Quadratic and quartic integrals using the method of brackets

We use the method of brackets to evaluate quadratic and quartic type integrals. We recall the operational rules of the method and give examples to illustrate its working. The method is then used to evaluate the quadratic type integrals which occur in entries 3.251.1,3,4 in the table of integrals by Gradshteyn and Ryzhik and obtain closed form expressions in terms of hypergeometric functions. The method is further used to evaluate the quartic integrals, entry 2.161.5 and 6 in the table. We also present generalization of both types of integrals with closed form expression in terms of hypergeometric functions.