I'm looking for a convenient upper bound on the integral $\int_y^\infty x^k \exp(-(x-\mu)^2/2) dx$ for (possibly large) positive integer k. This is equivalent to finding higher moments of a truncated normal distribution. A bound that works for non-integer k as well would be even better.
Of course "convenient" is in the eye of the beholder, but I'd like some sort of fairly simple expression that I can use in further calculations. For example, an upper bound of the form $f(x) \exp( -g(x))$ where where $f$ and $g$ are low-degree polynomials would be great. I'm more interested in simplicity of form than in obtaining the tightest possible bound.