You should use a two variable transform and then marginalize out the second variable. A good resource on this kind of technique is the Casella and Berger book.
Use a two variable transform $\varphi$ as follows:
$$ \begin{bmatrix}
U\\
V
\end{bmatrix}
= \varphi(X,Y) =
\begin{bmatrix}
X\\
X/(Y+c)
\end{bmatrix}$$
Then $\varphi$ is invertible and its inverse is
$$ \begin{bmatrix}
X\\
Y
\end{bmatrix}
= \varphi^{-1}(U,V) =
\begin{bmatrix}
\varphi^{-1}_1(U,V)\\
\varphi^{-1}_2(U,V)
\end{bmatrix}
=
\begin{bmatrix}
U\\
U/V - c
\end{bmatrix}$$
The joint PDF of $U,V$ is $p_{U,V}(u,v) = \lvert\det\mathrm{J}_{\varphi^{-1}}(u,v)\rvert p_{X,Y}(\varphi^{-1}_1(u,v), \varphi^{-1}_2(u,v))$.
I'll let you compute the jacobian $\mathrm{J}_{\varphi^{-1}}$ and its determinant. Then you can obtain the marginal $p_V$ by marginalisation:
$$p_V(v) = \int p_{U,V}(u,v)\,\mathrm{d}u$$