Evaluate $\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx$

Question

Is there a closed form for the integral $$\int_0^\infty x^{\lambda-1} \exp\left(-ax-b\sqrt x-\frac{c}{\sqrt x} - \frac{d}{x}\right) \: dx?$$

where $\lambda>0$, $a>0$, $d>0$ and where $b$, $c$ could be any real number. Any help would be greatly appreciated!

Answer 1

Hint:

$\int_0^\infty x^{\lambda-1}e^{-ax-b\sqrt x-\frac{c}{\sqrt x}-\frac{d}{x}}~dx$

$=\int_0^\infty(\sqrt x)^{\lambda-1}e^{-ax^2-bx-\frac{c}{x}-\frac{d}{x^2}}~d(\sqrt x)$

$=\dfrac{1}{2}\int_0^\infty x^{\frac{\lambda}{2}-1}e^{-ax^2-bx-\frac{c}{x}-\frac{d}{x^2}}~dx$

Answer 2

Due to Glasser's master theorem, in some non-trivial cases the integral can be represented in terms of Bessel $K$ functions, but I won't bet a penny on the existence of a nice closed form for any value of $(a,b,c,d,\lambda)$ in some open set.