Differential cross sections are presented for the prompt and non-prompt production of the hidden-charm states X(3872) and ψ(2S), in the decay mode J/ψπ+π−, measured using 11.4 fb−1 of pp collisions at s=8 TeV by the ATLAS detector at the LHC. The ratio of cross-sections X(3872)/ψ(2S) is also given, separately for prompt and non-prompt components, as well as the non-prompt fractions of X(3872) and ψ(2S). Assuming independent single effective lifetimes for non-prompt X(3872) and ψ(2S) production gives RB=ℬ(B→X(3872)+any)ℬ(X(3872)→J/ψπ+π−)/ℬ(B→ψ(2S)+any)ℬ(ψ(2S)→J/ψπ+π−)=(3.95±0.32(stat)±0.08(sys))×10−2 separating short- and long-lived contributions, assuming that the short-lived component is due to Bc decays, gives RB = (3.57 ± 0.33(stat) ± 0.11(sys)) × 10−2, with the fraction of non-prompt X(3872) produced via Bc decays for pT(X(3872)) > 10 GeV being (25 ± 13(stat) ± 2(sys) ± 5(spin))%. The distributions of the dipion invariant mass in the X(3872) and ψ(2S) decays are also measured and compared to theoretical predictions.[Figure not available: see fulltext.]