Consider the following integral:
$$\int\exp\left(ax+bx^2\right)x^{\eta}\mathrm{d}x$$
where $\eta\ge0$ is a real number, and $a$ and $b$ are also real numbers.
Can I express this integral in terms of some special functions, such as the incomplete Gamma function ($\Gamma(x;z_1,z_2) = \int_{z_1}^{z_2} t^{x-1}e^{-t}\mathrm{d}t$), or some other special function that is computable using standard numerical libraries?
If you complete the square using some constant value you multiply & divide from outside the integral, you could write it as the $\eta$ moment of the normal distribution : $$\int x^\eta e^{(x-\mu)^2/2\sigma^2} dx$$
However to evaluate this you will go through the Gamma function which seems unavoidable.
This is also called the raw moment.
Note : This only works if $\eta\in \mathbb N$.