Essentially correct, bit it’s a tad less simpler :
sage: f(x) = x^2*(log(x))^4/((x+1)*(1+x^2))
sage: list(map(lambda u:u.integrate(x, 0, 1), f(x).partial_fraction_decomposition()))
[-5/128*pi^5 + 45/64*zeta(5), 45/4*zeta(5)]
sage: list(map(lambda u:mathematica.Integrate(u, (x, 0, 1)).sage(), f(x).partial_fraction_decomposition()))
[-5/128*pi^5 + 45/128*zeta(5), 45/4*zeta(5)]
HTH
please post here the link to your report