\(\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}\)
- Started with
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
13.1
- Using strategy
rm 13.1
- Applied add-sqr-sqrt to get
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{red}{{x}^{\left(\frac{1}{3}\right)}} \leadsto {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - \color{blue}{{\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2}\]
13.3
- Using strategy
rm 13.3
- Applied flip3-- to get
\[\color{red}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2} \leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^{3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}}\]
13.2
- Applied simplify to get
\[\frac{\color{red}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^{3} - {\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^{3}}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)} \leadsto \frac{\color{blue}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}\]
13.1
- Applied taylor to get
\[\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^3 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)} \leadsto \frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}\]
13.1
- Taylor expanded around 0 to get
\[\frac{\color{red}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)} \leadsto \frac{\color{blue}{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}\]
13.1
- Applied simplify to get
\[\frac{{\left({\left(1 + x\right)}^{\frac{1}{3}}\right)}^3 - {\left({x}^{\frac{1}{3}}\right)}^3}{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 + \left({\left({\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)}^2 + {\left(x + 1\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(\sqrt{{x}^{\left(\frac{1}{3}\right)}}\right)}^2\right)} \leadsto \frac{1 + \left(x - x\right)}{{x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}\]
1.7
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{1 + \left(x - x\right)}{{x}^{\left(\frac{1}{3}\right)} \cdot \left({\left(1 + x\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}\right) + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}}} \leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{\left(\frac{1}{3}\right)} \cdot \left({x}^{\left(\frac{1}{3}\right)} + {\left(1 + x\right)}^{\left(\frac{1}{3}\right)}\right) + {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}}\]
1.7
- Removed slow pow expressions
