)*log(-(27^(1/4)*sqrt(2)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x +
2)*e*(e^(-4))^(1/4) - 3*sqrt(3)*(e^4*x^2 - 4*e^2)*sqrt(e^(-4)) + 3*sqrt(-3*e^2*x
^2 + 12)*(e*x + 2))/(e^2*x^2 - 4)) - 2*(2*e^2*x^2 + 17*e*x + 71)*(-3*e^2*x^2 + 1
2)^(3/4)*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((e*x+2)**(5/2)/(-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. \[ \int \frac{{\left (e x + 2\right )}^{\frac{5}{2}}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(1/4),x, algorithm="giac")

[Out]

integrate((e*x + 2)^(5/2)/(-3*e^2*x^2 + 12)^(1/4), x)