Fourier transform of $\exp( ic_1x^3 -c_2x^2 )$ [when $c_2=0$ it is the Airy function]

Question

I'm wondering if anybody has a clue on the following integral $$ \int_0^\infty \exp[i c_1 x^3 - c_2 x^2 + i x p] \mathrm{d}x$$ basically the Fourier transform of $\exp[i c_1 x^3 - c_2 x^2 ]$. I am interested in the case $c_1 > 0, \: c_2 >0$.

I know that when $c_2 = 0$ this reduced to the standard Airy function of $p$ (modulo some constants of course).

I've tried looking at integral tables but without luck...

Thank you for any possible advice :)