We introduce the fused sequential addition and migration (fSAM) algorithm for generating microstructures of fiber composites with long, flexible, nonoverlapping fibers and industrial volume fractions. The proposed algorithm is based on modeling the fibers as polygonal chains and enforcing, on the one hand, the nonoverlapping constraints by an optimization framework. The connectivity constraints, on the other hand, are treated via constrained mechanical systems of d'Alembert type. In case of straight, that is, nonflexible, fibers, the proposed algorithm reduces to the SAM (Comput. Mech., 59, 247–263, 2017) algorithm, a well-established method for generating short fiber-reinforced composites. We provide a detailed discussion of the equations governing the motion of a flexible fiber and discuss the efficient numerical treatment. We elaborate on the integration into an existing SAM code and explain the selection of the numerical parameters. To capture the fiber length distributions of long fiber reinforced composites, we sample the fiber lengths from the Gamma distribution and introduce a strategy to incorporate extremely long fibers. We study the microstructure generation capabilities of the proposed algorithm. The computational examples demonstrate the superiority of the novel microstructure-generation technology over the state of the art, realizing large fiber aspect ratios (up to 2800) and high fiber volume fractions (up to 32% for an aspect ratio of 150) for experimentally measured fiber orientation tensors.