Autonomous implantable bioelectronics requires efficient radiating structures for data transfer and wireless powering. The radiation of body-implanted capsules is investigated to obtain the explicit radiation optima for E- and B-coupled sources of arbitrary dimensions and properties. The analysis uses the conservation-of-energy formulation within dispersive homogeneous and stratified canonical body models. The results reveal that the fundamental bounds exceed by far the efficiencies currently obtained by conventional designs. Finally, a practical realization of the optimal source based on a dielectric-loaded cylindrical-patch structure is presented. The radiation efficiency of the structure closely approaches the theoretical bounds and shows a fivefold improvement over existing systems.