Integral of Airy function containing a second order polynom

I wonder if there is an analytic expression for

$ int_{-infty}^{infty} mathrm{Ai}(a x^2 + b x + c) dx$

As a Bonus:

$ int_{-infty}^{infty} e^{- d x^2} mathrm{Ai}(a x^2 + b x + c) dx$

where $a,b,c$ are Real and $d < 0$.