Integral Too Hard For Mathematica

This problem can be solved numerically by

s = ParametricNDSolveValue[{z'[t] == Exp[-2 t^2] t^(9/5) (x - t)^(4/5) 
    HypergeometricPFQ[{1/2, 1}, {7/5, 19/10}, -t^2], z[0] == 0}, z, {t, 0, x}, {x}];
Plot[s[x][x], {x, 0, 5}]

As noted by Marius Ladegård Meyer in a comment above, NDSolve itself is not sufficient, because x appears both in the integrand and as the upper limit of integration.