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sage: var('r phi')
(r, phi)
sage: integral(-sqrt(-2*r^2*cos(phi)^2+25), (phi,0,2*pi))
-integrate(sqrt(-2*r^2*cos(phi)^2 + 25), phi, 0, 2*pi)
sage: integral(-sqrt(-2*r^2*cos(phi)^2+25), phi)
-integrate(sqrt(-2*r^2*cos(phi)^2 + 25), phi)

So Sage/Maxima knows that the minus sign comes out. Given that Wolfram Alpha indicates that the answer is an elliptic integral of the second kind, it's not totally surprising that we don't get a symbolic answer here. Certain Maxima things will output elliptic integrals/functions, but not too many.

So what is your desired output? Keep in mind that many (most?) functions do not have antiderivatives expressible in terms of elementary functions.