AngouriMath

Navigation

Home Quick start Wiki Almanac Research What's new Try Use Angouri.org GitHub Report a bug ~~Donate~~ Contacts

SolveCubeRootsOfTwoLinearsBesideAWholePowerOfAThird(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)

Method (no overloads)

Summary

Cube roots of two linears whose powers add up to a whole number, beside a whole power
of a third linear, (a + b x)^(p/3) (c + d x)^(r/3) (g + h x)^n with p + r a multiple of 3, as the rational function the quotient of the two roots makes of it,
t = (a + b x)^(1/3)/(c + d x)^(1/3), with its compound coefficients named.

Remarks

With D = a d - b c, W = b - d t^3 and s = (p + r)/3: x is
(c t^3 - a)/W, c + d x = -D/W, g + h x = N/W for
N = (b g - a h) + (c h - d g) t^3, and dx = -3 D t^2 dt/W^2; and the two
roots are t^p (c + d x)^s. So the integrand is
3 (-D)^(s + 1) t^(p + 2) N^n / W^(s + n + 2), rational in t, and exactly so:
a whole power of the quotient of the principal roots is the quotient of their powers
for every complex x, as SolveByAQuotientOfTwoLinearRadicals(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) says.
That rule makes the same substitution and declined these, past the budget: its
rational function is written as the substitution leaves it, with
k (b - d t^3) + f (c t^3 - a) where N is, and the rational rules ran out
of time on t^3/((b k - a f + (c f - d k) t^3) (b - d t^3)^2) where
t^3/((p + q t^3) (b - d t^3)^2) is answered in a quarter of a second. Here
N and W are written as binomials in t^3 from the coefficients, and a
coefficient that is a compound of the symbols is named by a symbol of its own and
written back into the answer. Naming loses no relation the decomposition needs:
W and N have the resultant h D, zero only where the third linear
is constant or the two under the roots are one, and those, and a third linear that is
a multiple of one of the two, are declined, each decided as a value.
Only cube roots, and only a third linear below the bar where the denominator in
t is not squarefree, n <= -2 or W^2 beside N: Rubi's
(a + b x)^m (c + d x)^n (e + f x)^p rows with m, n thirds. Of the
rest, the third linear above the bar or below it once beside W at most once, the
substitution rule answers most, and they are left to it as they are.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

Table of contents

AngouriMath

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

SolveCubeRootsOfTwoLinearsBesideAWholePowerOfAThird(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean)