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The following code running with Sage rather than mpmath runs for me, but it says there are no zeros on the interval.


G_0 = 37.7
B = 0.36

def S(x_):
     return add([(12*A**4*x_**6*i**4-30*A**2*x_**3*i**2*(j**2+k**2)+3*(j**2+k**2)**2)/(2*(A**2*x_**3*i**2+j**2+k**2)**(7/2)) for i in range(1,N+1) for j in range(1,N+1) for k in range(1,N+1)])

def F(x_):
     return G_0 * (x_ - 1/(x_**2))

print(find_root(F(x) + B*A*sqrt(x)*S(x), 0.85,1) )