Encoding many qubits in different degrees of freedom (DOFs) of single photons is one of the routes towards enlarging the Hilbert space spanned by a photonic quantum state. Hyperentangled photon states (i.e. states showing entanglement in multiple DOFs) have demonstrated significant implications for both fundamental physics tests and quantum communication and computation. Increasing the number of qubits of photonic experiments requires miniaturization and integration of the basic elements and functions to guarantee the set-up stability. This motivates the development of technologies allowing the control of different photonic DOFs on a chip. Femtosecond laser writing on a glass makes possible to use both path and polarization of photon states enabling precise control of both degrees of freedom. We demonstrate the contextual use of path and polarization qubits propagating within a laser written integrated quantum circuit and use them to engineer a four qubit hyperentangled cluster state. We also characterized the cluster state and exploited it to perform the Grover's search algorithm following the one-way quantum computation model. In addition, we tested the non-local properties of the cluster state by using multipartite non-locality tests.