The numerator is what you get by differentiating the contents of the brackets (save for a factor of 2). So "guess" that the answer is something like (x^2+6x-5)^-1, and differentiate this back to get -(2x+6)(x^2+6x-5)^-2. So this is the original integrand times -2, so the guess was two times too high and the wrong sign. So we re-guess as -1/2(x^2+6x-5)^-1 which checks out upon differentiating.
Alternatively, if you think is too "guessy' you could just substitute u=x^2+6x-5 and see what turns up:
{du}/{dx}=2x+6,
or {dx}/{du}=1/(2(x+3) so the integral is:
int (x+3)/(x^2+6x-5)^2 {dx}/{du}du
=int cancel(x+3)/u^2 * 1/(2(cancel(x+3)))du
=(1/2)int1/u^2du
=-(1/2).(1/u)+C
=-1/(2(x^2+6x-5))+C
The key step is that all the x's cancelled out in the odd mixed-up expression with both x's and u's. This happened only because of the connection between the numerator and the denominator.
