Evaluating $\int e^{ax} x^b (1-x)^c \mathrm{dx}$

Question

Edit: clarify question

The integrand looks kind of like a gamma density function, and kind of like a beta density function, so maybe it has a somewhat nice solution?

$$\int e^{ax} x^b (1-x)^c \mathrm{dx}$$

Wolfram alpha does not want to do it.

Answer 1

You can expand out the $(1-x)^c$ to get terms of the form $\int e^{ax}x^n dx$. Wolfram Alpha then gives a solution in terms of the incomplete Gamma function. This is a form that can be integrated by parts-set $dv=e^{ax}dx, u=x^n$ and step down the exponents, giving $\int e^{ax}x^n dx=\frac {x^n e^{ax}}a -\frac na \int x^{n-1}e^{ax}dx$