third
The third order loss function, (1/6)Emax(X-x,0)^3.
Via partial expectations rather than a hand-expanded cubic. Since ug(u) for a gamma of shape a is ascale times the density of shape a+1, each factor of u raises the shape by one, so the kth partial expectation above x is a^(k) * scale^k * G0 under shape a+k, where a^(k) is the rising factorial.
No separate branch below zero: the complementary distribution functions are all 1 there and the expression collapses to the third moment about x on its own.