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i have this integral it must result in all constant values and the value of the integral its self which is Zeta[2/3] it can be done.
ClearAll["Global`*"]
e0 = (3*h^2)/(8*m*L^2)
ee = e0*n^2
ef = n^3/e0;
be[e_] := 1/(Exp[e/(k*t) - u/(k*t)] - 1);
g[e_] := Defer[(2/Sqrt[Pi])*((2*Pi*m)/h^2)^(3/2)*v*Sqrt[e]];
sub1 = {e/(k*t) -> x}
sub2 = {e -> x*k*t}
be[e] /. sub1;
g[e] /. sub2;
f[x_] := Refine[(g[e] /. sub2)*(be[e] /. sub1), Assumptions -> {u == 0}]
Integrate[f[x], {x, 0, Infinity}]
Because of Defer in your g[e] hold x in substitution so not evaluated
remove Defer in g[e] and Do substitution like this then you find your answer
ClearAll["Global`*"]
e0 = (3*h^2)/(8*m*L^2)
ee = e0*n^2
ef = n^3/e0;
be[e_] := 1/(Exp[e/(k*t) - u/(k*t)] - 1);
g[e_] := (2/Sqrt[Pi])*((2*Pi*m)/h^2)^(3/2)*v*Sqrt[e];
sub1 = {(e/(k t) ) -> x}
sub2 = {(e)^(1/2) -> (x k t )^(1/2)}
f[x_] := Refine[((g[e] /. sub2)*(be[e] /. sub1)),
Assumptions -> {u == 0}]
Integrate[f[x], {x, 0, Infinity}]
