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Single-source time-domain electric- and magnetic-field integral equations for analyzing scattering from homogeneous penetrable objects are presented. Their temporal discretization is effected by using shifted piecewise polynomial temporal basis functions and a collocation testing procedure, thus allowing for a marching-on-in-time (MOT) solution scheme. Unlike dual-source formulations, single-source equations involve space-time domain operator products, for which spatial discretization techniques developed for standalone operators do not apply. Here, the spatial discretization of the single-source time-domain integral equations is achieved by using the high-order divergence-conforming basis functions developed by Graglia et al. alongside the high-order divergence- and quasi curl-conforming (DQCC) basis functions of Valdés et al. The combination of these two sets allows for a well-conditioned mapping from div- to curl-conforming function spaces that fully respects the space-mapping properties of the space-time operators involved. Numerical results corroborate the fact that the proposed procedure guarantees accuracy and stability of the MOT scheme.