In the Wikipedia entry for "Inverse-Wishart distribution" (current revision) there is a formula for a Wishart distribution conditioned on a conjugate prior ( $\Psi,\nu$) in which the original Wishart scale matrix $\Sigma$ has been integrated out to yield the conditional distribution

$$ p(X\mid\Psi,\nu) = \frac { |\Psi|^{\nu/2}\Gamma_p( \frac {\nu + n}{2})}{\pi^{np/2}
|\Psi+X^TX|^{(\nu+n)/2} \Gamma_p (\frac{\nu}{2})} $$

where $A_{n\times n} $ is the observed data $X^TX $.

This doesn't appear to be a valid Wishart distribution - which I think should be something more like

$$ p(X) = \frac { |\Psi|^{\nu/2}\Gamma_p( \frac {\nu + n}{2})}{\pi^{np/2}
|\Psi+A|^{(\nu+n)/2} \Gamma_p (\frac{\nu}{2})} \exp \Big [-\operatorname{tr} \Big ((\Psi +\Sigma)^{-1}A \Big) \Big].$$
So am I misunderstanding this and the formula is a partial result for illustration purposes only?