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SolveASumOfPowersOfXTimesAPowerOfALinearInARootOfX(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A sum of powers of x times one power with a symbol in its exponent of a linear in a
root of x, or of the square of one written out, sum c_i x^(m_i) G with
G = B^p and B either L = alpha + beta u or gamma L^2 in
u = x^(1/k): the derivative of G L T for a sum T of powers of u,
where the sum is elementary.

Remarks

With s the power of L in B, (G L u^j)' in u is
G ((s p + 1 + j) beta u^j + j alpha u^(j - 1)), so the coefficients of T are read
off the integrand's, in u, from its highest power down, the pivot (s p + 1 + j) beta not zero for a symbolic p. Down to the lowest power or 0, whichever is lower, and one
equation is left over: j alpha t_j = 0 at that lowest power, which holds exactly
where the sum is elementary.
Each term on its own is not elementary -- x^m (a + b x^(1/k))^q with q symbolic and
k m + k at most 0 is a hypergeometric function -- and the sum was split. Rubi's
1.2.3.2 Q^p/x^2 - 2/3 b^3 (1 - 2p)(1 - p) p Q^p/(a^3 x) with
Q = a^2 + 2 a b x^(1/3) + b^2 x^(2/3) is (a + b x^(1/3)) Q^p times three
powers of x. After the rule for the derivative of a product of powers, which answers the
sums whose antiderivative is one. The answer is differentiated back at sampled points
before it is returned.
https://github.com/asc-community/AngouriMath/issues/233

AngouriMath

Table of contents

AngouriMath

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

SolveASumOfPowersOfXTimesAPowerOfALinearInARootOfX(AngouriMath.Entity,AngouriMath.Entity.Variable)