A rapidly inserted central catheter can include a catheter tube, a catheter hub, and one or more extension legs. The catheter tube can include a single-piece catheter tip coupled to a distal end portion of the catheter tube having a first section, a second section, and a third section. The first section of the catheter tip can have a uniform taper over an outer diameter thereof for dilating tissue around a needle tract from a size commensurate with an outer diameter of a needle shaft to a size commensurate with an outer diameter of the second section of the catheter tip. The third section of the catheter tip can have a non-uniform taper over an outer diameter thereof for dilating the tissue from the size commensurate with the outer diameter of the second section of the catheter tip to a size commensurate with an outer diameter of the catheter tube.