In this article, the distributed finite-time optimization problem is studied for integrator chain multiagent systems with mismatched and matched disturbances and quadraticlike local cost functions. The agents' models are permitted to be heterogeneous with different orders ranging from first-order to higher order forms. To solve the problem, a nonsmooth embedded control framework is established, which consists of two parts. In the first part, by using nonsmooth control theory and designing some distributed finite-time estimators to estimate the gradients of the agents' local cost functions, a distributed finite-time optimal signal generator with fractional powers is constructed, of which the output signals converge to the minimizer of the global function in finite time. In the second part, by embedding the generator into the feedback loop, taking its output signals as the local optimal reference outputs for the agents, and combining nonsmooth control and finite-time disturbance observer techniques together, some feedforward-feedback composite tracking controllers are designed for the agents to track their local optimal reference outputs in finite time. Under the proposed control framework, all the agents' outputs converge to the minimizer of the global cost function in finite time and the distributed finite-time optimization goal is achieved. Numerical simulations demonstrate the effectiveness of the proposed control framework.