I have questions about the integral $$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$ for $a,b,c>0$.

What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In particular, what is the asymptotics of $F(a,a^p,a^q)$ for positive $p,q$ when $a$ tends to $0$?

When $c=0$ this integral is expressible in terms of a modified Bessel function. Is $F(a,b,c)$ expressible in terms of special functions with known asymptotics in more general situations?

Are there any lower bounds for $F(a,b,c)$ in terms of elementary functions which are sharper than the bound given by Jensen's inequality? What would be good strategies to obtain them?