An optimization procedure is presented which is used to design two types of ultralow-noise DC superconducting quantum interference devices (SQUIDs) with integrated flux coupling circuits, a magnetometer, and a gradiometer. Essential to this design approach is the reduction of the parasitic capacitance, the choice of a layout which moves the input coil resonances well away from the desired operating frequency of the SQUID, and the proper damping of these resonances. It is shown that both the microwave and LC resonances can be damped in a nearly noise-free manner by placing all RC shunt in parallel with the input coil. This leads to smooth voltage-flux modulation characteristics and significantly improved noise performance. A simple, four-level Nb/Six/Ny/Nb Josephson junction technology for device fabrication is described. White flux noise levels of 5*10-7φ0/√Hz for the magnetometer and 7.9*10-7φ0/√Hz for the gradiometer were measured. The corresponding uncoupled energy resolutions are 100 h and 130 h, respectively. The 1/f noise of the magnetometer at 1 Hz is less than 4*10-6φ0/√Hz.