Panobinostat, Gemcitabine Hydrochloride, Busulfan, and Melphalan Before Stem Cell Transplant in Treating Patients With Refractory or Relapsed Multiple Myeloma

Inclusion Criteria: * Refractory or relapsed myeloma, defined as one or more of the following: * Treated with first-line therapy including at least 2 cycles of lenalidomide, bortezomib or thalidomide, and one or more of the following: * Less than partial response (PR) to first-line therapy * Relapse after first (1st) line therapy * High-risk cytogenetics, defined by deletion (del)(13q) by conventional cytogenetics, or by del(17p), t(4;14), t(14;16), t(14;20) or 1q+ by fluorescence in situ hybridization (FISH) * Relapse after a prior autologous stem cell transplant (ASCT) * Plasma cell leukemia * Soft tissue plasmacytoma * Serum creatinine =\< 1.8 mg/dL and/or estimated serum creatinine clearance \>= 50 ml/min * Serum glutamic oxaloacetic transaminase (SGOT) and/or serum glutamate pyruvate transaminase (SGPT) =\< 3 x upper limit of normal * Serum bilirubin =\< 2 x upper limit of normal, unless proven to be due to disease involvement * Alkaline phosphatase =\< 2 x upper limit of normal, unless proven to be due to disease involvement * Adequate pulmonary function with forced expiratory volume in 1 second (FEV1), forced vital capacity (FVC) and diffusion capacity of the lung for carbon monoxide (DLCO) \>= 50% of expected corrected for hemoglobin and/or volume * Adequate cardiac function with left ventricular ejection fraction \>= 40% * No uncontrolled arrhythmias or symptomatic cardiac disease * Clinically euthyroid; note: patients are permitted to receive th…