Personalized Kinase Inhibitor Therapy Combined With Chemotherapy in Treating Patients With Newly Diagnosed Acute Myeloid Leukemia and Acute Lymphoblastic Leukemia

Inclusion Criteria: * Patients must have histologically confirmed AML or ALL, excluding acute promyelocytic leukemia (APL); for histological confirmation, a bone marrow biopsy and aspirate must be reviewed at Oregon Health \& Science University (OHSU) * Patients must have newly diagnosed AML or ALL without previous treatment; hydroxyurea will be allowed to control peripheral blast count as clinically indicated but would need to be stopped prior to initiation of tyrosine kinase inhibitor \[TKI\]); all-trans retinoic acid (ATRA) is permitted during the period prior to ruling out a diagnosis of APL; previous radiation treatment is allowable; patients must be deemed eligible for treatment with cytotoxic induction chemotherapy with cytarabine and idarubicin for AML or hyper-cyclophosphamide, dexamethasone, doxorubicin, vincristine sulfate (CVAD) for ALL; patients newly diagnosed with ALL must have received no prior treatment for their ALL with the exception of steroids (i.e. prednisone, dexamethasone); intrathecal methotrexate or cytarabine, allowed prior to and throughout the enrollment period for AML and ALL * Subjects must be aged between \>= 18 years and =\< 64 for AML and \> 40 and =\< 64 for ALL * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Total bilirubin \< 2.0 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN and thought to be due to hepatocellular dysfunction * Potassium \> lower limit of normal (LLN) or correctable with supplements prior to first dose of study medication * Magnesium \> LLN or correctable with supplements prior to first dose of study medication * Total calcium (corrected for serum albumin) \>= LLN or correctable with supplements prior to first dose of study medication * Serum amylase and lipase =\< 1.5 x institutional ULN * International normalized ratio (INR) =\< 2.0 or correctable to 2.0 with vitamin K therapy * Corrected QT (QTc) =\< 450 msec. for men or QTc =\< 470 msec. for women * Creatinine \< 2.0 x ULN * No clinically significant uncontrolled infections as determined by investigator * Patients must be able to take oral medications * Persons of reproductive potential must agree to an adequate method of contraception throughout treatment and for at least 4 weeks after study drug is stopped * Women of childbearing potential must have a negative serum or urine pregnancy test within 8 days prior to start of study drug administration * Patients must be willing to accept blood product transfusions * Ability to understand and the willingness to sign a written informed consent document and Health Insurance Portability and Accountability Act (HIPAA) documentation * DRUG-SPECIFIC INCLUSION CRITERIA * DASATINIB * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 24 hours prior to the start of study drug * Women must not be breastfeeding * WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug, plus 30 days (duration of ovulatory cycle) for a total of 30 days post-treatment completion * Men who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug plus 90 days (duration of sperm turnover) for a total of 90 days post-treatment completion * Azoospermic males and WOCBP, who are not heterosexually active, are exempt from contraceptive requirements; however, WOCBP must still undergo pregnancy testing * Investigators shall counsel WOCBP and male subjects who are sexually active with WOCBP on the importance of pregnancy prevention and the implications of unexpected pregnancy; investigators shall advise WOCBP and male subjects who are sexually active with WOCBP on the use of highly effective contraception; highly effective methods of contraception have a failure rate of \< 1% when used consistently and correctly * At a minimum, subjects must agree to the use of two methods of contraception, with one method being highly effective and the other method being either highly effective or less effective Exclusion Criteria: * Newly diagnosed AML patients who are identified with FLT3-ITD or tyrosine kinase domain (TKD) point mutation in the codon for an aspartate (D835) or an isoleucine (I836) residue * Patients with a diagnosis of Philadelphia chromosome (Ph)+ ALL are not eligible * Subjects who are currently receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent or other agents used in the study * Drugs that affect the cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) systems are allowed but should be used with caution depending on specific kinase inhibitor used * All outside study medications and supplements will be reviewed and monitored by the inpatient pharmacy team; patients will be discouraged from taking herbals and additional supplements * Patients should not be prescribed concomitant medications that may contribute to prolonged QTc without consultation with the chemotherapy pharmacist; additional ECGs should be done at the investigator?s discretion to ensure the subject?s safety; drugs that are generally accepted to increase the risk of Torsades de Pointes, include (but not limited to): * Quinidine, procainamide, disopyramide * Amiodarone, ibutilide, dofetilide, sotalol * Erythromycin, clarithromycin * Chlorpromazine, mesoridazine, thioridazine, pimozide * Cisapride, bepridil, droperidol, methadone, arsenic, chloroquine, domperidone, halofantrine, levomethadyl, pentamidine, sparfloxacin, lidoflazine * Left ventricular ejection fraction \< 50% * Uncontrolled intercurrent illness including but not limited to, symptomatic New York Heart Association (NYHA) class III congestive heart failure, uncontrolled angina pectoris, myocardial infarction or stroke within 6 months prior to enrollment, or psychiatric illness/social situations that would limit compliance with study requirements * Patients with a known human immunodeficiency virus (HIV) diagnosis are excluded from the study * History of hypersensitivity to any of the kinase inhibitors included in this study * Pregnant or lactating women are excluded from the study * Diagnosed congenital long QT syndrome * Any history of significant bleeding disorder unrelated to cancer, including any congenital bleeding disorder or any acquired bleeding disorder within one year of start of study * Any history of clinically significant ventricular arrhythmia (such as ventricular tachycardia, ventricular fibrillation or torsade de pointes) * Patients must not have clinically significant malabsorption syndrome or history * All patients must discontinue anti-platelet agents or anticoagulants 7 days prior to initiation of study drug * All patients with unhealed wounds or fistulas should not be given vascular endothelial growth factor (VEGF) inhibiting TKIs * DRUG-SPECIFIC EXCLUSION CRITERIA * PONATINIB * Patients with cytogenetically ?favorable risk' AML (core-binding factor leukemias) will not be enrolled on the ponatinib arm; testing with cytogenetics and fluorescence in situ hybridization (FISH) can establish this subtype within 7 days of the diagnostic bone marrow biopsy * History of acute pancreatitis within 1 year of study or history of chronic pancreatitis * Uncontrolled hypertriglyceridemia (triglycerides \> 450 mg/dL) * Any history of myocardial infarction, stroke, or revascularization * Any history of venous thromboembolism including deep venous thrombosis or pulmonary embolism with the exception of upper Extremity/Line associated deep vein thrombosis (DVTs) which are adequately treated (line removed and/or patient anti-coagulated) * Uncontrolled hypertension (diastolic blood pressure \> 90 mm Hg; systolic \> 140 mm Hg); patients with hypertension should be under treatment on study entry to effect blood pressure control * DASATINIB * Any history of second or third degree heart block (may be eligible if the subject currently has a pacemaker) * Known pulmonary arterial hypertension * Patients may not have clinically significant pleural or pericardial effusion per provider discretion * SORAFENIB * Major surgery, open biopsy, or significant traumatic injury within 30 days * Non-healing wound, ulcer, or bone fracture * Thrombotic or embolic venous or arterial events, such as cerebrovascular accident, including transient ischemic attacks, arterial thrombosis, deep vein thrombosis and pulmonary embolism within the past 6 months * Upper Extremity/Line associated DVTs which are adequately treated (line removed and/or patient anticoagulated) are eligible * Uncontrolled hypertension * Active bleeding during screening * Hypersensitivity to sorafenib * Idelalisib * Ongoing drug-induced liver injury, chronic active hepatitis C (hepatitis C virus \[HCV\]), chronic active hepatitis B (hepatitis B virus \[HBV\]), alcoholic liver disease, non-alcoholic steatohepatitis, primary biliary cirrhosis, extrahepatic obstruction caused by cholelithiasis, cirrhosis of the liver, portal hypertension, or history of autoimmune hepatitis * Ongoing symptomatic pneumonitis * Ongoing inflammatory bowel disease or autoimmune colitis * Ongoing cytomegalovirus (CMV) infection, treatment, or prophylaxis within the past 28 days prior to the screening test for active CMV * History of serious allergic reaction including anaphylaxis and epidermal necrolysis * Ruxolitinib * Chronic active hepatitis C (HCV)