$\int \frac{x^4+1}{{\left(7x^{29}+3x^{25}+x^{32}\right)}^{1/4}}dx=\frac{a}{b}{\left(\frac{7x^4+x^7+3}{x^7}\right)}^{3/4}+c\text{ then }a+b$

$\begin{array}{r}I=\int \frac{x^4+1}{x^8{\left[\frac{7}{x^3}+\frac{3}{x^7}+1\right]}^{1/4}}dx\\ =\int \frac{\frac{1}{x^4}+\frac{1}{x^8}}{{\left(\frac{7}{x^3}+\frac{3}{x^7}+1\right)}^{1/4}}dx\end{array}$

$\begin{array}{l}\text{ Let }\frac{7}{x^3}+\frac{3}{x^7}+1=t\\ \frac{-21}{x^4}-\frac{21}{x^8}dx=dt\\ \left(\frac{1}{x^4}+\frac{1}{x^8}\right)dx=\frac{-1}{21}dt\\ I=\frac{-1}{21}\int \frac{1}{1/4}dt=\frac{-1}{21}\frac{t^{1-\frac{1}{4}}}{1-\frac{1}{4}}+C\end{array}$

$=\frac{-4}{63}{\left(\frac{7}{x^3}+\frac{3}{x^7}+1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+C$