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.
$\int e^{\frac{a}{x}+bx}x^\eta~dx$
$=\int_0^xx^\eta e^{\frac{a}{x}+bx}~dx+C$
$=\int_0^xt^\eta e^{\frac{a}{t}+bt}~dt+C$
$=\int_0^1(xt)^\eta e^{\frac{a}{xt}+bxt}~d(xt)+C$
$=x^{\eta+1}\int_0^1t^\eta e^{\frac{a}{xt}+bxt}~dt+C$
$=x^{\eta+1}\int_\infty^1\left(\dfrac{1}{t}\right)^\eta e^{\frac{at}{x}+\frac{bx}{t}}~d\left(\dfrac{1}{t}\right)+C$
$=x^{\eta+1}\int_1^\infty\dfrac{e^{\frac{at}{x}+\frac{bx}{t}}}{t^{\eta+2}}dt+C$
$=x^{\eta+1}K_{\eta+1}\left(-\dfrac{a}{x},-bx\right)+C$ (according to https://core.ac.uk/download/pdf/81935301.pdf)
