Moments of a quadratic and quartic exponential

Question

I am interested in the calculation of an integrals like \begin{equation} \int_{-\infty}^{+\infty} \mathrm d x \; e^{-ax^2-bx^4} x^n \end{equation} where $n\in\mathbb N$ and $a,b\in\mathbb R^+$. By expanding the quartic term in series, the result reduces to the series of momenta of a Gaussian integral. Is the whole series an analytic function? If so, how can it be computed?