The present work describes the design and the implementation of a protocol for arbitrary precision computation of Small Angle X-ray Scattering (SAXS) profiles, and its inclusion in a probabilistic framework for protein structure determination. This protocol identifies a set of maximum-likelihood estimators for the form factors employed in the Debye formula, a theoretical forward model for SAXS profiles. The resulting computation compares favorably with the state of the art tool in the field, the program CRYSOL in the suite ATSAS. A faster, parallel implementation on Graphical Processor Units (GPUs) is also provided. Empowered by data available from SAXS experiments, by this protocol as a forward model for Markov Chain Monte Carlo (MCMC) simulations, by a continuous model of the peptide bond (TorusDBN) and the conformations of side chains (COMPAS and BasiliskDBN), we are able to propose ensembles of protein structures all fitting the experimental data. For the first time, we describe in full atomic detail a set of different conformations attainable by flexible polypeptides in solution. This method is not limited by assumptions in shape or size of the samples. It allows therefore to investigate crucial biological targets difficult to study with high-resolution experimental methods, like flexible proteins in physiological conditions and large systems of multi-domain proteins.