sequences and series - Double sum trouble

Evaluate:
$$\sum_{j=1}^{\infty} \sum_{i=1}^{\infty} \frac{j^2i}{3^j(j3^i+i3^j)}$$

Honestly, I don't see where to start with this. I am sure that this is a trick question and I am missing something very obvious. I tried writing down a few terms for a fixed $j$ but I couldn't spot any pattern or some kind of easier series to handle.

Any help is appreciated. Thanks!

Answer

After symmetrization with respect to the exchange $i\leftrightarrow j$, the sum can be rewritten as
\begin{align}
\frac12\sum_{i,j=1}^{\infty} \left(\frac{j^2i}{3^j(j3^i+i3^j)}+\frac{i^2j}{3^i(j3^i+i3^j)}\right)=\frac12\sum_{i,j=1}^{\infty} \frac{i\cdot j}{3^i\cdot3^j}=\frac12\left(\sum_{i=1}^{\infty}\frac{i}{3^i}\right)^2=\frac{9}{32}.
\end{align}