When Mathematica returns the input as the output, it means that the calculation returned unevaluated. This often means that the function does not have the methods available to solve the problem symbolically, or it is mathematically impossible to obtain a symbolic solution (not all sums, integrals, or differential equations have symbolic solutions after all).
Simplifying equation ot7 for very simpler form:
Integrate[( E^-(x - a1)^2 + E^-(x - a2)^2 )^(5/2), {x, -Infinity, Infinity}, Assumptions -> {a1 > 0, a2 > 0}]
(*returned unevaluated*)
Or even simpler:
Integrate[( E^(-x^2/2) + 2*E^(-x^2/3) )^(5/2), {x, -Infinity, Infinity}]
(*returned unevaluated*)
Analytical solving seems impossible.