In this paper, we consider problem of estimating diffraction attenuation from approximation of a terrain using multiple bridged knife-edge model. This model can be considered as a generalization of well-known multiple knife-edge one where spaces between knife-edge are bridged by reflecting surfaces. A series based-standard solution is presented in the literature but suffers from its very high computational complexity. We, thus, propose to generalize the recursive Vogler algorithm developed for multiple knife-edge model to tackle this problem. To reduce complexity, the proposed algorithm exploit the recursive form to avoid repeated calculations of the existing solution. Moreover, we provide a complexity analysis of both the standard algorithm and the proposed algorithm based on number of computed integrals. Both theoretical and numerical results show that our proposed algorithm is faster than the original one while having identical accuracy, hence proving the effectiveness of our solution.