Integrate the incomplete gamma function of a whole order by parts against a reciprocal power

BREAKING-CHANGES.md

@@ -143,6 +143,8 @@ as `a - b = 0`, as a statement's own solve reads it: `"x + 1 = 3".Solve("x")` wa
 - **Exact forms.** A whole `a >= 1` and `a = 1/2` are elementary.
 - **Integrals.** Of a linear `u`, beside a polynomial, or beside any power of `x` but `1/x` where `u` is a multiple
   of `x`, by parts, in gamma functions of `u`.
+  And of a whole `a <= 0` over a power of another linear, by parts, where what is left is an exponential over
+  linears.
 ([#2075](https://github.com/asc-community/AngouriMath/issues/2075))
 
 | Input | Was (2.6.0) | Now |
@@ -153,6 +155,7 @@ as `a - b = 0`, as a statement's own solve reads it: `"x + 1 = 3".Solve("x")` wa
 | `"gamma(1/3, -2)".EvalNumerical()` | a `FunctionArgumentCountException` | `10.212552064243320672…`, real |
 | `"x^2*gamma(0, a*x)".ToEntity().Integrate("x")` | a `FunctionArgumentCountException` | `x ^ 3 / 3 * gamma(0, a * x) - 1 / (3 * a ^ 3) * 2 * e ^ (-a * x) * (1 + a * x + (a * x) ^ 2 / 2) + C` |
 | `"x^m*gamma(n, b*x)".ToEntity().Integrate("x")` | a `FunctionArgumentCountException` | `(x ^ (m + 1) * gamma(n, b * x) - x ^ m * (b * x) ^ (-m) * gamma(n + m + 1, b * x) / b) / (m + 1) + C` |
+| `"gamma(0, a + b*x)/(c + d*x)^2".ToEntity().Integrate("x")` | a `FunctionArgumentCountException` | `-gamma(0, a + b x)/(d (c + d x))` and two exponential integrals, of `a + b x` and of `b (c + d x)/d` |
 
 ### Numerical equality is to half the evaluation's digits
 

Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs

@@ -5039,6 +5039,11 @@ Logf or Entity.Arcsinf or Entity.Arccosf
                     or Entity.Arctanf or Entity.Arccotanf
                     or Entity.Arcsecantf or Entity.Arccosecantf => true,
                 _ when IsASpecialFunction(factor) => true,
+                // And the upper incomplete gamma function, whose derivative -u^(n - 1) e^(-u) u' leaves
+                // an exponential beside what was integrated: against 1/(c + d x)^2, a rational function
+                // of x times e^(-a - b x) for a whole n <= 0, which partial fractions and the exponential
+                // integral answer. https://github.com/asc-community/AngouriMath/issues/2075
+                Entity.IncompleteGammaf => true,
                 // And a whole power of one, which is the same function for this purpose:
                 // differentiating ln(x)^2 gives 2ln(x)/x, whose x cancels against the integrated
                 // polynomial exactly as ln(x)'s does, leaving x*ln(x) -- one step simpler, and

Sources/Tests/UnitTests/Calculus/IncompleteGammaIntegrationTest.cs

@@ -86,6 +86,18 @@ private static readonly (string Name, string Value)[] Pins =
         [InlineData("x*(x + 1)*gamma(n, a + b*x)")]
         public void BesideAPolynomialWithAnOffset(string integrand) => DifferentiatesBack(integrand, Positive);
 
+        /// <summary>
+        /// Of a whole order <c>n &lt;= 0</c> over a power of another linear, by parts with the function differentiated: the
+        /// remainder is <c>(a + b x)^(n - 1) e^(-a - b x)</c> over a power of <c>c + d x</c>, a rational function times an
+        /// exponential, which partial fractions and the exponential integral answer. Rubi's 8.6.
+        /// </summary>
+        [Theory]
+        [InlineData("gamma(0, a + b*x)/(c + d*x)^2")]
+        [InlineData("gamma(0, a + b*x)/(c + d*x)^4")]
+        [InlineData("gamma(-1, a + b*x)/(c + d*x)^3")]
+        [InlineData("gamma(-3, a + b*x)/(c + d*x)^2")]
+        public void AWholeOrderOverAPowerOfAnotherLinear(string integrand) => DifferentiatesBack(integrand, Positive);
+
         /// <summary>
         /// As the function it is, rather than expanded: <c>Gamma(100, a x)</c> is a polynomial of degree 99 times
         /// <c>e^(-a x)</c>, and a hundred rounds of parts.