3.5 \(\int e^x \left (a+b e^x\right )^n \, dx\)

Optimal. Leaf size=20 \[ \frac{\left (a+b e^x\right )^{n+1}}{b (n+1)} \]

[Out]

(a + b*E^x)^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0373475, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (a+b e^x\right )^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[E^x*(a + b*E^x)^n,x]

[Out]

(a + b*E^x)^(1 + n)/(b*(1 + n))

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Rubi in Sympy [A]  time = 6.26518, size = 14, normalized size = 0.7 \[ \frac{\left (a + b e^{x}\right )^{n + 1}}{b \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*exp(x))**(n + 1)/(b*(n + 1))

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Mathematica [A]  time = 0.022402, size = 19, normalized size = 0.95 \[ \frac{\left (a+b e^x\right )^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]  Integrate[E^x*(a + b*E^x)^n,x]

[Out]

(a + b*E^x)^(1 + n)/(b + b*n)

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Maple [A]  time = 0.002, size = 20, normalized size = 1. \[{\frac{ \left ( a+b{{\rm e}^{x}} \right ) ^{1+n}}{b \left ( 1+n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(x)*(a+b*exp(x))^n,x)

[Out]

(a+b*exp(x))^(1+n)/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^x + a)^n*e^x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247985, size = 30, normalized size = 1.5 \[ \frac{{\left (b e^{x} + a\right )}{\left (b e^{x} + a\right )}^{n}}{b n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e^x + a)*(b*e^x + a)^n/(b*n + b)

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Sympy [A]  time = 2.7679, size = 56, normalized size = 2.8 \[ \begin{cases} \frac{e^{x}}{a} & \text{for}\: b = 0 \wedge n = -1 \\a^{n} e^{x} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + e^{x} \right )}}{b} & \text{for}\: n = -1 \\\frac{a \left (a + b e^{x}\right )^{n}}{b n + b} + \frac{b \left (a + b e^{x}\right )^{n} e^{x}}{b n + b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23383, size = 26, normalized size = 1.3 \[ \frac{{\left (b e^{x} + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e^x + a)^(n + 1)/(b*(n + 1))