Small wonder you are having difficulty. That is non-trivial, to be sure. I have to ask a few things.
1) Why do you think you need to compute this integral?
2) Why do you think the answer exists in any convenient form?
3) Do you get to use Complex Analysis? Actually, is this the original intent?
4) Are you comfortable with Erf?
5) If you are far enough along in mathematics to be contemplating such an integral, you should have learned to pay much better attention to your notation. It is not clear at all what your intent is in that exponent. I took it to mean a quadratic in x all divided by 4z^2. That is absolutely NOT what you have writtten. What did you really mean?