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LargestPowerSolvedAtOnce

Field

Summary

The reduction of SolveAPolynomialOverAPowerOfAQuadraticBesideTheRootOfAnother(AngouriMath.Entity,AngouriMath.Entity.Variable) as one linear system over the polynomials in the symbols: P/(A^n sqrt(B)) is
(Q sqrt(B)/A^(n - 1))' + (g + h x)/(A sqrt(B)) + sigma/sqrt(B) with Q of
degree 2n - 3, which is
2 P = 2 Q' A B + Q B' A - 2 (n - 1) Q A' B + 2 (g + h x) A^(n - 1) + 2 sigma A^n,
one equation for each power of x up to 2n in as many unknowns. Null where a
coefficient is not a polynomial in at most seven symbols, or the solve declines.

Remarks

The decomposition is the one the reduction a power at a time makes, since it is
unique where A and B have no root in common; written so, each unknown
is one quotient of polynomials, where a step at a time carried each step's quotients
through the next as written. Solved by fraction-free elimination, the values are put
over their least common denominator in lowest terms, and that is written as the
powers of the discriminant of A, the resultant of A and B and the
symbols it is, Rubi's e^2 - 4 d f and
(c d - a f)^2 - (b d - a e) (c e - b f). A's content comes out first: over
a monic quadratic the integrand arrives over d f + e f x + f^2 x^2. The two
last pieces are asked with g, h and sigma as symbols of their own.
A step at a time, Rubi's 1.2.1.5 (a + b x + c x^2)^(3/2)/(d + e x + f x^2)^3 was
declined past LargestReducedCoefficients, and
(a + b x + c x^2)^(3/2)/(d + e x + f x^2)^2 was 140,668 characters; they are 2,160
and 5,686.
https://github.com/asc-community/AngouriMath/issues/233

Summary

The highest power of the quadratic below the bar that
ReducedOverThePolynomialsInItsSymbols(AngouriMath.Entity[],System.Int32,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity) takes: with six symbols the fifth
ran past twenty seconds in the elimination, where the reduction a power at a time
declines it in half a second.

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AngouriMath

LargestPowerSolvedAtOnce