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This is because in the first case, Maxima is not able to decide whether 5/2*a - 1/2*sqrt(25*a^2 + 40) is negative, and when it does not know, it answers False:

sage: bool(5/2*a - 1/2*sqrt(25*a^2 + 40) < 0)
False
sage: bool(5/2*a - 1/2*sqrt(25*a^2 + 40) > 0)
False

to be compared to:

sage: bool(3*a < sqrt(9*a^2 + 10))
True
sage: bool(3*a > sqrt(9*a^2 + 10))
False

The fun thing, it seems that what Maxima is not able to do is to multiply everything by 2 to get the correct result:

sage: bool(5*a - 1*sqrt(25*a^2 + 40) < 0)    
True
sage: bool(5*a - 1*sqrt(25*a^2 + 40) > 0)
False