(As I’ve asked on the Math StackExchange and on a related previous question), I am interested in getting a complete symbolic solution to the integral of an expression with a lot of unassigned variables. If you combine some of the variables, the integral can be reduced to the form:

$$int_{-infty}^infty frac{text{A} Delta +text{B}}{left(Delta ^2+W^2right) left(text{C}+text{D}Delta +text{E}Delta ^2 right)}dDelta$$

Mathematica claims that the solution to this integral is:

$$frac{pi (text{B}-i text{A} W)}{W (text{C}-W (text{E} W+i text{D}))} text{if: } Imleft(frac{Epmsqrt{E^2-4 C E}}{E}right)<0 $$

Shown as code:

```
Integrate(
(A1 Δ + B1)/((W^2 + Δ^2) (C1 + D1 Δ + E1 Δ^2)),
{Δ, -∞, ∞}, Assumptions -> {W > 0})
```

Which returns:

```
ConditionalExpression((π (B1 - I A1 W))/(
W (C1 - W (I D1 + E1 W))),
Im((D1 - Sqrt(D1^2 - 4 C1 E1))/E1) < 0 &&
Im((D1 + Sqrt(D1^2 - 4 C1 E1))/E1) < 0 && Re(W) > 0)
```

Mathematica generates a conditional-expression, but doesn’t specify if this is a “full” answer. For instance what if we consider the integral under the domain of parameters with the opposite inequalities: $Imleft(frac{Epmsqrt{E^2-4 C E}}{E}right)>0$? Is there a solution in this domain of parameters?

I can try to force Mathematica to spit out an answer under different conditions. For example:

```
Integrate(
(A1 Δ + B1)/((W^2 + Δ^2) (C1 + D1 Δ + E1 Δ^2)),
{Δ, -∞, ∞},
Assumptions -> {W > 0, Im((D1 - Sqrt(D1^2 - 4 C1 E1))/E1) > 0,
Im((D1 + Sqrt(D1^2 - 4 C1 E1))/E1) < 0})
```

I can get another symbolic answer for this new parameter space:

$$frac{i pi left(text{A1} D W+(-i) text{B1} sqrt{D^2-4 C E}-2 text{B1} E Wright)}{W sqrt{D^2-4 C E} left(C+W left(E W+i sqrt{D^2-4 C E}right)right)}$$

Is there an option to do this automatically and generate a solution for the entire set of possible combinations in the domain space? I’m honestly pretty surprised that it does not automatically return a combined piece-wise function with these different integrated results.