if x < 1.1034064f+17

1. Started with
   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
   6.9
2. Using strategy rm
   6.9
3. Applied flip-- to get
   \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
   6.9
4. Applied simplify to get
   \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   6.9
5. Using strategy rm
   6.9
6. Applied frac-sub to get
   \[\frac{\color{red}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1 \cdot \left(1 + x\right) - x \cdot 1}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   5.6
7. Applied simplify to get
   \[\frac{\frac{\color{red}{1 \cdot \left(1 + x\right) - x \cdot 1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{\color{blue}{1}}{x \cdot \left(1 + x\right)}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   0.4
8. Applied simplify to get
   \[\frac{\frac{1}{\color{red}{x \cdot \left(1 + x\right)}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\frac{1}{\color{blue}{(x * x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   0.3

if 1.1034064f+17 < x

1. Started with
   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]
   12.3
2. Using strategy rm
   12.3
3. Applied flip-- to get
   \[\color{red}{\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}} \leadsto \color{blue}{\frac{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
   12.3
4. Applied simplify to get
   \[\frac{\color{red}{{\left(\frac{1}{\sqrt{x}}\right)}^2 - {\left(\frac{1}{\sqrt{x + 1}}\right)}^2}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\frac{1}{x} - \frac{1}{1 + x}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   12.3
5. Applied taylor to get
   \[\frac{\frac{1}{x} - \frac{1}{1 + x}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   9.5
6. Taylor expanded around inf to get
   \[\frac{\color{red}{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   9.5
7. Applied taylor to get
   \[\frac{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   9.5
8. Taylor expanded around 0 to get
   \[\frac{\color{red}{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\color{blue}{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
   9.5
9. Applied simplify to get
   \[\frac{\left(\frac{1}{{x}^2} + \frac{1}{{x}^{4}}\right) - \frac{1}{{x}^{3}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}} \leadsto \frac{\left(\frac{1}{{x}^{4}} + \frac{\frac{1}{x}}{x}\right) - \frac{1}{{x}^3}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{1 + x}}}\]
   7.0

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10. Applied final simplification