I need help to evaluate $$\int \left[\sqrt{\int x\mathrm{d}x+\int x^{2}\mathrm{d}x+\int x^{\frac{7}{2}}\mathrm{d}x+\int x^{4}\mathrm{d}x} \right]\mathrm{d}x$$ I am trying this question by evaluating separately each of
$$\int x\mathrm{d}x\qquad\int x^{2}\mathrm{d}x\qquad\int x^{\frac{7}{2}}\mathrm{d}x\qquad\text{and}\qquad\int x^{4}\mathrm{d}x$$
Now $$\int x\mathrm{d}x = \frac{x^{2}}{2}+C\quad \int x^{2}\mathrm{d}x=\frac{x^{3}}{3}+C_1\quad \int x^{\frac{7}{2}}\mathrm{d}x= \frac{x^{\frac{9}{2}}}{\frac{9}{2}}+C_2\quad\text{ and }\quad\int x^{4}\mathrm{d}x=\frac{x^{5}}{5}+C_3$$
Now the final integration is becoming $$\int \left[\sqrt{\frac{x^{2}}{2}+C+\frac{x^{3}}{3}+C_1+\frac{x^{\frac{9}{2}}}{\frac{9}{2}}+C_2+\frac{x^{5}}{5}+C_3}\right]\mathrm{d}x$$ But how to integrate further? Please help me out.
Your integral is of the form
$$\int \sqrt{f(x)}\, dx$$
where $f(x)$ is a quintic polynomial. If the polynomial $f$ has degree 3 or 4 and has no repeated roots, you have an elliptic integral, a highly studied type of function. In your case, the polynomial may or may not have repeated roots (depending on the constant term), but even if there are no repeated roots, it has a higher degree. I would think of this as a `cousin' of an elliptic integral. Presumably it doesn't have a name as Wolfram couldn't find it.