Numerical and Asymptotic Consequences of Rewriting Antiderivatives of the Form ∫x(ax + b) p/q dx : General Results and a Case Study

We analyze the numerical and asymptotic consequences of rewriting a family of elementary antiderivatives, with particular attention to the limiting regimes as the variable approaches zero and infinity. Although algebraically equivalent expressions exist, their floating-point evaluations can diverge substantially in both accuracy and stability. We show that the severity of this divergence is governed by the \emph{cancellation ratio}$|n{+}1|/|n{+}2|$, where $n=p/q$ is the exponent: forms whose terms share the same asymptotic growth rate lose $\log_2(|n{+}2|/|n{+}1|)$ bits of precision. For the case study $\int x(ax{+}b)^{-9/10}\,dx$ with $(a,b)=(3,1)$, the cancellation-prone form $F_2$ exhibits roughly $10\times$ larger relative error than $F_1$ across the large-$x$ regime (3.5 bits lost), while more extreme exponents such as $n=-99/100$ show ratios exceeding $100\times$ (6.7 bits). We derive condition number estimates showing that error amplifies as $\mathcal{O}(\epsilon_{\mathrm{mach}}/|\delta|)$ near the critical point $x^\star = b/[a(n{+}1)]$. Through rigorous error analysis, cancellation maps, and sensitivity studies, we demonstrate how rewriting alters asymptotic behavior and the propagation of roundoff errors. We further propose an adaptive runtime algorithm that selects the numerically stable antiderivative form based on asymptotic regime and conditioning. A focused case study illustrates these general stability phenomena and motivates guidelines for representation choice in symbolic--numeric contexts. MSC: 65G50 65D30 65Y20