Roots of $3x^5 - 15x +5$ Let $f(x) =3x^5 - 15x +5$. By  Eisenstein’s Criterion we can show that $f(x)$ irreducible over $\mathbb{Q}$(Since $5 \nmid 3,  5 \vert -15,5$ and $5^2 \nmid 15$). Since $g$ is continuous and $ f( -2 ) =  -61, f( -1 ) =  17 ,f( 0 ) =  5,f( 1 ) =  -7,f( 2 ) =  71$. So by intermediate value theorem  we can say that $f$ has three real roots. Clearly $f'(x) > 0 $ for all $x > 2$ and $f'(x) < 0$ for all $x < -2$. So $f(x)$ is monotone and so $f$ does not have zeroes in these regions. Suppose $f(x)$ has $4$ real zeroes then by using Rolle's theorem we can say that $f'(x)$ should contain at least $3$ zeroes between the roots of $f$. But $f'(x) = 15(x^4 -1)$ does not have $3$ real zeroes. So $f$ has two other  complex roots. Let $K$ be the smallest subfield of complex numbers containing $\mathbb{Q}$ and $5$ roots of $f(x)$. Then using fundamental theorem of Galois theory we can say that $Gal(K/\mathbb{Q}) \approx S_5$, the symmetric group of five letters. Since $S_5$ is not solvable, by a theorem of Galois we can  conclude that $f(x)$ is not solvable by radicals.
That is  each zero of the polynomial $f(x)$ cannot  be written as an expression  involving elements of $\mathbb{Q}$ combined by the operations of addition, subtraction, multiplication, division, and extraction of roots.
How do the roots of $f$ look like? We have information about the location of real roots but I think that information may not help in finding some expression for roots. Precisely, my question is that that does there exist a series, continued fractions, or some integral which represent the roots of $f(x)$?.

The solutions are algebraic numbers. Because the Galois group of your algebraic equation isn't solvable, the soultions cannot be represented as radical expressions.
They can't be represented in terms of elementary functions either. See
closed-form expression for roots of a polynomial
Polynomials with degree $5$ solvable in elementary functions?
Additionally, your equation
$$3x^5-15x+5=0$$
is a trinomial equation. see Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104
A closed-form solution can be obtained also using confluent Fox-Wright Function $\ _1\Psi_1$. see Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106
