[Rafael-SOWNet]

Every step of the rational integrator wants a proper fraction — a numerator of lower
degree than the denominator — and each of the three declines an improper one rather than
dividing it out. So x^2/(x + 1) had no antiderivative, although it is x - 1 + 1/(x + 1)
and every piece of that has been integrable throughout.

"x^2/(x + 1)".Integrate("x")

was  integral(x ^ 2 / (x + 1), x)
is   x ^ 2 / 2 + -x + ln(x + 1) + C

The division was already written

TreeAnalyzer.PolynomialLongDivision has done this all along, for the simplifier's own
PolynomialLongDivision rule set. The integrator never asked it. The diff to the library is
one call, placed before the three decompositions rather than after them — no new algorithm,
and nothing about the division itself changed.

That is worth saying plainly because the gap read like a missing capability from the outside:
three separate places in the code say an improper fraction "has to be divided out first, which
is not done here", and the thing they describe existed one namespace away.

What it delivers

As written:

integrand	now
x^2/(x + 1)	x^2/2 - x + ln(x + 1) + C
x^3/(1 + x^2)	answered
x^4/(x^2 + 1)	answered
(x^5 + 2)/(x^2 + 1)	answered
(x^2 + 3x + 5)/(x + 2)	answered
(x^3 + 1)/(x + 1), (x^2 - 1)/(x - 1)	answered — exact division, proper part zero

Through a substitution, which is where it matters more, because there the improper fraction is
produced rather than written by anyone:

integrand	becomes	now
tan(x)^2	u^2/(1 + u^2)	tan(x) - arctan(tan(x)) + C
tan(x)^3	u^3/(1 + u^2)	answered
sqrt(x)/(x + 1)	2u^2/(1 + u^2)	answered

int tan(x)^2 comes back as tan(x) - arctan(tan(x)) + C rather than the textbook
tan(x) - x. Those are the same function on the principal branch, and the second is what the
rewrite would have to assert about which branch x is on, so the first is the honest form.

The declined lists caught this, which is what they are for

All three of tan(x)^2, tan(x)^3 and sqrt(x)/(x + 1) were on "what is still declined"
theories added earlier today, each naming the improper-fraction gap as its reason. All three
failed the moment this landed, and moved to the positive cases with a note saying where they
came from. A boundary recorded as a test fails when the boundary moves; recorded as prose it
just goes quietly wrong.

What is still declined

A proper fraction is untouched — the helper answers nothing for one, so the three steps
below it see exactly what they saw before. int 1/(x + 1) is still ln(x + 1) + C and
int x/(x^2 + 1) still ln(x^2 + 1)/2 + C, both asserted.

A symbolic leading coefficient, and this corrects something I wrote on #180 earlier today.
Item 18 there is x^2/(a + b*x), which I said should come with this work. It does not: the
division would have to divide by b, which is not decidably non-zero, and at b = 0 the
quotient is x^2/a, whose antiderivative is not the limit of the divided form. x^2/(x + a)
is declined too, for the narrower reason that the helper does not divide a polynomial whose
other coefficients are symbolic even when the leading one is 1. Both are pinned by a test,
and I will correct the comment on #180.

Handling either would want the Piecewise-on-the-degenerate-case treatment that
IntegrateRationalQuadratic already uses for a = 0, which is its own change with its own
soundness argument.

Verification

Full suite 9356 passed, 0 failed, 14 skipped. Corpus still 40 of 40.

Every new answer is checked by differentiating it back and comparing at sampled points, through
a helper that first asserts the antiderivative contains no integral( — without that guard the
check passes vacuously, since d/dx of an unevaluated integral(f, x) is f.

Part of #180.