ψ[c_,r_]:=r^2Exp[-c r^2]

H=-ℏ^2/(2 μ) (1/r^2 D[r^2 D[#,r],r])-e^2/((4 Pi ϵ) r) #&

VariationalNumerator = Simplify[ψ[c,r]H@ψ[c,r]]

VariationalDenominator=Simplify[ψ[c,r] ψ[c,r]]

EnergyExpression=Integrate[VariationalNumerator r^2, {r,0,Infinity},Assumptions-> {c∈Reals, c>0}]/Integrate[VariationalDenominator r^2, {r,0,Infinity}, Assumptions -> {c∈Reals, c>0}]

Solve[D[%,c]==0,c]

EnergyExpression /.%