Initial program 30.0
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
Initial simplification30.0
\[\leadsto \sqrt[3]{1 + x} - \sqrt[3]{x}\]
- Using strategy
rm Applied add-cube-cbrt30.0
\[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}}} - \sqrt[3]{x}\]
Applied cbrt-prod30.1
\[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}} - \sqrt[3]{x}\]
- Using strategy
rm Applied flip3--30.1
\[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right)}^{3} - {\left(\sqrt[3]{x}\right)}^{3}}{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) \cdot \sqrt[3]{x}\right)}}\]
Taylor expanded around inf 0.6
\[\leadsto \frac{\color{blue}{1}}{\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}} \cdot \sqrt[3]{\sqrt[3]{1 + x}}\right) \cdot \sqrt[3]{x}\right)}\]
Final simplification0.6
\[\leadsto \frac{1}{\left(\sqrt[3]{x} \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)}\]
