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

Method (no overloads)

Summary

A rational function of a quadratic Q beside a half-odd power of it,
R(Q) Q^(k/2) with R's factors Q + delta, under t = Q'/sqrt(Q): with
D = 4 a c - b^2, Q = D/(4 c - t^2) and dx/sqrt(Q) = 2 dt/(4 c - t^2), so it
is rational in t. Rubi's 1/((d + b x + c x^2)^3 sqrt(a + b x + c x^2)) is
2 (4 c - t^2)^2/(D + 4 c (d - a) - (d - a) t^2)^3.

Remarks

t^2 = (b + 2 c x)^2/Q = 4 c - D/Q, which gives Q, and dt/dx = D/(2 Q^(3/2)); each
holds wherever Q is not zero, on either sign of it. The quadratics below the bar differ from
the radicand by a constant, so they are linear in 1/(4 c - t^2), and the integral in t is
a polynomial over a power of a binomial in t^2, which is answered at once. After the
reduction over a power of the quadratic beside the root of another, which answers the third
and fourth powers in a third of the characters and declines the fifth; ahead of it, this
wrote 4620 characters where that writes 1439. Only the third power below the bar or past it,
the lower ones being answered as they are; the answer is differentiated back at sampled points.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

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AngouriMath

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

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