I'm not sure what method this system uses for definite integration, but it is likely one of Simpsons, Gaussian, and Clenshaw-Curtis. These are numerical methods that, for the best accuracy, require lots of steps of evaluation. Depending on the error bounds you want, it could take a while.

Edit: also the nonlinear square root term may cause difficulty.

People might find this interesting: Maple uses Clenshaw Curtis as the default, but will switch dynamically to Gaussian if singularities are detected.