or did I miss anything somewhere?
how did you factor out the 8/6 like that? the 8 and the 6 were being added, you can't just pull it out randomly. you have the right idea with making 2x+3 = 2x-5+8. split up the integral into 2x-5 over the denominator + 8 over the denominator. the first integral is a simple u-substitution and will end up being the natural log of the denominator. the second integral, use partial fractions if since x^2 - 5x + 6 can be factored as (x-3)(x-2). you can also try completing the square and applying a formula if you don't like partial fractions.