3.679 \(\int \left (a+b e^x\right )^3 \, dx\)

Optimal. Leaf size=40 \[ a^3 x+3 a^2 b e^x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]

[Out]

3*a^2*b*E^x + (3*a*b^2*E^(2*x))/2 + (b^3*E^(3*x))/3 + a^3*x

_______________________________________________________________________________________

Rubi [A]  time = 0.0393371, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ a^3 x+3 a^2 b e^x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^x)^3,x]

[Out]

3*a^2*b*E^x + (3*a*b^2*E^(2*x))/2 + (b^3*E^(3*x))/3 + a^3*x

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} \log{\left (e^{x} \right )} + 3 a^{2} b e^{x} + 3 a b^{2} \int ^{e^{x}} x\, dx + \frac{b^{3} e^{3 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*exp(x))**3,x)

[Out]

a**3*log(exp(x)) + 3*a**2*b*exp(x) + 3*a*b**2*Integral(x, (x, exp(x))) + b**3*ex
p(3*x)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0101908, size = 40, normalized size = 1. \[ a^3 x+3 a^2 b e^x+\frac{3}{2} a b^2 e^{2 x}+\frac{1}{3} b^3 e^{3 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^x)^3,x]

[Out]

3*a^2*b*E^x + (3*a*b^2*E^(2*x))/2 + (b^3*E^(3*x))/3 + a^3*x

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 36, normalized size = 0.9 \[{\frac{{b}^{3} \left ({{\rm e}^{x}} \right ) ^{3}}{3}}+{\frac{3\,a{b}^{2} \left ({{\rm e}^{x}} \right ) ^{2}}{2}}+3\,{a}^{2}b{{\rm e}^{x}}+{a}^{3}\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*exp(x))^3,x)

[Out]

1/3*b^3*exp(x)^3+3/2*a*b^2*exp(x)^2+3*a^2*b*exp(x)+a^3*ln(exp(x))

_______________________________________________________________________________________

Maxima [A]  time = 0.746033, size = 45, normalized size = 1.12 \[ a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^3,x, algorithm="maxima")

[Out]

a^3*x + 1/3*b^3*e^(3*x) + 3/2*a*b^2*e^(2*x) + 3*a^2*b*e^x

_______________________________________________________________________________________

Fricas [A]  time = 0.257728, size = 45, normalized size = 1.12 \[ a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^3,x, algorithm="fricas")

[Out]

a^3*x + 1/3*b^3*e^(3*x) + 3/2*a*b^2*e^(2*x) + 3*a^2*b*e^x

_______________________________________________________________________________________

Sympy [A]  time = 0.13468, size = 37, normalized size = 0.92 \[ a^{3} x + 3 a^{2} b e^{x} + \frac{3 a b^{2} e^{2 x}}{2} + \frac{b^{3} e^{3 x}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*exp(x))**3,x)

[Out]

a**3*x + 3*a**2*b*exp(x) + 3*a*b**2*exp(2*x)/2 + b**3*exp(3*x)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220614, size = 45, normalized size = 1.12 \[ a^{3} x + \frac{1}{3} \, b^{3} e^{\left (3 \, x\right )} + \frac{3}{2} \, a b^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^3,x, algorithm="giac")

[Out]

a^3*x + 1/3*b^3*e^(3*x) + 3/2*a*b^2*e^(2*x) + 3*a^2*b*e^x