Integral hangs the physical Prime

[Albert Chan]

(05-27-2023 06:14 AM)C.Ret Wrote:  \( I=\int_{-3}^{0}\: \sqrt[4]{x^2+1}\cdot\sqrt[5]{x+2}\: \mathrm{d}x \)

...

\( I=\int_{-3}^{0}\: \frac{1}{\left((x+2)^{1/5}\right)^4}\cdot(x+2)\cdot(x^2+1)^{1/4}\: \mathrm{d}x \)

Why? Why? Why?

Final rewrite is close, but not quite right. (integrand first term turned complex if x < -2)
Let y = x+2

if y > 0, then ⁵√y = (+y)^(1/5) = y / (y^4)^(1/5)
if y < 0, then ⁵√y = -(-y)^(1/5) = y / (-y)^(4/5) = y / (y^4)^(1/5)

Note that if y<0, (y^4)^(1/5) ≠ (y^(1/5))^4

On HP Prime old emulator, 2.1.14181 (2018 10 16), we don't have this (display?) bug.

Cas> int((4 NTHROOT (x^2+1)) * (5 NTHROOT (x+2)), x, -3, 0)

\(\displaystyle \int _{-3}^{0}\frac{5\cdot (x+2) \left(\left(x+2\right)^{2}-4\cdot (x+2)+5\right)^{\frac{1}{4}}}{5} \cdot \left(5\mbox{ NTHROOT }(x+2)\right)^{-4}\, dx\)      ✔

Cas> float(Ans)

0.8841671971