integral with sqrt*sqrt unsolved

[rwst]

sage: integral(sqrt(x-1)*sqrt(1/x-1/4), x)
integrate(sqrt(x - 1)*sqrt(1/x - 1/4), x)

This came up in
http://ask.sagemath.org/question/27237/another-problem-with-integral/

Component: calculus

Keywords: integral

Issue created by migration from https://trac.sagemath.org/ticket/18822

[sagetrac-mafra]

comment:1

I believe this description is wrong, as

sqrt(x-1)*sqrt(1/x-1/4)

is not equal to (note the x^2 outside of the square root)

sqrt(-1/4*x^2 + 5/4*x - 1)/x^2

but it is equal to

sqrt((-1/4*x^2 + 5/4*x - 1)/x)

where x is inside the square root (and not the other way around).

According to Mathematica, sqrt(x-1)*sqrt(1/x-1/4) integrates to a complicated function that involves elliptic integrals.

[rwst]

Description changed:

--- 
+++ 
@@ -2,9 +2,6 @@
 ```
 sage: integral(sqrt(x-1)*sqrt(1/x-1/4))
 integrate(sqrt(x - 1)*sqrt(1/x - 1/4), x)
-
-sage: integrate(sqrt(-1/4*x^2 + 5/4*x - 1)/x^2)
--sqrt(-1/4*x^2 + 5/4*x - 1)/x + 1/2*arcsin(-2/3*x + 5/3) + 5/8*arcsin(5/3*x/abs(x) - 8/3/abs(x))
 ```
 This came up in
 http://ask.sagemath.org/question/27237/another-problem-with-integral/

[rwst]

comment:2

That's right. Elliptic E and F functions with argument containing inverse trig function may already appear as solution to integral(1/sqrt(a+b*x^3), x).

[fchapoton]

Changed keywords from none to integral

[fchapoton]

Description changed:

--- 
+++ 
@@ -1,6 +1,6 @@
 
 ```
-sage: integral(sqrt(x-1)*sqrt(1/x-1/4))
+sage: integral(sqrt(x-1)*sqrt(1/x-1/4), x)
 integrate(sqrt(x - 1)*sqrt(1/x - 1/4), x)
 ```
 This came up in

[mkoeppe]

comment:5

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.