ϕ[α_, x_] = x ( b - x) + α x^2 ( b - x)^2

e[α_] := - h^2 / (2 m) ((∫_0^b)ϕ[α, x]D[ϕ[α, x], {x, 2}]\[DifferentialD]x) / (∫_0^b)ϕ[α, x] ϕ[α, x]\[DifferentialD]x

Simplify[e[α]]

optα = Solve[D[e[α], α] == 0, α]

Simplify[e[α]/.optα]

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