Evaluate $\displaystyle\int \dfrac{dx}{\sqrt{x + 1} + \sqrt[3]{x + 1}}$
Let $I = \displaystyle\int \dfrac{dx}{\sqrt{x + 1} + \sqrt[3]{x + 1}}$
Put $x + 1 = p^6$ $\;\;\; \cdots$ (1)
Then, $dx = 6 p^5 dp$
Further, $\sqrt{x + 1} = \left(p^6\right)^{1/2} = p^3$ and $\sqrt[3]{x + 1} = \left(p^6\right)^{1/3} = p^2$
$\begin{aligned}
\therefore \; I & = \int \dfrac{6 \; p^5 \; dp}{p^3 + p^2} \\\\
& = 6 \int \dfrac{p^5 \; dp}{p^2 \left(p + 1\right)} \\\\
& = 6 \int \dfrac{p^3 \; dp}{p + 1}
\end{aligned}$
Put $p + 1 = t$ $\implies$ $p = t - 1$ $\;\;\; \cdots$ (2)
Differentiating equation (2) gives $dp = dt$
Also, from equation (2), $p^3 = \left(t - 1\right)^3 = t^3 - 3 t^2 + 3 t - 1$
$\begin{aligned}
\therefore \; I & = 6 \int \dfrac{t^3 - 3 t^2 + 3 t - 1}{t} \; dt \\\\
& = 6 \left\{\int t^2 \; dt - 3 \int t \; dt + 3 \int dt - \int \dfrac{dt}{t} \right\} \\\\
& = 6 \left\{\dfrac{t^3}{3} - \dfrac{3 t^2}{2} + 3 t - \log \left|t\right| \right\} + k \\\\
& = 2 t^3 - 9 t^2 + 18 t - 6 \log \left|t\right| + k \\\\
& = 2 \left(p + 1\right)^3 - 9 \left(p + 1\right)^2 + 18 \left(p + 1\right) - 6 \log \left|p + 1\right| + k \;\;\; \cdots (3)
\end{aligned}$
Now, from equation (1),
$p = \left(x + 1\right)^{1/6}$ $\;\;\; \cdots$ (4)
$\begin{aligned}
\therefore \; \left(p + 1\right)^3 & = \left[\left(x + 1\right)^{1/6} + 1\right]^3 \\\\
& = \left[\left(x + 1\right)^{1/6}\right]^3 + 3 \left[\left(x + 1\right)^{1/6}\right]^2 + 3 \left(x + 1\right)^{1/6} + 1 \\\\
& = \left(x + 1\right)^{1/2} + 3 \left(x + 1\right)^{1/3} + 3 \left(x + 1\right)^{1/6} + 1 \;\;\; \cdots (5)
\end{aligned}$
$\begin{aligned}
\left(p + 1\right)^2 & = \left[\left(x + 1\right)^{1/6} + 1\right]^2 \\\\
& = \left[\left(x + 1\right)^{1/6}\right]^2 + 2 \left(x + 1\right)^{1/6} + 1 \\\\
& = \left(x + 1\right)^{1/3} + 2 \left(x + 1\right)^{1/6} + 1 \;\;\; \cdots (6)
\end{aligned}$
Substituting equations (4), (5) and (6) in equation (3) gives
$\begin{aligned}
I & = 2 \left[\left(x + 1\right)^{1/2} + 3 \left(x + 1\right)^{1/3} + 3 \left(x + 1\right)^{1/6} + 1\right] \\\\
& - 9 \left[\left(x + 1\right)^{1/3} + 2 \left(x + 1\right)^{1/6} + 1\right] + 18 \left[\left(x + 1\right)^{1/6} + 1\right] - 6 \log \left|\left(x + 1\right)^{1/6} + 1\right| + k \\\\
& = 2 \left(x + 1\right)^{1/2} - 3 \left(x + 1\right)^{1/3} + 6 \left(x + 1\right)^{1/6} - 6 \log \left|\left(x + 1\right)^{1/6} + 1\right| + c
\end{aligned}$
where $11 + k = c = \text{constant}$