  Combining sum factorization, weighted quadrature, and row-based assembly
enables efficient higher-order computations for tensor product splines. We aim
to transfer these concepts to immersed boundary methods, which perform
simulations on a regular background mesh cut by a boundary representation that
defines the domain of interest. Therefore, we present a novel concept to divide
the support of cut basis functions to obtain regular parts suited for sum
factorization. These regions require special discontinuous weighted quadrature
rules, while Gauss-like quadrature rules integrate the remaining support. Two
linear elasticity benchmark problems confirm the derived estimate for the
computational costs of the different integration routines and their
combination. Although the presence of cut elements reduces the speed-up, its
contribution to the overall computation time declines with h-refinement.
