Does $\int \frac{exp( -b\sqrt{a+x})}{\sqrt{x}} dx$ have a solution?

Question

Is there a solution for the following integral:

$$ \int_0^{\infty} \frac{\exp( -b\sqrt{a+x})}{\sqrt{x}} dx $$

where $a$ and $b$ are constants. If it is not, what is the best approximation? Especially in the limit as $b \to \infty$.

In case of $a=0$, there is an analytic solution.

Answer 1

Letting $x=a\sinh^2t$, we have $I=2\sqrt a~K_1\Big(b\sqrt a\Big):$ see Bessel function for more information.