AbstractWe report on a series of detailed Breit-Pauli and Dirac B-spline R-matrix (DBSR) differential cross section (DCS) calculations for excitation of the$$5\,^2\textrm{S}_{1/2} \rightarrow 5\,^2\textrm{P}_{1/2}$$$5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{S}}_{1/2}\to 5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{P}}_{1/2}$and$$5\,^2\textrm{S}_{1/2}\rightarrow 5\,^2\textrm{P}_{3/2}$$$5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{S}}_{1/2}\to 5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{P}}_{3/2}$states in rubidium by 40 eV incident electrons. The early BP computations shown here were carried out with both 5 states and 12 states, while the DBSR models coupled 150 and 325 states, respectively. We also report corresponding results from a limited set of DCS measurements on the unresolved$$5\,^2\textrm{P}_{1/2,3/2}$$$5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{P}}_{1/2,3/2}$states, with the experimental data being restricted to the scattered electron angular range 2–$$10^\circ $$${10}^{\circ}$. Typically, good agreement is found between our calculated and measured DCS for excitation of the unresolved$$5\,^2\textrm{P}_{1/2,3/2}$$$5{\phantom{\rule{0ex}{0ex}}}^{2}{\text{P}}_{1/2,3/2}$states, with best accord being found between the DBSR predictions and the measured data. The present theoretical and experimental results are also compared with predictions from earlier 40 eV calculations using the nonrelativistic Distorted-Wave Born Approximation and a Relativistic Distorted-Wave model.