A constant of integration is matched to a linear divisor with a symbol in it

[Rafael-SOWNet]

By parts against a polynomial chooses the polynomial's antiderivative so that what the other factor's derivative divides by divides it too. Against x ln(1 + x), the antiderivative of x is (x^2 - 1)/2, and the remainder is a polynomial. That was done only where the division left a number over. a + b x leaves a symbol: x^2/2 over a + b x leaves a^2/(2 b^2). So x Shi(a + b x)^2 kept x^2/(a + b x) in its remainder, and nothing read it.

The antiderivative less its value at the root -a/b is divisible by the linear, and it is now written as the linear times the quotient. The linear then cancels as written, and x Shi(a + b x)^2 and x Ei(a + b x)^2 are two rounds of parts each, as x Shi(b x)^2 is (#1639).

A number left over is read as before, so the constant changes only for a linear divisor with a symbol in it. The answers it touches are special functions of a + b x beside a power of x: x Ei(a + b x) reads Ei(a + b x) (a + b x)(x/(2b) - a/(2b^2)) - ..., the same function as before.

Measured

Input	Was (2.5.0)	Now
"x*Shi(a+b*x)^2".Integrate("x")	a ^ 2 * Shi * x ^ 2 / 2 + a * b * 2 * Shi * x ^ 3 / 3 + b ^ 2 * Shi * x ^ 4 / 4 + C, with Shi a variable	an antiderivative in Shi(a + b x) and Ei(2 (a + b x)), Ei(-2 (a + b x))
"x*Ei(a+b*x)^2".Integrate("x")	a ^ 2 * Ei * x ^ 2 / 2 + a * b * 2 * Ei * x ^ 3 / 3 + b ^ 2 * Ei * x ^ 4 / 4 + C, with Ei a variable	an antiderivative in Ei(a + b x) and Ei(2 (a + b x))

• Rubi, master at 6c94652f and this change on it, run side by side, 0 wrong everywhere:
  • Family 8: 373 → 376 of 420. The three are x Ei, x Shi and x Chi of a + b x, squared.
  • Family 2: 544 of 650 on both.
  • The independent suites: 1756 of 1814 on both.
  • Families 1 and 3–7 at five a file: 800 of 897 on both.
• Run alone, the three are 0 of 3 on master and 3 of 3 here. The fourth problem that moved, 2.3 Issue 619 Bug: Solving a system of equations... #622, is 22 s unevaluated on both.
• The unit tests pass: 14,259, none failed (net10.0). On master with A Gaussian below the bar is read as its negated exponent above it #1641 the classes this touches pass (136), and the native AOT publish of the C++ wrapper builds with no trim or AOT warnings.
• The performance gate passes on d480366e, which is this change on 6c94652f. Allocation matches the baseline on all 19 gated benchmarks.

SpecialFunctionsByPartsTest has two rows with a shifted argument, each differentiated back with its parameters pinned.

Still declined, 18 problems of family 8's shifted squares:

• Alone: Shi(a + b x)^2, Si(a + b x)^2, and erf, erfc and erfi of a + b x squared.
• Beside x: x Si(a + b x)^2 and x Ci(a + b x)^2.
• Beside x^2: Ei, Shi, Chi, Si and Ci of a + b x squared.
• Beside (c + d x) and (c + d x)^2: the error functions of a + b x squared.

The term-by-term asking of #1639 is held to a multiple of x, and these need it, or the polynomial divided by the linear first.

Part of #1501.

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