I'm calculating an expectation of the form $$\mathbb E\left[1\over aX+b\right]$$where $X\sim\chi^2_1(0)$ obeys an central chi squared distribution with 1 degree of freedom. The integral formula is $$\frac{1}{\sqrt{2\pi}}\int_0^\infty{x^{-1/2}e^{-x/2}\over ax+b}dx$$and I can not solve this integral. Also I tried the integral calculator but it still offers no solution.
How to calculate this integral? I'm longing toward the answer!