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IntegrateRationalOverPowerOfQuadratic​(AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity,​System.​Int32,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

∫ N(x)/(a x^2 + b x + c)^n dx for a numerator of any degree, or null where the fraction is improper and this is not the rule for it.

Remarks

Dividing the numerator by the quadratic writes N = qQ + r with r linear
at worst, so N/Q^n is q/Q^(n-1) + r/Q^n — one power off the denominator
and two degrees off the numerator at each step, which ends. The remainder is answered
by the two helpers above and the quotient goes round again.
Improper fractions are declined rather than divided out here. The recursion
bottoms out at a single power, and a numerator still of degree two or more there is
exactly the improper case, which
SolveByPartialFractions(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) already opens with
PolynomialLongDivision(AngouriMath.Entity,AngouriMath.Entity,System.Boolean,AngouriMath.Entity.Variable). Declining hands it there instead of
answering it twice, in two shapes, from two places.
 This is not only reach. TryStandardIntegrals runs before every search, so a
shape answered here never reaches the candidate exploration that re-integrates
rewritten forms — which is where this family's cost has always been. Both
x^2/(x^2 + 2)^2 and (x^2 + 1)/(x^2 + 2)^2 were tens of seconds spent to
return the integral unevaluated.

























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