How to solve for recurrence with substitution or other methods?
Use the Akra-Bazzi theorem. Let $p$ be the solution to $2\cdot(1/3)^p + (1/2)\cdot(2/3)^p= 1$, namely $p = 1$. Compute
$$ S(n) = \int_1^n \frac{x^2}{x^{p+1}} \, dx = n-1= \Theta(n). $$
Then $T(n) = \Theta(x^p S(n)) = \Theta(n^2)$.