$\int \frac{{\left(\sqrt{x}\right)}^3}{{\left(\sqrt{x}\right)}^5+x^4}dx=a\log \left(\frac{{\displaystyle x^k}}{{\displaystyle x^k+1}}\right)+c$ then the value of a & k  respectively are

$\int \frac{{\left(\sqrt{x}\right)}^3}{{\left(\sqrt{x}\right)}^5+x^4}dx$

$=\int \frac{x^{3/2}}{x^{5/2}+x^4}dx$

$$\begin{gathered} =\int \frac{x^{3/2}}{x^{5/2}\left(1+x^{3/2}\right)}dx \\ =\int \frac{1}{x^{5/2}\left(x^{-3/2}+1\right)}dx \\ =\int \frac{1}{t}\left(\frac{-2}{3}dt\right) Put x^{-3/2}+1=t \\ =\frac{-2}{3}\log t+C \frac{-3}{2}{x}^{-5/2}dx=dt \\ =\frac{-2}{3}\log \left|x^{-3/2}+1\right|+C \\ =\frac{-2}{3}\log \left|\frac{1+x^{3/2}}{x^{3/2}}\right|+C \\ =\frac{2}{3}\log \left|\frac{x^{3/2}}{1+x^{3/2}}\right|+C \\ a=2/3, K=3/2 \end{gathered}$$