Catalyzes the addition of GlcNAc or GlcUA monosaccharides to the nascent hyaluronan polymer. Therefore, it is essential to hyaluronan synthesis a major component of most extracellular matrices that has a structural role in tissues architectures and regulates cell adhesion, migration and differentiation. This is one of three isoenzymes responsible for cellular hyaluronan synthesis. {ECO:0000269|PubMed:23303191, ECO:0000269|PubMed:25795779, ECO:0000269|PubMed:26883802}. This is the function of HAS3 (hyaluronan synthase 3, ENSG00000103044).