We study the behavior of microorganisms utilizing planar and spiral bending waves to swim in a fluid. More specifically, spermatozoa and bacteria species encounter different fluid environments filled with mucus, cells, hormones, and other large proteins. These networks of materials are assumed to be stationary and of low volume fraction. They act as friction, possibly preventing or enhancing forward progression of the swimmers. The Brinkman equation is used to model the average fluid flow where a flow dependent term, a resistance parameter, models the resistive effects of the fibers on the fluid. To understand the effects of the resistance, we study the asymptotic swimming speeds of an infinite-length swimmer propagating planar and spiral bending waves in a Brinkman fluid. We find that, up to the second order expansion, the swimming speeds are enhanced as the resistance increases. The analytical solutions are compared with numerical results of finite-length swimmers obtained from the method of Regularized Brinkmanlets (MRB). In addition, we develop a grid-free numerical method to study the bend and twist of an elastic rod immersed in a Brinkman fluid. The rod is discretized using a Kirchhoff Rod model. The linear and angular velocity of the rod are derived using the MRB. The method is also validated by comparing results to asymptotic swimming speeds. For large amplitude planar bending, our model results show a non-monotonic change in swimming speed with respect to the resistance parameter. The non-monotonicity is due to the emergent waveforms; as the resistance parameter increases, the swimmer becomes incapable of achieving the amplitude of its preferred configuration.