Primitive of $ \exp(-a \times x- \frac bx)$

Question

I want to calculate the primitive of $\exp\left(-a\times x-\frac bx\right)$

ie. $\int_0^c \exp\left(-ax-\frac bx\right)$

I found only this post.

It is for $c=+\infty$

Answer 1

$\int_0^ce^{-ax-\frac{b}{x}}~dx$

$=\int_0^1e^{-acx-\frac{b}{cx}}~d(cx)$

$=c\int_0^1e^{-acx-\frac{b}{cx}}~dx$

$=c\int_\infty^1e^{-\frac{ac}{x}-\frac{bx}{c}}~d\left(\dfrac{1}{x}\right)$

$=c\int_1^\infty\dfrac{e^{-\frac{bx}{c}-\frac{ac}{x}}}{x^2}~dx$

$=cK_1\left(\dfrac{b}{c},ac\right)$ (according to https://core.ac.uk/download/pdf/81935301.pdf)