Approach 1. Let  u = x + 3. This is the substitution most of us would try. It is good only in the sense that it works. The key trick is to solve it for  x = u – 3  so that the  x  in the numerator is also expressed in  u.
 
 
 
Approach 2. The next approach is not widely taught at least in the U.S. only because not many math teachers know it. What we should appreciate in the following approach is that it avoids rational exponents altogether. Let us start with:
 

Differentiate the middle expression with respect to  x. The Chain Rule then gives us  3u²du = dx. So,
 

This is the same result as in Approach 1.
 
The main spirit behind Approach 2 is to aim for as big a substitution as possible, so that we would not have to deal with rational exponents until the final step.