# [Maxima] Macsyma elliptic integrals

Milan Lukic milan at eskimo.com
Sat Oct 20 00:37:52 CDT 2007

```* Raymond Toy (RT/EUS) &lt;raymond.toy at ericsson.com&gt; [2007-10-19 16:53:29 -0400]:

> Richard Fateman wrote:
> > Unless there is some other code not called by integrate,  only the 3rd of
> > these examples is integrated. The others return the noun form.
> > After ascertaining that neither a nor b is zero, the answer to item 3 comes
> > out as the sum of 3 asinh..
> >
> >
> > B^2*ASINH(2*ABS(B)*T/(ABS(A)*ABS(2*T+2*ABS(B)))
> > 	   -2*A^2/(ABS(A)*ABS(2*T+2*ABS(B))))
> >  /(2*SQRT(B^2+A^2)*ABS(B))
> >  +B^2*ASINH(2*ABS(B)*T/(ABS(A)*ABS(2*T-2*ABS(B)))
> > 	     +2*A^2/(ABS(A)*ABS(2*T-2*ABS(B))))
> >   /(2*SQRT(B^2+A^2)*ABS(B))-ASINH(T/ABS(A))\$
> >
> >
>
> Hmm.  Oops.  I typed that wrong.  It should be
>
> integrate(t^2/sqrt((t^2+a^2)*(t^2+b^2)),t)
>
> the sqrt of a quartic.
>
> So, Macsyma doesn't know how to do any elliptic integrals?
>
> On a related note, does any have a reference on how to factor a real
> quartic into two quadratics?  I know I can do it by solving the quartic
> and then recombining the roots together to make the quadratic factors,
> but that seems expensive.
>

I don't expect that the following reference is what you are looking
for, but nevertheless, you might enjoy taking a look at how Euler did
it. See "Euler, The Master of Us All", by William Dunham.
Euler reduces A*y^4+B*y^3+C*y^2+D*y+E=0 to x^4+a*x^2+b*x+c=0 (page
104). He then shows how to factor the later into
(x^2+u*x+v)*(x^2-u*x+w), pages 111-115.

hth,
Milan

> Ray
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