Analytical solution for Bayesian linear regression

I have an informative prior from a constrained quadratic regression (c <0
where cx^2) on prior data.
The 'actual' data is constrained similarly.

I am looking to use the analytical method decribed here:

http://limnology.wisc.edu/regime/appendix_14jul03.pdf

to form the posterior parameters.

As the parameters have not been achieved by using (X'*X)^{-1}
directly, is there still an analytical solution or must I use simulation.

Thanks