3.29 \(\int e^{-n x} \left (a+b e^{n x}\right )^3 \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^3 e^{-n x}}{n}+3 a^2 b x+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0812827, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a^3 e^{-n x}}{n}+3 a^2 b x+\frac{3 a b^2 e^{n x}}{n}+\frac{b^3 e^{2 n x}}{2 n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(n*x))^3/E^(n*x),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3*exp(-n*x)/n + 3*a**2*b*log(exp(n*x))/n + 3*a*b**2*exp(n*x)/n + b**3*Integr
al(x, (x, exp(n*x)))/n

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Mathematica [A]  time = 0.0264373, size = 48, normalized size = 0.92 \[ \frac{-2 a^3 e^{-n x}+6 a^2 b n x+6 a b^2 e^{n x}+b^3 e^{2 n x}}{2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^(n*x))^3/E^(n*x),x]

[Out]

((-2*a^3)/E^(n*x) + 6*a*b^2*E^(n*x) + b^3*E^(2*n*x) + 6*a^2*b*n*x)/(2*n)

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Maple [A]  time = 0.01, size = 57, normalized size = 1.1 \[{\frac{{b}^{3} \left ({{\rm e}^{nx}} \right ) ^{2}}{2\,n}}+3\,{\frac{a{b}^{2}{{\rm e}^{nx}}}{n}}-{\frac{{a}^{3}}{n{{\rm e}^{nx}}}}+3\,{\frac{{a}^{2}b\ln \left ({{\rm e}^{nx}} \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*exp(n*x))^3/exp(n*x),x)

[Out]

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

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Maxima [A]  time = 0.805785, size = 63, normalized size = 1.21 \[ 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244733, size = 65, normalized size = 1.25 \[ \frac{{\left (6 \, a^{2} b n x e^{\left (n x\right )} + b^{3} e^{\left (3 \, n x\right )} + 6 \, a b^{2} e^{\left (2 \, n x\right )} - 2 \, a^{3}\right )} e^{\left (-n x\right )}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*a^2*b*n*x*e^(n*x) + b^3*e^(3*n*x) + 6*a*b^2*e^(2*n*x) - 2*a^3)*e^(-n*x)/n

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Sympy [A]  time = 0.37586, size = 73, normalized size = 1.4 \[ 3 a^{2} b x + \begin{cases} \frac{- 2 a^{3} n^{2} e^{- n x} + 6 a b^{2} n^{2} e^{n x} + b^{3} n^{2} e^{2 n x}}{2 n^{3}} & \text{for}\: 2 n^{3} \neq 0 \\x \left (a^{3} + 3 a b^{2} + b^{3}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*exp(n*x))**3/exp(n*x),x)

[Out]

3*a**2*b*x + Piecewise(((-2*a**3*n**2*exp(-n*x) + 6*a*b**2*n**2*exp(n*x) + b**3*
n**2*exp(2*n*x))/(2*n**3), Ne(2*n**3, 0)), (x*(a**3 + 3*a*b**2 + b**3), True))

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GIAC/XCAS [A]  time = 0.224016, size = 63, normalized size = 1.21 \[ 3 \, a^{2} b x + \frac{b^{3} e^{\left (2 \, n x\right )}}{2 \, n} + \frac{3 \, a b^{2} e^{\left (n x\right )}}{n} - \frac{a^{3} e^{\left (-n x\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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