This isn't really a partial fractions problem. Those generally involve a product of terms in the denominator.

What you can do on this problem as you've noted is that it can be rewritten as

$\dfrac {4x-3}{2x+1} = \dfrac{2(2x+1)-5}{2x+1}=2-\dfrac{7}{2x+1}$

which is easier to integrate.