Problem: Evaluate the integral
Solution: This integral does not fit any particular box, so we have to try some substitution and hope for the best. There are two reasonable possibilities here, the exponential itself or the whole expression under the square root. We try the latter:
This does not look too bad, but we cannot simplify it any further, so we have to recall the box "integrals with roots" and take the square root of y as another substitution (or use an indirect substitution as recommended there):
If you now have a feeling that the calculations could have been done in a more elegant way, you are right. A smarter substitution would lead us to the stage that we are at right now in one step, it would go like this. In any case, we now have to use the partial fractions decomposition, but it is going to be easy - we have linear factors. (Actually, if we have a list of integrals handy, the remainder after the long division should be there.)
If you want to see some details of the decomposition, the necessary linear substitutions, or see what happens if we use the substitution when we denote just the exponential by y, click here.