I would have done:
sage: var('m k')
(m, k)
sage: assume(m, 'complex')
sage: assume(k, 'complex')
sage: assume(imag_part(k*m) < 0)
sage: assumptions()
[m is complex,
k is complex,
imag_part(m)*real_part(k) + imag_part(k)*real_part(m) < 0]
but this seems not to work in your case. With decomposing a little more:
sage: var('mi mr ki kr')
(mi, mr, ki, kr)
sage: assume(mr, 'real')
sage: assume(mi, 'real')
sage: assume(kr, 'real')
sage: assume(ki, 'real')
sage: m = mr + I*mi
sage: k = kr + I*ki
sage: assume((k*m).imag_part() < 0)
sage: assumptions()
[mr is real,
mi is real,
kr is real,
ki is real,
-imag_part(kr)*imag_part(mi) - imag_part(ki)*imag_part(mr) - imag_part(mi)*real_part(ki) + imag_part(mr)*real_part(kr) - imag_part(ki)*real_part(mi) + real_part(kr)*real_part(mi) + imag_part(kr)*real_part(mr) + real_part(ki)*real_part(mr) < 0]
seems not to work either. If we replace the last assumption by:
sage: assume(ki*mr+kr*mi < 0)
it complains that some assumptions are missing, then:
sage: assume(kr>0)
sage: assume(ki>0)
sage: assume(mr>0)
sage: assume(mi<0)
at his point Maxima accepts to work with your integral but provides nothing interesting:
sage: assumptions()
[mr is real,
mi is real,
kr is real,
ki is real,
kr*mi + ki*mr < 0,
kr > 0,
mr > 0,
ki > 0,
mi < 0]
sage: integral(f, x, 0, infinity)
integrate(((1/x + 1)^0.5 - 1)*e^((2*ki - 2*I*kr)*(I*mi + mr)*x), x, 0, +Infinity)
sage: assume(kr>0)
Sorry for not helping more, it seems that Maxima is not able to solve this integral.
You can try with sympy as folows, but i do not think the result is more useful:
sage: sage: assume(m, 'complex')
sage: sage: assume(k, 'complex')
sage: sage: assume(imag_part(k*m) < 0)
sage: sympy.integrate(f, (x, 0, infinity))
Piecewise((-0.282094791773878*meijerg(((0.5,), ()), ((-1, 0), ()), 2*exp_polar(I*pi/2)*polar_lift(k)*polar_lift(m)) + I/(2*k*m), Abs(periodic_argument(exp_polar(I*pi/2)*polar_lift(k)*polar_lift(m), oo)) < pi/2), (Integral(((1 + 1/x)**0.5 - 1)*exp(-2*I*k*m*x), (x, 0, oo)), True))
