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

 Method (no overloads)

Summary

An integrand holding a fractional power of something linear in the variable,
turned into a rational function by u^q = a*x + b.

Remarks

Setting u = (a*x + b)^(1/q) gives x = (u^q - b)/a and
dx = (q/a) u^(q-1) du, and every occurrence of the radical becomes a power of
u. A quotient of polynomials in x and that radical is then a quotient of
polynomials in u, which the rational integrator answers.
Why the general substitution does not already do this, which is the point.SolveBySubstitution(AngouriMath.Entity,AngouriMath.Entity.Variable,System.Boolean) rewrites sub-expressions: it finds
1 + x inside x/sqrt(1 + x), replaces that, and is left with a bare
x it cannot express, so it declines. This substitutes for x itself, so
there is nothing left behind to fail on. x/sqrt(1 + x) and
x/sqrt(2 - 3x) had no antiderivative for exactly that reason.
One q for the whole integrand. Where several radicals share a base —
sqrt(x + 1) beside (x + 1)^(1/3) — the exponent denominators are taken
together by their least common multiple, so one substitution clears both. Radicals over
different linear bases are not handled: sqrt(1 - x) + sqrt(1 + x) needs
two substitutions at once and is declined rather than half-rewritten.
No condition is owed by the rewrite itself.a is non-zero, since a zero
coefficient is not a linear expression in x and the reader below rejects it,
and u^q = a*x + b is invertible wherever the radical it came from is defined.
The answer inherits the radical's own domain and adds nothing -- except where the
even-root step below has read u as a non-negative real, which it is only
where the radical is real. Beyond that, u is imaginary and the integrand can
still be real: x sqrt(c - a c x) / e^(3 atanh(a x)) above a x = 1 is the
product of two imaginary factors, and the answer built for a real u is not its
antiderivative there (Rubi 7.3.6). So the answer says what it used: it is given
provided a x + b >= 0 exactly when that step changed the integrand.
 https://github.com/asc-community/AngouriMath/issues/718

























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