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SolveAPolynomialOverAPowerOfALinearBesideTwoRoots(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A polynomial over a power of a linear beside the square roots of two others,
P(x) (a + b x)^(m/2) (c + d x)^(n/2) / (g + h x)^k with m and n odd and neither below -1, and k >= 0, by undetermined coefficients:
algebraic terms times sqrt(a + b x) sqrt(c + d x), and a multiple of each of
two integrals that are not algebraic.

Remarks

Write S = sqrt(a + b x) sqrt(c + d x) and Q = (a + b x)(c + d x). Over
S the integrand is T(x)/((g + h x)^k S), where
T = P (a + b x)^((m + 1)/2) (c + d x)^((n + 1)/2) is a polynomial, and
IntegrateOverARootOfAQuadratic(AngouriMath.Entity,AngouriMath.Entity.Variable,AngouriMath.Entity[],AngouriMath.Entity,AngouriMath.Entity,AngouriMath.Entity,System.Func{AngouriMath.Entity,AngouriMath.Entity},AngouriMath.Functions.Algebra.IndefiniteIntegralSolver.ALinearBesideTheRoot,AngouriMath.Entity,System.Nullable{System.Int32},System.Int32) takes that to algebraic terms and two
integrals, as it does for the root of one quadratic.
The two integrals left. With t = sqrt(a + b x)/sqrt(c + d x),
dx/S = 2 dt/(b - d t^2), so int 1/S is 2 artanh(sqrt(r) t)/(b sqrt(r)) with r = d/b. And dx/(y S) = -2 dt/(p t^2 + q) with y = g + h x,
p = g d - h c and q = h a - g b, so int 1/(y S) is
-2 arctan(sqrt(rho) t)/(q sqrt(rho)) with rho = p/q. Each derivative
needs only sqrt(z)^2 = z, so each holds for every value of the symbols that the
divisions allow. Each is real for one sign of r or rho and the other
function for the other sign: the one for the sign of a number, and both, each where
it holds, for a quantity with symbols in it. The slopes are divided by, which is the
generic case, as everywhere in the integrator.
The substitution SolveByAQuotientOfTwoLinearRadicals(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) answers the same
integrands. With symbols in the linears, though, its rational function in t is
over a power of b - d t^2 that grows with the degree of P. Rubi's 1.1.1.6
(A + B x + C x^2) sqrt(c + d x) sqrt(e + f x) took more than 4 GB that way,
where this is a triangular system.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

SolveAPolynomialOverAPowerOfALinearBesideTwoRoots​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveAPolynomialOverAPowerOfALinearBesideTwoRoots(AngouriMath.Entity,AngouriMath.Entity.Variable)