AngouriMath
SolveATrinomialInTheSquareBesideTheRootOfAQuadratic(AngouriMath.Entity,AngouriMath.Entity.Variable)
Method (no overloads)
Summary
A rational function with a trinomial in the square below the bar, beside a root of a
quadratic, taken apart over the trinomial's roots in the square:a + b x^2 + c x^4 is c (x^2 - r) (x^2 - s) with r, s = (-b -+ sqrt(b^2 - 4 a c))/(2c) , and each
factor is a quadratic the rule before this one splits the integrand over and closes
beside the root. Rubi's 1.2.2.4sqrt(d + e x^2)/(a + b x^2 + c x^4) was declined.
quadratic, taken apart over the trinomial's roots in the square:
factor is a quadratic the rule before this one splits the integrand over and closes
beside the root. Rubi's 1.2.2.4
Remarks
Asked with r and s symbols of their own and written back after: the
integrand so written is an integrand for every value of them, and the one asked for at
theirs. Named, each stays one atom through the split: with
(sqrt(b^2 - 4 a c) - b)/(2c) in place of r the split declined
x^2 sqrt(d + e x^2)/(a + b x^2 + c x^4) (#1877 was the
same with a quadratic). The same writing where the roots are a conjugate pair: each
piece overx^2 - r is an arctangent of w x/sqrt(d + e x^2) with
w^2 = (d + e r)/(-r) , which holds on either branch of w , and for r off the real line w x/sqrt(d + e x^2) stays off the arctangent's cuts, so the
two conjugate pieces add to a real, continuous antiderivative. Written instead over
the two real quadraticsx^2 -+ p x + m , the answer to
x^2 sqrt(d + e x^2)/(a + b x^2 + c x^4) was 29,139 characters where this one is
under 2,000. Not where the discriminant vanishes, the generic case's. With a symbol in
the trinomial only, and in a function of x^2 only.
https://github.com/asc-community/AngouriMath/issues/1885
integrand so written is an integrand for every value of them, and the one asked for at
theirs. Named, each stays one atom through the split: with
same with a quadratic). The same writing where the roots are a conjugate pair: each
piece over
two conjugate pieces add to a real, continuous antiderivative. Written instead over
the two real quadratics
under 2,000. Not where the discriminant vanishes, the generic case's. With a symbol in
the trinomial only, and in a function of x^2 only.
https://github.com/asc-community/AngouriMath/issues/1885
Angouri © 2019-2023 · Project's repo · Site's repo · Octicons · Transparency · 4862 pages online