Eligible participants are HIV positive and plan to be treated by high dose chemotherapy followed by an autologous stem cell transplant (ASCT). Participants are a minimum of 15 years of age with Karnofsky performance status greater than or equal to 70% that have primary refractory or recurrent diffuse large B-cell, immunoblastic, plasmablastic, high grade, Burkitt, primary effusion lymphoma, or classical Hodgkin lymphoma. Participants must have received 2 or 3 prior treatment regimens, including an induction chemotherapy and 1 or 2 salvage regimens. Monoclonal antibody therapy and local radiation will not be counted as prior therapies. Participants must have chemosensitive disease as demonstrated by complete or partial response to induction or most recent salvage chemotherapy. Participants cannot have had prior autologous, allogeneic HCT, or CART-cell therapy. Participants must initiate conditioning therapy within 3 months of stem cell mobilization or bone marrow harvest. Blood cell mobilization or bone marrow harvest will be carried out per institutional guidelines. Participants may not have HIV refractory to pharmacologic therapy. Patients must not have an uncontrolled infection. Participants must not have received previous cellular therapy  Age 15 years old or older at time of enrollment. Receiving antiretroviral therapies (ART) with HIV viral load below the limit of detection by standard commercial assay. A single plasma HIV-1 RNA measurement that is ≥ the limit of quantification of the FDA-approved commercial assay but <500 copies is allowed if this is followed by an HIV-1 RNA measurement below the limit of detection. Diagnosis of refractory or recurrent diffuse large B-cell lymphoma, composite lymphoma with greater than 50% diffuse large B-cell lymphoma, mediastinal B-cell lymphoma, immunoblastic, plasmablastic, Burkitt or high grade B cell lymphoma or classical Hodgkin lymphoma. Participants with aggressive B-cell lymphoma that is transformed from follicular lymphoma are eligible for the study, pending fulfillment of other criteria. Two or three prior regimens of chemotherapy over the entire course of their disease treatment (induction chemotherapy and salvage chemotherapies). Monoclonal antibody therapy and involved field radiation therapy will not be counted as prior therapies All participants must have chemosensitive disease as demonstrated by at least a partial response (as defined by the criteria in Chapter 3) to induction or salvage therapy. Absolute Lymphocyte Count (ALC) greater than or equal to 1000/µL.  Participants with adequate organ function as measured by:  Cardiac: Participants must have a left ventricular ejection fraction at rest greater than or equal to 40% demonstrated by MUGA or echocardiogram. b) Hepatic: i. Bilirubin less than or equal to 2.0 mg/dL (except for isolated hyperbilirubinemia attributed to Gilbert syndrome or antiretroviral therapy as specified in Appendix D) and ALT and AST less than or equal to 3x the upper limit of normal. ii. Concomitant Hepatitis: Participants with chronic hepatitis B or C may be enrolled on the trial providing the above criteria are met. In addition, they must not have evidence of active viral replication by PCR, and no clinical or pathologic evidence of irreversible chronic liver disease. Renal: Creatinine clearance (calculated creatinine clearance is permitted based on institutional practice) greater than 40 mL/min. Pulmonary DLCO (corrected for hemoglobin), FEV1, FVC greater than or equal to 45% of predicted. Plan to treat participant with high dose chemotherapy and autologous hematopoietic stem cell transplantation (ASCT). Voluntary written consent or assent obtained prior to enrollment on study with the understanding that consent or assent may be withdrawn by the participant at any time without prejudice to future medical care.  Exclusion Criteria:  Karnofsky performance score less than 70%. Participant is known to have an HIV subtype other than B. Participant has documented raltegravir or protease inhibitor resistance. Myocardial infarction within 6 months prior to enrollment or New York Heart Association (NYHA) Class III or IV heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, or electrocardiographic evidence of acute ischemia. Uncontrolled bacterial, viral or fungal infection (currently taking medication and with progression or no clinical improvement). Participant has active CNS involvement. Participants with prior malignancies except resected non-melanoma skin cancer or treated cervical carcinoma in situ. Cancer treated with curative intent greater than or equal to 5 years previously will be allowed. Cancer treated with curative intent less than 5 years BMT CLINICAL TRIALS NETWORK HIV T-Cell - Protocol 1903 Version 1.0 Dated February 24, 2021 2-4 Confidential previously may be eligible must be reviewed and approved by the Protocol Officer or Chairs. Female participants that are pregnant as per institutional definition or breastfeeding. Fertile men or women unwilling to use contraceptive techniques from the time of initiation of mobilization until six-months post-transplant. Prior autologous or allogeneic HCT, or prior therapy with chimeric antigen receptor (CAR) T-cells. Participants with evidence of MDS/AML or abnormal cytogenetic analysis indicative of MDS on the pre-transplant bone marrow examination. Pathology report documentation need not be submitted. Steroids greater than 0.5 mg/kg/day prednisone equivalents. Bone marrow involvement by lymphoma at time of workup. Prior history of bone marrow involvement is allowed if cleared prior to ASCT.