The orderp command computes the order at p of the p-adic expansion of a rational function ex in x.

The lcoeffp command computes the leading coefficient  at p of the p-adic expansion of a rational function ex in x.

$\mathrm{with}\left(\mathrm{padic}\right)\:$

$\mathrm{expansion}\left(\frac{x^3+1}{x^2+3x+5}\,x^2+2\,x\right)$

${\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_1+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_2}{\left(x^2+2\right)}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_3}{{\left(x^2+2\right)}^2}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_4}{{\left(x^2+2\right)}^3}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_5}{{\left(x^2+2\right)}^4}+\frac{{\left({\mathrm{p\_adic}\left(x^2+2\,0\,\left[-\frac{x}{3}-\frac{1}{3}\,\frac{4}{9}\,-\frac{4}{81}+\frac{4x}{81}\,\frac{4x}{729}-\frac{16}{729}\,-\frac{8x}{6561}-\frac{4}{6561}\,-\frac{20x}{59049}+\frac{44}{59049}\right]\right)}_3\right)}_6}{{\left(x^2+2\right)}^5}+\mathrm{O}\left({\left(x^2+2\right)}^6\right)$

$\mathrm{orderp}\left(\frac{x^3+1}{x^2+3x+5}\,x^2+2\,x\right)$

$0$

$\mathrm{lcoeffp}\left(\frac{x^3+1}{x^2+3x+5}\,x^2+2\,x\right)$

$-\frac{x}{3}-\frac{1}{3}$

$\mathrm{expansion}\left(\frac{x^3+1}{x^2+3x+5}\,\frac{1}{x}\,x\right)$

$\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_1}{\left(\frac{1}{x}\right)}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_2}{{\left(\frac{1}{x}\right)}^2}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_3}{{\left(\frac{1}{x}\right)}^3}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_4}{{\left(\frac{1}{x}\right)}^4}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_5}{{\left(\frac{1}{x}\right)}^5}+\frac{{\left({\mathrm{p\_adic}\left(\frac{1}{x}\,\mathrm{-1}\,\left[1\,\mathrm{-3}\,4\,4\,\mathrm{-32}\,76\right]\right)}_3\right)}_6}{{\left(\frac{1}{x}\right)}^6}+\mathrm{O}\left({\left(\frac{1}{x}\right)}^5\right)$

$\mathrm{orderp}\left(\frac{x^3+1}{x^2+3x+5}\,\frac{1}{x}\,x\right)$

$\mathrm{-1}$

$\mathrm{lcoeffp}\left(\frac{x^3+1}{x^2+3x+5}\,\frac{1}{x}\,x\right)$

$1$