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FWIW :

sage: mathematica.Integrate(f, (x, -infinity, infinity))
ConditionalExpression[(Sqrt[Pi]*Gamma[-l])/(Sqrt[sigma^(-2)]*Gamma[1/2 - l]), 
 Re[l] < 0 && (Re[sigma^2] >= 0 || NotElement[sigma^2, Reals])]

which does not (yet) translate automatically to Sage for lack of symbolic logical functions (one can whip up a cases equivalent manually..).

sage: import sympy
sage: sympy.integrate(*map(sympy.sympify, (f(x), (x, -oo, oo))))
Piecewise((sqrt(pi)*sigma*gamma(-l)/gamma(1/2 - l), ((re(l) < 0) & (Abs(arg(sigma)) < pi/2)) | ((re(l) < 0) & Ne(sigma**(-2), 0) & (Abs(arg(sigma)) < pi/2) & Ne(Abs(arg(sigma)), pi/2))), (Integral((1 + x**2/sigma**2)**(l - 1/2), (x, -oo, oo)), True))

which does not (yet) translate to Sage (ditto...).