3.680 \(\int \left (a+b e^x\right )^4 \, dx\)

Optimal. Leaf size=53 \[ a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*x

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Rubi [A]  time = 0.050668, antiderivative size = 53, 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 \[ a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^x)^4,x]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} \log{\left (e^{x} \right )} + 4 a^{3} b e^{x} + 6 a^{2} b^{2} \int ^{e^{x}} x\, dx + \frac{4 a b^{3} e^{3 x}}{3} + \frac{b^{4} e^{4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*exp(x))**4,x)

[Out]

a**4*log(exp(x)) + 4*a**3*b*exp(x) + 6*a**2*b**2*Integral(x, (x, exp(x))) + 4*a*
b**3*exp(3*x)/3 + b**4*exp(4*x)/4

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Mathematica [A]  time = 0.0123917, size = 53, normalized size = 1. \[ a^4 x+4 a^3 b e^x+3 a^2 b^2 e^{2 x}+\frac{4}{3} a b^3 e^{3 x}+\frac{1}{4} b^4 e^{4 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^x)^4,x]

[Out]

4*a^3*b*E^x + 3*a^2*b^2*E^(2*x) + (4*a*b^3*E^(3*x))/3 + (b^4*E^(4*x))/4 + a^4*x

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Maple [A]  time = 0.003, size = 48, normalized size = 0.9 \[{\frac{{b}^{4} \left ({{\rm e}^{x}} \right ) ^{4}}{4}}+{\frac{4\,a{b}^{3} \left ({{\rm e}^{x}} \right ) ^{3}}{3}}+3\,{a}^{2}{b}^{2} \left ({{\rm e}^{x}} \right ) ^{2}+4\,{a}^{3}b{{\rm e}^{x}}+{a}^{4}\ln \left ({{\rm e}^{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*exp(x))^4,x)

[Out]

1/4*b^4*exp(x)^4+4/3*a*b^3*exp(x)^3+3*a^2*b^2*exp(x)^2+4*a^3*b*exp(x)+a^4*ln(exp
(x))

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Maxima [A]  time = 0.762892, size = 61, normalized size = 1.15 \[ a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^4,x, algorithm="maxima")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x

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Fricas [A]  time = 0.241483, size = 61, normalized size = 1.15 \[ a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^4,x, algorithm="fricas")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x

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Sympy [A]  time = 0.166968, size = 51, normalized size = 0.96 \[ a^{4} x + 4 a^{3} b e^{x} + 3 a^{2} b^{2} e^{2 x} + \frac{4 a b^{3} e^{3 x}}{3} + \frac{b^{4} e^{4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*exp(x))**4,x)

[Out]

a**4*x + 4*a**3*b*exp(x) + 3*a**2*b**2*exp(2*x) + 4*a*b**3*exp(3*x)/3 + b**4*exp
(4*x)/4

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GIAC/XCAS [A]  time = 0.301992, size = 61, normalized size = 1.15 \[ a^{4} x + \frac{1}{4} \, b^{4} e^{\left (4 \, x\right )} + \frac{4}{3} \, a b^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, a^{3} b e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^x + a)^4,x, algorithm="giac")

[Out]

a^4*x + 1/4*b^4*e^(4*x) + 4/3*a*b^3*e^(3*x) + 3*a^2*b^2*e^(2*x) + 4*a^3*b*e^x