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SolveAsTheDerivativeOfAProductOfPowers(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A product of powers times a sum that is the derivative of the product with some of
the powers raised by one: e^x x^2 ln(x)^2 (3 + (3 + x) ln(x)) is
(e^x x^3 ln(x)^3)', and Rubi's
F^(c (a + b x)) x^m ln(d x)^n (p + p n + p (1 + m + b c x ln F) ln(d x)) is
(p F^(c (a + b x)) x^(m + 1) ln(d x)^(n + 1))', for symbols m and n.

Remarks

The product rule on G = K prod f_i^(e_i) gives G' = G sum e_i f_i'/f_i,
so an integrand of that shape is a product of the same powers, each lowered where
its base's derivative divides it, times a sum. Read the other way: the sum in the
integrand names the powers to raise. Every subset of the powers with a base holding
x and an exponent free of it is tried raised by one, the exponentials kept,
and the integrand divided by the candidate's derivative must be a constant: decided
at sampled points first, with every symbol pinned, and where it is, the constant is
the quotient of the two sums simplified -- p above -- and the answer is checked
by differentiating it back. Nothing else reads a symbolic exponent: the Risch-Norman
ansatz wants whole powers of its monomials, and splitting the sum loses it, since
F^(c(a + bx)) x^m ln(dx)^n on its own is not elementary.
Where a power is not a whole one, the sum may be missing -- the product of powers is
then a constant times its own raised product's derivative -- and x may be raised from
nothing, a power of it the integrand does not have.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

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AngouriMath

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

SolveAsTheDerivativeOfAProductOfPowers(AngouriMath.Entity,AngouriMath.Entity.Variable)