We obtain the optimal order of high-dimensional integration complexity in the quantum computation model in anisotropic Sobolev classes Wr∞ ([0, 1]d) and Holder Nikolskii classes Hr∞([0, 1]d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n-1 and over randomized classical methods due to factor n-1/2. Moreover, we give an estimation for optimal query complexity in the class H∧∞ (D) whose smoothness index is the boundary of some complete set in Zd+.