Carboplatin and Paclitaxel or Oxaliplatin and Capecitabine With or Without Bevacizumab as First-Line Therapy in Treating Patients With Newly Diagnosed Stage II-IV or Recurrent Stage I Epithelial Ovarian or Fallopian Tube Cancer

Inclusion Criteria: * Patients with a histologic diagnosis of mucinous adenocarcinoma of the ovary or fallopian tube with either optimal (=\< 1 cm residual disease) or suboptimal residual disease following initial surgery; patients may have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 or no measurable disease * All patients must have had appropriate surgery including appendectomy (unless patient has history of prior appendectomy) for ovarian or fallopian tube carcinoma with appropriate tissue available for histologic evaluation to confirm diagnosis and stage * Patients must have stage II-IV disease (new or recurrent-chemonaïve; no brain metastasis) or recurrent stage I disease (chemonaïve) * Newly diagnosed patients must begin protocol therapy within 10 weeks of primary debulking; for stage I recurrent patients (chemonaïve), they should begin protocol therapy within 14 days of randomization * Patients must have a negative colonoscopy within 1 year of enrolling in the study * Absolute neutrophil count (ANC) \>= 1,500/mcl * White blood cell (WBC) count \>= 3,000/mcl * Platelets \>= 100,000/mcl * Hemoglobin (Hgb) \>= 10 g/dl (can be post transfusion) * Creatinine less than or equal to 1.5 x institutional upper limit normal (ULN) OR creatinine clearance \> 50 cc/min * Bilirubin =\< 1.5 x ULN * Serum glutamic oxaloacetic transaminase (SGOT) =\< to 2.5 x ULN * Alkaline phosphatase =\< to 2.5 x ULN * Neuropathy (sensory and motor) =\< C…