3.6 \(\int e^{d x} \left (a+b e^{c+d x}\right )^n \, dx\)

Optimal. Leaf size=32 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(b*d*E^c*(1 + n))

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Rubi [A]  time = 0.114404, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[E^(d*x)*(a + b*E^(c + d*x))^n,x]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(b*d*E^c*(1 + n))

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Rubi in Sympy [A]  time = 8.51479, size = 22, normalized size = 0.69 \[ \frac{\left (a + b e^{c + d x}\right )^{n + 1} e^{- c}}{b d \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(d*x)*(a+b*exp(d*x+c))**n,x)

[Out]

(a + b*exp(c + d*x))**(n + 1)*exp(-c)/(b*d*(n + 1))

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Mathematica [A]  time = 0.0438236, size = 31, normalized size = 0.97 \[ \frac{e^{-c} \left (a+b e^{c+d x}\right )^{n+1}}{b d n+b d} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(d*x)*(a + b*E^(c + d*x))^n,x]

[Out]

(a + b*E^(c + d*x))^(1 + n)/(E^c*(b*d + b*d*n))

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Maple [A]  time = 0.006, size = 31, normalized size = 1. \[{\frac{ \left ( a+b{{\rm e}^{dx}}{{\rm e}^{c}} \right ) ^{1+n}}{d{{\rm e}^{c}}b \left ( 1+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(d*x)*(a+b*exp(d*x+c))^n,x)

[Out]

1/d*(a+b*exp(d*x)*exp(c))^(1+n)/exp(c)/b/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253913, size = 49, normalized size = 1.53 \[ \frac{{\left (b e^{\left (d x\right )} + a e^{\left (-c\right )}\right )}{\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e^(d*x) + a*e^(-c))*(b*e^(d*x + c) + a)^n/(b*d*n + b*d)

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Sympy [A]  time = 164.253, size = 114, normalized size = 3.56 \[ \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge d = 0 \wedge n = -1 \\\frac{a^{n} e^{d x}}{d} & \text{for}\: b = 0 \\x \left (a + b e^{c}\right )^{n} & \text{for}\: d = 0 \\\frac{e^{- c} \log{\left (\frac{a}{b} + e^{c} e^{d x} \right )}}{b d} & \text{for}\: n = -1 \\\frac{a \left (a + b e^{c} e^{d x}\right )^{n}}{b d n e^{c} + b d e^{c}} + \frac{b \left (a + b e^{c} e^{d x}\right )^{n} e^{c} e^{d x}}{b d n e^{c} + b d e^{c}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(d*x)*(a+b*exp(d*x+c))**n,x)

[Out]

Piecewise((x/a, Eq(b, 0) & Eq(d, 0) & Eq(n, -1)), (a**n*exp(d*x)/d, Eq(b, 0)), (
x*(a + b*exp(c))**n, Eq(d, 0)), (exp(-c)*log(a/b + exp(c)*exp(d*x))/(b*d), Eq(n,
-1)), (a*(a + b*exp(c)*exp(d*x))**n/(b*d*n*exp(c) + b*d*exp(c)) + b*(a + b*exp(
c)*exp(d*x))**n*exp(c)*exp(d*x)/(b*d*n*exp(c) + b*d*exp(c)), True))

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GIAC/XCAS [A]  time = 0.22833, size = 41, normalized size = 1.28 \[ \frac{{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1} e^{\left (-c\right )}}{b d{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e^(d*x + c) + a)^(n + 1)*e^(-c)/(b*d*(n + 1))