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IntegrateAPolynomialOverAPowerOfATrinomial​(AngouriMath.​Entity,​AngouriMath.​Entity,​AngouriMath.​Entity.​Variable,​System.​Boolean)

 Method (no overloads)

Summary

A polynomial over a power of a trinomial in x^n,
P(x)/(a + b x^n + c x^(2n))^k with n >= 2 and k >= 2, a power at
a time down to a polynomial over the trinomial itself.

Remarks

For a monomial, with w = x^n, T = a + b w + c w^2 and D = 4 a c - b^2,
the derivative of x^(m+1) (beta0 + beta1 w) T^(p+1) is
x^m T^p + delta0 x^m T^(p+1) + delta1 x^(m+n) T^(p+1), where
beta0  = (b^2 - 2 a c)/(a n (p + 1) D),       beta1 = b c/(a n (p + 1) D),
delta0 = -((2 a c - b^2)(m + 1) + n (p + 1) D)/(a n (p + 1) D),
delta1 = b c (m + 2 n p + 3 n + 1)/(a n (p + 1) D),

which is what matching the powers of w in
(m + 1)(beta0 + beta1 w) T + n beta1 w T + (p + 1) n w (b + 2 c w)(beta0 + beta1 w) against 1 + (delta0 + delta1 w) T gives. So int x^m T^p is that algebraic
term, less delta0 int x^m T^(p+1) and delta1 int x^(m+n) T^(p+1): two
monomials a power up. What reaches the first power is gathered into one polynomial
over T and integrated once.
 After the Hermite reduction, which answers the square with symbols in it in one linear
solve and keeps those answers; its system grows with the degree, and
x^4/(a + b x^2 + c x^4)^3 was declined. Rubi's 1.2.2.2 and 1.2.2.4 have these by the
dozen. a and D are divided by, which is the generic case, as everywhere in
the integrator: D = 0 is a square, which the rules for one take.
https://github.com/asc-community/AngouriMath/issues/718

























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