<s>[INST] Evaluation of $\int \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx$ Evaluation of $\displaystyle \int \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx$
$\bf{My\; Try::}$ Let $x=t^4\;,$ Then $dx = 4t^3dt$
So Integral is $\displaystyle \int\frac{\sqrt[3]{t^4+t}}{t^2} \cdot 4t^3dt$
So Integral is $\displaystyle 4\int t^{\frac{7}{3}}\cdot (1+t^{-3})^{\frac{1}{3}}$
Now How can i solve after that
Help me
Thanks
 [/INST] As Pranav Arora showed, there is no nice answer to this antiderivative. Personally, the only way I can think about it is a Taylor expansion of the integrand followed by a term by term integration.
As, you did, starting with $x=t^4$,we have $$\frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}=t^{-\frac{5}{3}} \sqrt[3] {1+t^3}=t^{-\frac{5}{3}} \Big(1+\frac{t^3}{3}-\frac{t^6}{9}+\frac{5 t^9}{81}-\frac{10
   t^{12}}{243}+O\left(t^{13}\right)\Big)$$
Then, replacing $t$ by $\sqrt[4] x$, we can find  that $$\frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}=\frac{1}{x^{5/12}}+\frac{\sqrt[3]{x}}{3}-\frac{x^{13/12}}{9}+\frac{5
   x^{11/6}}{81}-\frac{10 x^{31/12}}{243}+\frac{22 x^{10/3}}{729}-\frac{154
   x^{49/12}}{6561}+\frac{374 x^{29/6}}{19683}+O\left(x^{61/12}\right)$$ Now integration leads to 
$$\int \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx=\frac{12 x^{7/12}}{7}+\frac{x^{4/3}}{4}-\frac{4 x^{25/12}}{75}+\frac{10
   x^{17/6}}{459}-\frac{40 x^{43/12}}{3483}+\frac{22 x^{13/3}}{3159}-\frac{616
   x^{61/12}}{133407}+\frac{748 x^{35/6}}{229635}+O\left(x^{73/12}\right)$$
If we compare the exact and approximate solutions for $$\int_0^a \frac{\sqrt[3]{x+\sqrt[4]{x}}}{\sqrt{x}}dx$$ they match quite well for $0 \leq a \leq 2$.
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