Methods from Rationalising substitutions

Rationalise an integrand containing surds

Replacing the radical is not enough. After rewriting both the integrand and $dx$, every occurrence of the new variable must have an integer exponent; only then do polynomial division and partial fractions apply. Forgetting to rewrite $dx$ in terms of $du$ gives the wrong powers of $u$.

When several rational powers of $x$ appear, set $x=u^N$ where $N$ is a common multiple of their denominators, so every power becomes an integer power of $u$. Work on an interval where the original real powers and the chosen inverse are defined; when $N$ is even, $x=u^N$ covers only $x\ge0$.

The trigonometric and hyperbolic table clears $\sqrt{a^2\pm x^2}$ and $\sqrt{x^2-a^2}$ by a named identity. Use a rationalising substitution for a surd that is not one of those three shapes.