In accordance with this statement obtained by Чебышёв (1853), each of
simplify(int(x^(1/2)*(x^2 + 1)^(-3/4), x), symbolic);
simplify(int((x^(1)*(1 - x^2))^(1/3), x), symnolic);
simplify(int(x^(-1)*(x^6 + 1)^(-1/6), x), symnolic);
simplify(int(x^(17/2)*(x^2 + 1)^(1/4), x), symnolic);
can be reduced to an integral of rational functions, which can be expressed in terms of elementary functions. But it appears that Maple 2023.0 is still unable to completely calculate them. For instance:
However, closed-form (and readable) solutions in elementary forms exist (cf. Regression reports for Computer Algebra Independent Integration Tests. Summer 2022 version (12000.org)); in fact, Mathematica returns:
So, why can't Maple find these compact antiderivatives (expressed by elementary functions) directly here? In other words, is there a way to resolve them in Maple without applying some change of the variable to these indefinite integrals manually?