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# sympy.integrals.prde.prde_special_denom

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```"""Most of these tests come from the examples in Bronstein's book."""
from sympy import Poly, Matrix, S, symbols, I
from sympy.integrals.risch import DifferentialExtension
from sympy.integrals.prde import (prde_normal_denom, prde_special_denom,
prde_linear_constraints, constant_system, prde_spde, prde_no_cancel_b_large,
```
```    G = [(Poly(t, t), Poly(1, t)), (Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))]
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert prde_special_denom(a, ba, bd, G, DE) == \
(Poly(t + 1, t), Poly(t**2, t), [(Poly(t, t), Poly(1, t)),
(Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))], Poly(1, t))
G = [(Poly(t, t), Poly(1, t)), (Poly(1, t), Poly(t, t))]
assert prde_special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), G, DE) == \
```
```    DE.decrement_level()
G = [(Poly(t, t), Poly(t**2, t)), (Poly(2*t, t), Poly(t, t))]
assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3 + 2, t), G, DE) == \
(Poly(5*x*t + 1, t), Poly(0, t), [(Poly(t, t), Poly(t**2, t)),
(Poly(2*t, t), Poly(t, t))], Poly(1, x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly((t**2 + 1)*2*x, t)]})
G = [(Poly(t + x, t), Poly(t*x, t)), (Poly(2*t, t), Poly(x**2, x))]
assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3, t), G, DE) == \
```

```"""Most of these tests come from the examples in Bronstein's book."""
from sympy import Poly, Matrix, S, symbols, I
from sympy.integrals.risch import DifferentialExtension
from sympy.integrals.prde import (prde_normal_denom, prde_special_denom,
prde_linear_constraints, constant_system, prde_spde, prde_no_cancel_b_large,
```
```    G = [(Poly(t, t), Poly(1, t)), (Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))]
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly(t, t)]})
assert prde_special_denom(a, ba, bd, G, DE) == \
(Poly(t + 1, t), Poly(t**2, t), [(Poly(t, t), Poly(1, t)),
(Poly(t**2, t), Poly(1, t)), (Poly(t**3, t), Poly(1, t))], Poly(1, t))
G = [(Poly(t, t), Poly(1, t)), (Poly(1, t), Poly(t, t))]
assert prde_special_denom(Poly(1, t), Poly(t**2, t), Poly(1, t), G, DE) == \
```
```    DE.decrement_level()
G = [(Poly(t, t), Poly(t**2, t)), (Poly(2*t, t), Poly(t, t))]
assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3 + 2, t), G, DE) == \
(Poly(5*x*t + 1, t), Poly(0, t), [(Poly(t, t), Poly(t**2, t)),
(Poly(2*t, t), Poly(t, t))], Poly(1, x))
DE = DifferentialExtension(extension={'D': [Poly(1, x), Poly((t**2 + 1)*2*x, t)]})
G = [(Poly(t + x, t), Poly(t*x, t)), (Poly(2*t, t), Poly(x**2, x))]
assert prde_special_denom(Poly(5*x*t + 1, t), Poly(t**2 + 2*x**3*t, t), Poly(t**3, t), G, DE) == \
```