Based on these

In[1]:= Integrate[E^(a x) BesselI[0, a2 x] BesselI[0, a3 (z - x)], {x, 0, z}]

Out[1]= Integrate[E^(a x) BesselI[0, a2 x] BesselI[0, a3 (-x + z)], {x, 0, z}]

In[2]:= Integrate[E^(1 x) BesselI[0, 1 x] BesselI[0, 1 (z - x)], {x, 0, z}]

Out[2]= Integrate[E^(x) BesselI[0, x] BesselI[0, -x + z], {x, 0, z}]

In[3]:= Integrate[BesselI[0, 1 x] BesselI[0, 1 (z - x)], {x, 0, z}]

Out[3]= Integrate[BesselI[0, x] BesselI[0, -x + z], {x, 0, z}]

In[4]:= Integrate[E^(1 x) BesselI[0, 1 (z - x)], {x, 0, z}]

Out[4]= Integrate[E^(1 x)] BesselI[0, -x + z], {x, 0, z}]

I don't think Mathematica is going to find a solution without outside help. Asserting your coefficients are negative does not help.

Note: When the output is the same as the input this indicates that Mathematica was not able to find a solution to the problem.