We considered the Distributed Time--Fractional Ordinary Differential Inequality with initial data assumed to be positive. Inside the distributed derivative on left side of inequality, we introduced the Generalized Caputo fractional derivative of order in between 0 and 1. On right side, we have Fractional integral with polynomial terms like |u|^p and |u|^r where exponents are bigger or equal to 1. Fractional derivatives and integrals are well known in analysis because of their memory effect. We include a nonnegative locally integrable weight measure in (0,1) to make our problem more challenging. We obtained the most suitable test function, have enough regularity to provide us the well defined weak formulation of the model equation. We used Pohozaev--Mitidieri Test--Function Method well adapted to our Distributed Fractional Model. Using this method, our analysis is to establish the nonexistence of nontrivial weak solutions using fractional properties and nonlinear estimates for the memory term. It will be shown that we end up with a differential inequality of Fujita type. By a careful scaling analysis of the test functions, we determine an adequate explicit Fujita--type index that tells us the range when solution does not exist. Our results extend and generalized several Caputo--type evolution problems through, not only the kind of intended fractional derivative, but also through the form of the nonlinearity. Actually, our argument will unify the treatment of several prior works on same type of fractional derivatives and nonlinearities and others as well. Some numerical simulations/computations can make it more interesting and challenging.