Can the following integral:
$$\int\exp\left(\frac{a}{x}+bx\right)x^{\eta}\mathrm{d}x$$
be expressed in terms of special functions, like the Gaussian hyper geometric function $_2F_1(a,b;c;z)$, or the incomplete Gamma function? Ideally, use only special functions that are available as routines in standard numerical libraries (such as GSL or Numerical Recipes).
Here $\eta\ge0$ is a real number, and $a$ and $b$ are also real numbers.