I'm trying to find a solution or approximation of $$\int_0^r \exp(-g\sqrt{x^2+2bx+c})dx$$ where g > 0, c > 0 and r is from 0 to infinity.
I found that $$\frac{1.5}{b*\sqrt{c+1.5}+c+1.5}$$ Works well enough when g = 1, r = infinite and b < 5, however I also want to include r and g. Using a Taylor expansion doesn't result in anything that I can use.
Is there a solution to this integral and if not, is there a better approximation?