Opened 5 years ago

# integral with sqrt*sqrt unsolved

Reported by: Owned by: rws major sage-6.8 calculus N/A

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

### comment:1 Changed 23 months ago by mafra

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.

### comment:2 Changed 23 months ago by rws

• Description modified (diff)
• Summary changed from integral with sqrt*sqrt unsolved while solved when expanded to integral with sqrt*sqrt unsolved

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)`.

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