if x < 6.236616f+09

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   15.2
2. Applied taylor to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}\]
   15.2
3. Taylor expanded around 0 to get
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\frac{1}{3}}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{x}^{\frac{1}{3}}}\]
   15.2
4. Applied simplify to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\frac{1}{3}}} \leadsto \color{blue}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}}\]
   1.8
5. Using strategy rm
   1.8
6. Applied flip3-- to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \sqrt[3]{x}} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}}\]
   1.8
7. Applied simplify to get
   \[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
   1.8
8. Using strategy rm
   1.8
9. Applied flip3-- to get
   \[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - x}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)} \leadsto \frac{\color{blue}{\frac{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^{3} - {x}^{3}}{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^2 + \left({x}^2 + {\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 \cdot x\right)}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}\]
   1.9
10. Applied associate-/l/ to get
    \[\color{red}{\frac{\frac{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^{3} - {x}^{3}}{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^2 + \left({x}^2 + {\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 \cdot x\right)}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)}} \leadsto \color{blue}{\frac{{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^{3} - {x}^{3}}{\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left(\sqrt[3]{x}\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot \sqrt[3]{x}\right)\right) \cdot \left({\left({\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3\right)}^2 + \left({x}^2 + {\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 \cdot x\right)\right)}}\]
    1.9

if 6.236616f+09 < x

1. Started with
   \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
   20.0
2. Using strategy rm
   20.0
3. Applied add-sqr-sqrt to get
   \[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}} - {x}^{\left(\frac{1}{3}\right)} \leadsto \color{blue}{{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2} - {x}^{\left(\frac{1}{3}\right)}\]
   29.9
4. Applied taylor to get
   \[{\left(\sqrt{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)}}\right)}^2 - {x}^{\left(\frac{1}{3}\right)} \leadsto \left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)\]
   21.1
5. Taylor expanded around inf to get
   \[\color{red}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)} \leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right)}\]
   21.1
6. Applied simplify to get
   \[\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {x}^{\frac{-1}{3}}\right) \leadsto \left(\sqrt[3]{\frac{1}{x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right) - \left({x}^{\frac{-1}{3}} - \frac{1}{3} \cdot \sqrt[3]{\frac{1}{{x}^{4}}}\right)\]
   28.6

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7. Applied final simplification