Let $Y$ and $X$ be two vector-valued random variables and let $Z=f(X)$ implicitly define a third random variable from $X$. $X$ and $Z$ exist in the space $\Omega$. For simplicity, we assume $f(x)=Ax$ with non-degenerate matrix $A$. The goal is to compute the marginal pdf
$$ p_Y(y)=\int_\Omega p_{Y|Z}(y|z) p_X(x) dx $$
Assume that for the case where $A=I$ s.t. $Z=X$, the integral is computable. E.g. the two pdfs are conjugate exponential families, leaving an integral with a known solution.
We can use transformation of variables to write (using $J_f$ to mean the Jacobian of $f$)
$$ p_Z(z)=p_X(f^{-1}(z))\left|J_{f^{-1}}\right|=p_X(f^{-1}(z))|A^{-1}| $$
Since $f^{-1}(Z)=X$ we should be able to make the substitution
$$ p_Y(y)=\int_\Omega p_{Y|Z}(y|z) p_Z(z) \left|A\right|dx=\int_\Omega p_{Y|Z}(y|z) p_Z(z) dz $$
This puts the equation in a form where the assumed integration rule (e.g. conjugacy) can then be employed, but the result will see $Z$ marginalized out meaning that $A$ will have had no effect at all on the integral which seems unreasonable and leads me to believe there is a mistake in the above.