  We study the microlocal properties of the geodesic X-ray transform
$\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate
points. Assuming that there are no self-intersecting geodesics and all
conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} =
\mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a
pseudodifferential operator of order $-1$ and a sum of Fourier integral
operators. We also apply this decomposition to prove inversion of $\mathcal{X}$
is only mildly ill-posed in dimension three or higher.
