Simulink: integration with mixed variables in function argument

I would like to implement in Simulink the following dynamic system.

System equation:

Functions and are known functions of the following form:

is the input signal and it has the following form:

for and for the other time instants.

are known constants.

I have implemented the system as in the following picture. I run it through a MATLAB code where I generate timeseries for u, h and g.

The results that I get are different from the reference I have. I think that I am not carrying out the integration correctly. More in particular I guess that function h and function g should not be timeseries, but some other kind of functions. However I do not know what to use in order to represent the two integrations and correctly.