Six-body inelastic reactions are calculated microscopically including the full six-nucleon final state interaction. The total cross sections of [Formula presented] are considered as examples. The Lorentz integral transform method and the effective interaction approach for the hyperspherical formalism are employed. While [Formula presented] has a single broad giant resonance peak, there are two well separated peaks for [Formula presented] corresponding to the breakup of the neutron halo and the [Formula presented] core, respectively. The comparison with the few available experimental data is discussed.