Initial program 29.6
\[\sqrt{x + 1} - \sqrt{x}\]
- Using strategy
rm Applied add-cube-cbrt29.6
\[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt{x}\]
Applied sqrt-prod29.6
\[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt{x}\]
- Using strategy
rm Applied flip--29.5
\[\leadsto \color{blue}{\frac{\left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}\right) - \sqrt{x} \cdot \sqrt{x}}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}} + \sqrt{x}}}\]
Taylor expanded around inf 0.3
\[\leadsto \frac{\color{blue}{1}}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}} + \sqrt{x}}\]
- Using strategy
rm Applied add-sqr-sqrt0.3
\[\leadsto \frac{1}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1}}}} \cdot \sqrt{\sqrt[3]{x + 1}} + \sqrt{x}}\]
Applied cbrt-prod0.3
\[\leadsto \frac{1}{\sqrt{\sqrt[3]{x + 1} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right)}} \cdot \sqrt{\sqrt[3]{x + 1}} + \sqrt{x}}\]
Final simplification0.3
\[\leadsto \frac{1}{\sqrt{x} + \sqrt{\sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right)}}\]
