  We consider the long-time dynamics of focusing energy-critical Schr\"odinger
equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with
inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$
\begin{equation}\label{NLS-ab} \begin{cases}
i\partial_tu-\mathcal{L}_au=-|u|^{\frac{4}{d-2}}u+|u|^{\frac{4}{d-1}}u, \quad
(t,x)\in\mathbb{R}\times\mathbb{R}^d,\tag{CNLS$_a$},\\ u(0,x)=u_0(x)\in
H^1_a(\mathbb{R}^d), \end{cases} \end{equation} where
$\mathcal{L}_a=-\Delta+a|x|^{-2}$ and the energy is below and equal to the
threshold $m_a$, which is given by the ground state $W_a$ satisfying
$\mathcal{L}_aW_a=|W_a|^{\frac{4}{d-2}}W_a$. When the energy is below the
threshold, we utilize the concentration-compactness argument as well as the
variatonal analysis to characterize the scattering and blow-up region. When the
energy is equal to the threshold, we use the modulation analysis associated to
the equation \eqref{NLS-ab} to classify the dynamics of $H_a^1$-solution. In
both regimes of scattering results, we do not need the radial assumption in
$d=4,5$. Our result generalizes the scattering results of [31-33] and [3] in
the setting of standard combined NLS.
