Minimum edge-weighted directed subgraph in polynomial time

If you don't want to implement a specialized algorithm, you can solve the problem via mixed integer linear programming as follows. Let $T$ be the set of terminal nodes. For each directed arc $(i,j)\in A$, let nonnegative variable $x_{ij}$ be the flow from $i$ to $j$, and let binary variable $y_{ij}$ indicate whether arc $(i,j)$ is used. The problem is to minimize $$\sum_{(i,j)\in A} c_{ij} y_{ij}$$ subject to \begin{align} \sum_{(i,j)\in A} x_{ij} - \sum_{(j,i)\in A} x_{ji} &= \begin{cases} |T| &\text{if $i=v$} \\ -1 &\text{if $i\in T$} \\ 0 &\text{otherwise} \end{cases} \\ x_{ij} &\le |T|y_{ij} \end{align}