I want to find the closed form of this integral:
$$ I_5=\int \frac{1}{\sqrt{(x-a) (x-b) (x-c) (x-d) (x-e)}} \, \mathrm{d}x $$
where $a<b<c<d<e$.
My attempt
My idea is to find a transformation so that the part in the root sign can be reduced.
I solved the three and four cases with this idea.
For
$$ I_3=\int \frac{1}{\sqrt{(x-a) (x-b) (x-c)}} \, \rm{d}x $$
let
$$t^2 = \dfrac{b - a}{x - a}$$
then
$$ \begin{aligned} x&=\frac{a t^2-a+b}{t^2}\\ \rm{d}x&=\frac{2 (a-b)}{t^3}\,\rm{d}t \end{aligned} $$
and then
$$ \begin{aligned} I_3 &=\int\frac{-2\,\rm{d}t}{\sqrt{(1-t^2) \left(b-a-(c-a) t^2\right)}}\\ &=-\frac{2}{\sqrt{b-a}}\int\frac{1}{\sqrt{\left(1-t^2\right) \left(1-\dfrac{c-a}{b-a} t^2\right)}}\,\rm{d}t\\ &=-\frac{2}{\sqrt{b-a}}F\left(\arcsin t,\frac{c-a}{b-a}\right)\\ &=-\frac{2}{\sqrt{b-a}}F\left(\arcsin\sqrt{\frac{a-b}{a-x}},\frac{a-c}{a-b}\right) \end{aligned} $$
So for
$$ I_4=\int \frac{1}{\sqrt{(x-a) (x-b) (x-c)(x-d)}} \, \rm{d}x $$
the same let
$$ \begin{aligned} t^2&=\frac{(b-d)(x-a)}{(a-d) (x-b)}\\ x &= \frac{b t^2 (d-a)+a (b-d)}{t^2 (d-a)+b-d}\\ \rm{d}x&=\frac{2 t (a-b) (a-d) (b-d)}{\left(t^2 (d-a)+b-d\right)^2}\, \rm{d}t \end{aligned} $$
then
$$ \begin{aligned} I_4 &=\int\frac{-2\,\rm{d}t}{\sqrt{\left(1-t^2\right) \left((a-c) (b-d)- (a-d) (b-c)t^2\right)}}\\ &=-\frac{2}{\sqrt{(a-c) (b-d)}}F\left(\arcsin t,\frac{(a-d) (b-c)}{(a-c) (b-d)}\right)\\ &=-\frac{2}{\sqrt{(a-c) (b-d)}}F\left(\arcsin\left(\sqrt{\frac{(b-d) (x-a)}{(a-d) (x-b)}}\right),\frac{(a-d) (b-c)}{(a-c) (b-d)}\right) \end{aligned} $$
I have not found such a transformation for $I_5$ yet.
I tried the undetermined coefficient method
let
$$t^2=\frac{m x+n}{u x+v}$$
then this expanded part needs to be a polynomial with respect to $t^2$
$$P_5=\frac{2 t \sqrt{m-t^2 u} (m v-n u)}{\sqrt{-\left(\left(a \left(m-t^2 u\right)+n+t^2 (-v)\right) \left(b \left(m-t^2 u\right)+n+t^2 (-v)\right) \left(c \left(m-t^2 u\right)+n+t^2 (-v)\right) \left(d \left(m-t^2 u\right)+n+t^2 (-v)\right) \left(e \left(m-t^2 u\right)+n+t^2 (-v)\right)\right)}}$$
Can't solve suitable $(m,n,u,v)$