This randomized phase III trial studies rituximab with bendamustine hydrochloride or ibrutinib to see how well they work compared to ibrutinib alone in treating older patients with previously untreated chronic lymphocytic leukemia. Monoclonal antibodies, such as rituximab, may block cancer growth in different ways by targeting certain cells. Drugs used in chemotherapy, such as bendamustine hydrochloride, work in different ways to stop the growth of cancer cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Ibrutinib may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. It is not yet known whether rituximab with bendamustine hydrochloride may work better than rituximab and ibrutinib or ibrutinib alone in treating chronic lymphocytic leukemia.