Solving $T(n)=3T\bigl(\bigl\lfloor \frac{n}{3}\bigr\rfloor\bigr) +2n\log n$ without the Master Theorem
Let's suppose $n=3^k$. Then we have:
$$T(n)=3^kT(1)+2n\log \frac{n^k}{3^{k-1}\cdots3^0}=n+2n\log 3^{k^2-\frac{k(k-1)}{2}} =\\
=n+2n\log n \cdot \frac{\log_3 n+1}{2}   $$
Hope it helps.