Inclusion Criteria:
* Patients must have primary refractory or first relapse of cluster of differentiation 30 (CD30)+ Hodgkin lymphoma
* Patients must have measurable disease defined as lesions that can be accurately measured in two dimensions by computed tomography (CT), magnetic resonance imaging (MRI), medical photograph (skin or oral lesion), plain x-ray, or other conventional technique and a greatest transverse diameter of 1 cm or greater; or palpable lesions with both diameters \>= 2 cm; further, at least 1 of these lesions must be positive by positron emission tomography (PET) scan (i.e., Deauville score of 4 or more); Note: CT scans remain the standard for evaluation of nodal disease
* Patients must have a CT of chest, abdomen, and pelvis with PET within 28 days of enrollment; patients with evidence of lymphadenopathy in the neck must have a dedicated CT of neck
* Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 (performance status of 2 will be allowed if poor performance status is thought to be directly secondary to patient's Hodgkin lymphoma \[HL\])
* Absolute neutrophil count (ANC) \>= 1,500/uL, performed within 28 days prior to registration
* Platelets \>= 100,000/uL (without transfusion or growth factor support), performed within 28 days prior to registration
* Serum creatinine \< 1.5 mg/dl or creatinine clearance (CrCl) \> 60 mL/min, performed within 28 days prior to registration
* Total bilirubin \< 2 times upper limit of normal (unless due to Gilbert's syndrome), performed within 28 days prior to registration
* Aspartate aminotransferase (AST) \< 2.5 times upper limit of normal, performed within 28 days prior to registration
* All patients must be informed of the investigational nature of this study and have given written consent in accordance with institutional and federal guidelines
* Patients must be anticipated to complete 2 cycles of chemotherapy
Exclusion Criteria:
* Patients known to be positive for human immunodeficiency virus (HIV)
* Pregnant or nursing women; men or women of reproductive potential may not participate unless they have agreed to use an effective contraceptive method
* Patients with other prior malignancies except for adequately treated basal cell carcinoma, squamous cell carcinoma of the skin, breast or cervical cancer in situ, or other cancer from which the patient has been disease-free for 5 years or greater, unless approved by the protocol chair or co-chair
* Patients with known allergy, intolerance, or resistance (i.e., remission duration less than 6 months or lack of response) to ifosfamide, carboplatin, or etoposide
* Patients with evidence of active central nervous system lymphoma
* Patients with prior receipt of brentuximab vedotin
* Patients with peripheral neuropathy of \> grade 1
* Patients who have other medical conditions that would contraindicate treatment with aggressive chemotherapy (including active infection, uncontrolled hypertension, congestive heart failure, unstable angina pectoris, myocardial infarction within the past 6 months, or uncontrolled arrhythmia); if the patient's cardiac history is questionable, a measurement of left ventricular ejection fraction should be obtained within 42 days prior to registration; patients with left ventricular ejection fraction \< 50% are not eligible
* Prior failed (\< 5 x 10\^6 CD34/kg) peripheral blood stem cell (PBSC) collection
* Patients who had pelvic radiation within 12 months
* Previous chemotherapy/immunotherapy within 3 weeks before study entry
* Concurrent use of other anti-cancer agents or experimental treatments
