Problem that integrates the special function with the assumption.

I found different results when integrating a special function (see below), depending on where I put my guess (x> 0, x0> 0). The problem is that the two solutions are not compatible.

In fact, if I perform

FullSimplify[Integrate[(Exp[-((x + x0)/t)] Sqrt[x0/x] BesselI[1, (2 Sqrt[x x0]) / t]) / t, {t, 0, [Infinity]}, Assumptions -> {x> 0, x0> 0}]]

results

(x + x0 - Abs[x - x0]) / (2 x)

Whereas if I evaluate

FullSimplify[Integrate[(Exp[-((x + x0)/t)] Sqrt[x0/x] BesselI[1, (2 Sqrt[x x0]) / t]) / t, {t, 0, [Infinity]}], Assumptions -> {x> 0, x0> 0}]

then the result is

(x + x0) / (2 x)

Do you have any idea, why? Thank you