$ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx $

That is not a gaussian integral/integrand. That is: $$\int\limits_{c}^{\infty}e^{-a(x+b)^2}\,\mathrm{d}x=-\dfrac{\sqrt{{\pi}}\left(\operatorname{erf}\left(\sqrt{a}\left(c+b\right)\right)-1\right)}{2\sqrt{a}}$$ But if that's really what you meant, I don't think there is a closed form for this. If you choose $b=0$, one can write $$\int\limits_{c}^{\infty}\frac{e^{-ax^2}}{x}\,\mathrm{d}x= \dfrac{\operatorname{\Gamma}\left(0,ac^2\right)}{2}.$$ If we were alloud to set $b=d/x$ we could also look at the integral $$\int\limits_{c}^{\infty}\frac{e^{-ax^2+b}}{x}\,\mathrm{d}x= \dfrac{\operatorname{\Gamma}\left(0,ac^2\right)\mathrm{e}^d}{2}.$$