The surge of video traffic is a challenge for service providers that need to maximize Quality of Experience (QoE) while optimizing the cost of their infrastructure. In this paper, we address the problem of routing multiple HTTP-based Adaptive Streaming (HAS) sessions to maximize QoE. We first design a QoS-QoE model incorporating different QoE metrics which is able to learn online network variations and predict their impact on representative classes of adaptation logic, video motion and client resolution. Different QoE metrics are then combined into a QoE score based on ITU-T Rec. P.1202.2. This rich score is used to formulate the routing problem. We show that, even with a piece-wise linear QoE function in the objective, the routing problem without controlled rate allocation is non-linear. We therefore express a routing-plus-rate allocation problem and make it scalable with a dual subgradient approach based on Lagrangian relaxation where subproblems select a single path for each request with a trivial search, thereby connecting explicitly QoE, QoE and HAS bitrate. We show with ns-3 simulations that our algorithm provides values for HAS QoE metrics (quality, rebufferings, variation) equivalent to MILP and better than QoS-based approaches.