A general mixed boundary element approach based on displacement and traction integral equations for anisotropic media is presented. Integration of the singular and hypersingular kernels along general quadratic line elements is carried out by analytical transformation of the integrals into regular ones, which are numerically evaluated, plus simple singular integrals with known analytical solution. This is achieved by the simple election of an integration variable, which is consistent with that of the anisotropic fundamental solution. The generality of the method allows for the use of curved elements and discontinuous quarter-point elements to represent Fracture Mechanics problems. Stress Intensity Factors are accurately computed from the crack opening displacement at the nodes of the quarter-point element. Several examples, including curved crack geometries and different material properties are presented.