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Probably Maxima just doesn't know this integral. They do get some hypergeometric functions as a result of certain integrals, but it is not as strong in HG functions as Mma. I do get some results if I assume c>1, like

sage: integrate((a/x+x^c)^3,x)
3*a^2*x^(c - 1)/(c - 1) - 1/2*a^3/x^2 + 3/2*a*x^(2*c)/c + x^(3*c + 1)/(3*c + 1)

but Maxima asks redundant and other questions on this a bunch.

(%i8) display2d:false;

(%o8) false
(%i9) integrate((a/x+b*x^c)^3,x); 

Is  c-1  zero or nonzero?

nonzero;
Is  3*c+1  zero or nonzero?

nonzero;
Is  c  zero or nonzero?

nonzero;
(%o9) b^3*x^(3*c+1)/(3*c+1)+3*a*b^2*x^(2*c)/(2*c)+3*a^2*b*x^(c-1)/(c-1)-a^3
                                                                        /(2
                                                                         *x^2)

And once you replace the exponent with a d or some noninteger exponent, it doesn't seem to know what to do.