if x < 4.879869339815519e-09

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   37.1
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto \left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)\]
   37.1
3. Taylor expanded around 0 to get
   \[\color{red}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)} \leadsto \color{blue}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)}\]
   37.1
4. Applied simplify to get
   \[\color{red}{\left(1 + \frac{1}{3} \cdot x\right) - \left(\frac{1}{9} \cdot {x}^2 + {x}^{\frac{1}{3}}\right)} \leadsto \color{blue}{\left(1 - \sqrt[3]{x}\right) + \left(\frac{1}{3} - \frac{1}{9} \cdot x\right) \cdot x}\]
   0.1

if 4.879869339815519e-09 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   57.6
2. Using strategy rm
   57.6
3. Applied flip3-- to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
   57.2
4. Applied simplify to get
   \[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   57.3
5. Applied taylor to get
   \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   6.1
6. Taylor expanded around inf to get
   \[\frac{\color{red}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   6.1
7. Applied simplify to get
   \[\frac{{\left({\left(1 + \frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3 - {\left({\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right)}\]
   5.9

---

8. Applied final simplification
9. Applied simplify to get
   \[\color{red}{\frac{\left(\frac{1}{x} + 1\right) - \frac{1}{x}}{{\left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + {x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right)}} \leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) \cdot {x}^{\left(\frac{1}{3}\right)}}}\]
   5.9