if x < 7.172506652626118e+17

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   1.5
2. Using strategy rm
   1.5
3. Applied flip3-- to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}}\]
   1.3
4. Applied simplify to get
   \[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({x}^{\left(\frac{1}{3}\right)}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   1.3
5. Using strategy rm
   1.3
6. Applied pow-to-exp to get
   \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\color{red}{\left({x}^{\left(\frac{1}{3}\right)}\right)}}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\color{blue}{\left(e^{\log x \cdot \frac{1}{3}}\right)}}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   1.3
7. Applied simplify to get
   \[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\color{red}{\log x \cdot \frac{1}{3}}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)} \leadsto \frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(e^{\color{blue}{\frac{\log x}{3}}}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {x}^{\left(\frac{1}{3}\right)}\right)}\]
   1.3

if 7.172506652626118e+17 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   45.2
2. Using strategy rm
   45.2
3. Applied add-sqr-sqrt to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\]
   53.7
4. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2 \leadsto \left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)\]
   46.5
5. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)} \leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}\]
   46.5
6. Applied simplify to get
   \[\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) \leadsto \left({x}^{\frac{-1}{3}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right) - \left(\sqrt[3]{\frac{1}{x}} - \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)\]
   60.5

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7. Applied final simplification