Bound on integral

I need lower and upper bounds (as tight as possible) on the following integral: $$\int_0^\infty x^n\exp\left(-\frac{(x-x_0)^2}2\right)\, dx$$

$n$ is a real number greater than $0$, and $x_0>0$. I am guessing the bounds will be of the form $B(n)x_0^n+C(n)$, but I don't have a way of showing it (it might not be true either).

EDIT: I guess for the cases $n>1$ and $n<1$, one can use the convexity/concavity of the $x^n$ and make estimates of the form $x_0^n+nxx_0^{n-1}\le(x+x_0)^n\le2^{n-1}(x^n+x_0^n)$.

EDIT: $n$ is not an integer. More importantly, I would like an actual bound on the integral in terms of elementary functions.

Solution 1:

$$f(x)=\exp\left(-\frac{(x-x_0)^2}{2}\right)$$ $$f'(x)=(x_0-x)f(x)$$ $$xf(x)=x_0f(x)-f'(x)$$

Hence

$$I_n(x_0)=\int_0^{\infty}x^nf(x)=x_0I_{n-1}(x_0)-\int_0^{\infty}x^{n-1}f'(x)= x_0I_{n-1}(x_0)+(n-1)I_{n-2}(x_0)$$

And $I_0(x_0)$ can be computed from the usual error function. So you get a quick way to compute the results.

Solution 2:

If I am not mistaken, for $n>0$, your integral is bounded by: $$2^{1/2 - n/2} e^{-x_0^2/2} \Gamma(1 + n)$$ This follows by actually calculating the integral which involves hypergeometric functions.