# 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 $$d < 0$$.