code: def test_gosper_sum_AeqB_part3():
      f3a = 1 / n**4
      f3b = (6 * n + 3) / (
          4 * n**4 + 8 * n**3 + 8 * n**2 + 4 * n + 3
      )
      f3c = 2**n * (n**2 - 2 * n - 1) / (n**2 * (n + 1) ** 2)
      f3d = n**2 * 4**n / ((n + 1) * (n + 2))
      f3e = 2**n / (n + 1)
      f3f = (
          4
          * (n - 1)
          * (n**2 - 2 * n - 1)
          / (n**2 * (n + 1) ** 2 * (n - 2) ** 2 * (n - 3) ** 2)
      )
      f3g = (
          (n**4 - 14 * n**2 - 24 * n - 9)
          * 2**n
          / (n**2 * (n + 1) ** 2 * (n + 2) ** 2 * (n + 3) ** 2)
      )
  
      # g3a -> no closed form
      g3b = m * (m + 2) / (2 * m**2 + 4 * m + 3)
      g3c = 2**m / m**2 - 2
      g3d = Rational(2, 3) + 4 ** (m + 1) * (m - 1) / (m + 2) / 3
      # g3e -> no closed form
      g3f = -(
          Rational(-1, 16) + 1 / ((m - 2) ** 2 * (m + 1) ** 2)
      )  # the AeqB key is wrong
      g3g = Rational(-2, 9) + 2 ** (m + 1) / (
          (m + 1) ** 2 * (m + 3) ** 2
      )
  
      g = gosper_sum(f3a, (n, 1, m))
      assert g is None
      g = gosper_sum(f3b, (n, 1, m))
      assert g is not None and simplify(g - g3b) == 0
      g = gosper_sum(f3c, (n, 1, m - 1))
      assert g is not None and simplify(g - g3c) == 0
      g = gosper_sum(f3d, (n, 1, m))
      assert g is not None and simplify(g - g3d) == 0
      g = gosper_sum(f3e, (n, 0, m - 1))
      assert g is None
      g = gosper_sum(f3f, (n, 4, m))
      assert g is not None and simplify(g - g3f) == 0
      g = gosper_sum(f3g, (n, 1, m))
      assert g is not None and simplify(g - g3g) == 0
  
imports: from sympy.core.numbers import Rational, pi
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol
  from sympy.functions.combinatorial.factorials import binomial, factorial
  from sympy.functions.elementary.exponential import exp, log
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.special.gamma_functions import gamma
  from sympy.polys.polytools import Poly
  from sympy.simplify.simplify import simplify
  from sympy.concrete.gosper import gosper_normal, gosper_sum, gosper_term
  from sympy.abc import a, b, j, k, m, n, r, x
label_desc: Use image node_id 11 for calling a global function with example usage: test_gosper_sum_AeqB_part3() without return types