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You can make the substitution after integration:

sage: var('a b c d y')

sage: ii=integrate((a/x+b*x^c)^d,x,algorithm='mathematica_free')

sage: b=function('b',y)

sage: ii(b=b(y))

-(x^cb(y) + a/x)^dxhypergeometric2f1(-(d - 1)/(c + 1), -d, -(d - 1)/(c + 1) + 1, -x^(c + 1)b(y)/a)/((d - 1)(x^(c + 1)b(y)/a + 1)^d)

but I don't think that sage understands hypergeometric2f1

You can make the substitution after integration:

sage: var('a b c d y')y') sage: ii=integrate((a/x+b*x^c)^d,x,algorithm='mathematica_free') sage: b=function('b',y) sage: ii(b=b(y)) sage: ii=integrate((a/x+b*x^c)^d,x,algorithm='mathematica_free') sage: b=function('b',y) sage: ii(b=b(y)) -(x^cb(y) -(x^c*b(y) + a/x)^dxhypergeometric2f1(-(d a/x)^d*x*hypergeometric2f1(-(d - 1)/(c + 1), -d, -(d - 1)/(c + 1) + 1, -x^(c + 1)b(y)/a)/((d 1)*b(y)/a)/((d - 1)(x^(c 1)*(x^(c + 1)b(y)/a 1)*b(y)/a + 1)^d)1)^d)

but I don't think that sage understands hypergeometric2f1