Reduction of an integral to Legendre normal form F(phi,k), E(phi,k)

Find the substitution by means of which the integral $\int R\left(x ; \sqrt{P_{3}(x)}\right) d x$ can be transformed into the integral

$R(x ; y), R_{1}(t ; u)$ — rational functions, $P_{3}(x), P_{4}(x)$ — polynomials of the third and fourth degree.

Prove that, using a substitution of the form $x = (\alpha t+\beta ) /(t+1)$, the integral $\int R\left(x ; \sqrt{P_{4}(x)}\right) d x$ can be transformed into the integral

Prove that any integral of the form $\int R\left(x ; \sqrt{P_{4}(x)}\right) d x$ can be expressed in terms of elementary functions and the integral

where $R_{1}(u)$ is a rational function.

Prove that the integral

using one of the following substitutions:

$\sqrt{\left|\lambda_{1}\right|} x = t, \quad \sqrt{\left|\lambda_{1}\right|} x = \sqrt{1-t^{2}}, \quad \sqrt{\left|\lambda_{1}\right|} x = \frac{t}{\sqrt{1-t^{2}}}$,

is reduced to the form $\int \frac{R_{1}\left(t^{2}\right) d t}{\sqrt{\left(1-t^{2}\right)\left(1-k^{2} t^{2}\right)}}, k \in (0 ; 1)$.

Express the elliptic integral in terms of elementary functions and the functions $F(\varphi ; k)$ and $E(\varphi ; k)$ (see (13), (14)):

Express, in terms of $F(x ; k)$ and $E(x ; k)$, i.e. in terms of elliptic integrals of the first and second kind in Legendre's form (§ 3), and in terms of elementary functions, the integral: