Complete the square on both the numerator and the denominator. It is possible that one or both of these may actually be a perfect square in disguise, in which case you can cancel out the square root and have a much simpler problem.
Perform any u-substitution in both the numerator and denominator to simplify. Remember that rearranging a substitution can sometimes enable you to use a variation of it. For example, u = x - 2 can be rearranged to x = u + 2, if you need to substitute a lone "x." Don't forget that sometimes you may need to make multiple substitutions, first using a u-substitution and then a v-substitution, until the integral is simple enough.
Perform any trig substitutions you see. Most of these problems involve a trig substitution, so remember to also keep an eye out for identities like tan^2(theta) + 1 = sec^2(theta).
Integrate, using power rules, integration by parts or integral tables. Undo any substitutions you made until everything is back in terms of "x." For indefinite integrals, don't forget to write "+ c" at the end, and for definite integrals, evaluate the solution for the limits given.