I am not sure of why but this used to work in Maxima 5.18.1.
(%i1) assume(E[n]>0,mu>0,m>0,n>0)$
(%i2) sqrt(2)*sqrt(m)*'integrate(sqrt(E[n]-mu*x^4),x,-E[n]^(1/4)/mu^(1/4),E[n]^(1/4)/mu^(1/4))=%pi*hbar*(n-1/2)$
(%i3) changevar (%,z-(mu/E[n])^(1/4)*x, z, x);
->> 0=%pi*hbar*(n-1/2)
This is wrong, in 5.18.1 I used to get
(%i3 changevar (%,z-(mu/E[n])^(1/4)*x, z, x);
->> (sqrt(2)*sqrt(m)*E[n]^(3/4)*integrate(sqrt(1-z)*sqrt(z+1)*sqrt(z^2+1),z,-1,1))/mu^(1/4)=%pi*hbar*(n-1/2)
Rich