code: def test_prde_normal_denom():
      DE = DifferentialExtension(
          extension={"D": [Poly(1, x), Poly(1 + t**2, t)]}
      )
      fa = Poly(1, t)
      fd = Poly(x, t)
      G = [
          (Poly(t, t), Poly(1 + t**2, t)),
          (Poly(1, t), Poly(x + x * t**2, t)),
      ]
      assert prde_normal_denom(fa, fd, G, DE) == (
          Poly(x, t, domain="ZZ(x)"),
          (Poly(1, t, domain="ZZ(x)"), Poly(1, t, domain="ZZ(x)")),
          [
              (
                  Poly(x * t, t, domain="ZZ(x)"),
                  Poly(t**2 + 1, t, domain="ZZ(x)"),
              ),
              (
                  Poly(1, t, domain="ZZ(x)"),
                  Poly(t**2 + 1, t, domain="ZZ(x)"),
              ),
          ],
          Poly(1, t, domain="ZZ(x)"),
      )
      G = [
          (Poly(t, t), Poly(t**2 + 2 * t + 1, t)),
          (Poly(x * t, t), Poly(t**2 + 2 * t + 1, t)),
          (Poly(x * t**2, t), Poly(t**2 + 2 * t + 1, t)),
      ]
      DE = DifferentialExtension(
          extension={"D": [Poly(1, x), Poly(t, t)]}
      )
      assert prde_normal_denom(Poly(x, t), Poly(1, t), G, DE) == (
          Poly(t + 1, t),
          (Poly((-1 + x) * t + x, t), Poly(1, t, domain="ZZ[x]")),
          [
              (Poly(t, t), Poly(1, t)),
              (Poly(x * t, t), Poly(1, t, domain="ZZ[x]")),
              (Poly(x * t**2, t), Poly(1, t, domain="ZZ[x]")),
          ],
          Poly(t + 1, t),
      )
  
imports: from sympy.integrals.risch import DifferentialExtension, derivation
  from sympy.integrals.prde import prde_normal_denom, prde_special_denom, prde_linear_constraints, constant_system, prde_spde, prde_no_cancel_b_large, prde_no_cancel_b_small, limited_integrate_reduce, limited_integrate, is_deriv_k, is_log_deriv_k_t_radical, parametric_log_deriv_heu, is_log_deriv_k_t_radical_in_field, param_poly_rischDE, param_rischDE, prde_cancel_liouvillian
  from sympy.polys.polymatrix import PolyMatrix
  from sympy.core.numbers import Rational
  from sympy.core.singleton import S
  from sympy.core.symbol import symbols
  from sympy.polys.domains.rationalfield import QQ
  from sympy.polys.polytools import Poly
  from sympy.abc import x, t, n
label_desc: Use image node_id 1 for calling a global function with example usage: test_prde_normal_denom() without return types