  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.
