Catalyzes the polymerization of hyaluronan, a polysaccharide composed of a repeating disaccharide of N-acetylglucosamine (GlcNAc) and glucuronic acid (GlcUA) units. Each unit has the composition in beta-(1->4)-GlcUA-beta-(1->3)-GlcNAc. [hyaluronan](n) + UDP-N-acetyl-alpha-D-glucosamine = H(+) + N-acetyl-beta-D-glucosaminyl-(1->4)-[hyaluronan](n) + UDP N-acetyl-beta-D-glucosaminyl-(1->4)-[hyaluronan](n) + UDP-alpha-D-glucuronate = [hyaluronan](n+1) + H(+) + UDP 3-O-(beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)-beta-D-Gal-(1->3)-beta-D-Gal-(1->4)-beta-D-Xyl)-L-seryl-[protein] + UDP-alpha-D-glucuronate = 3-O-(beta-D-GlcA-(1->3)-beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)-beta-D-Gal-(1->3)-beta-D-Gal-(1->4)-beta-D-Xyl)-L-seryl-[protein] + H(+) + UDP 3-O-([beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)](n)-beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)-beta-D-Gal-(1->3)-beta-D-Gal-(1->4)-beta-D-Xyl)-L-seryl-[protein] + UDP-alpha-D-glucuronate = 3-O-(beta-D-GlcA-(1->3)-[beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)](n)-beta-D-GalNAc-(1->4)-beta-D-GlcA-(1->3)-beta-D-Gal-(1->3)-beta-D-Gal-(1->4)-beta-D-Xyl)-L-seryl-[protein] + H(+) + UDP Belongs to the glycosyltransferase 2 family. CS/HAS subfamily.