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 thyroid hormone supplements to treat underlying hypothyroidism * Zubrod performance status \< 2 * Negative beta-human chorionic gonadotropin (HCG) test in a woman of child-bearing potential, defined as not post-menopausal for 12 months or no previous surgical sterilization * Availability of \>= 2.5 million cluster of differentiation (CD)34+ cells/kg previously apheresed * Ability to provide written informed consent Exclusion Criteria: * Prior whole brain irradiation * Having received radiation therapy to head and neck (excluding eyes), and internal organs of chest, abdomen or pelvis in the month prior to enrollment * Active hepatitis B, either active carrier (hepatitis B surface antigen positive \[HBsAg +\]) or viremic (hepatitis B virus \[HBV\] deoxyribonucleic acid \[DNA\] \>= 10,000 copies/mL, or \>= 2,000 IU/mL) * Evidence of either cirrhosis or stage 3-4 liver fibrosis in patients with chronic hepatitis C or positive hepatitis C serology * Active infection requiring parenteral antibiotics * Known positivity for human immunodeficiency virus (HIV) * Autologous stem-cell transplant in the previous six months * Needing valproic acid for any medical condition during the study or within 5 days prior to first panobinostat treatment * Impairment of gastrointestinal (GI) function or GI disease that may significantly alter the absorption of panobinostat * Other concurrent severe and/or uncontrolled medical conditions (e.g., uncontrolled diabetes or active or uncontrolled infection) including abnormal laboratory values, that could cause unacceptable safety risks or compromise compliance with the protocol * Impaired cardiac function or clinically significant cardiac diseases, including any one of the following: * History or presence of sustained ventricular tachyarrhythmia; (patients with a history of atrial arrhythmia are eligible but should be discussed with Secura Bio, Inc (Secura Bio) prior to enrollment) * Any history of ventricular fibrillation or torsade de pointes * Bradycardia defined as heart rate (HR) \< 50 beats per minute (bpm); patients with pacemakers are eligible if HR \>= 50 bpm * Screening electrocardiogram (ECG) with a corrected QT (QTc) \> 470 msec * Right bundle branch block + left anterior hemiblock (bifascicular block) * Myocardial infarction or unstable angina =\< 12 months prior to starting study drug * Other clinically significant heart disease (e.g., congestive heart failure \[CHF\] New York \[NY\] Heart Association class III or IV , uncontrolled hypertension, history of labile hypertension, or history of poor compliance with an antihypertensive regimen) * Have undergone major surgery =\< 4 weeks prior to starting study drug or who have not recovered from side effects of such therapy * Prior malignancy with in the last 5 years (except for basal or squamous cell carcinoma, or in situ cancer of the cervix) * Any significant history of non-compliance to medical regimens or unwilling or unable to comply with the instructions given to him/her by the study staff * Received targeted agents within 2 weeks or within 5 half-lives of the agent and active metabolites (whichever is longer) and who have not recovered from side effects of those therapies * Having received immunotherapy or chemotherapy within 2 weeks; or radiation therapy to \> 30% of marrow-bearing bone within =\< 2 weeks prior to starting study treatment; or who have not yet recovered from side effects of such therapies * Grade \>= 3 nonhematological toxicity from prior therapy that has not resolved to =\< grade 1