To solve the integral using partial fraction decomposition, we first decompose the fraction:

$\frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)}.$

Step-by-Step Partial Fraction Decomposition

1. Factor the Denominator:

   The denominator is already partially factored:

   $x(5x^4 + 8x^2 + 16).$

   Now, we consider the expression $5x^4 + 8x^2 + 16$ as a quadratic in terms of $x^2$. Let $y = x^2$, then we have:

   $5y^2 + 8y + 16.$

   Since this quadratic has no real roots (its discriminant $8^2 - 4 \cdot 5 \cdot 16 = -264 < 0$), it is irreducible over the reals. Thus, we treat $5x^4 + 8x^2 + 16$ as an irreducible quadratic in $x^2$.

2. Set Up the Partial Fractions:

   Now, express the fraction as a sum of partial fractions:

   $\frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)} = \frac{A}{x} + \frac{Bx + C}{5x^4 + 8x^2 + 16}.$

   Multiply through by the denominator to clear the fractions:

   $2x^3 + 5x^2 + 16 = A(5x^4 + 8x^2 + 16) + (Bx + C)x.$

   Expand and collect like terms:

   $2x^3 + 5x^2 + 16 = (5A)x^4 + (8A + B)x^3 + (16A + C)x^2.$

3. Equate Coefficients:

   Now, match coefficients for each power of $x$:

   • Coefficient of $x^4$: $5A = 0 \implies A = 0$.
   • Coefficient of $x^3$: $8A + B = 2 \implies 0 + B = 2 \implies B = 2$.
   • Coefficient of $x^2$: $16A + C = 5 \implies 0 + C = 5 \implies C = 5$.

   So, we have $A = 0$, $B = 2$, and $C = 5$.

   The decomposition becomes:

   $\frac{2x^3 + 5x^2 + 16}{x(5x^4 + 8x^2 + 16)} = \frac{2x + 5}{5x^4 + 8x^2 + 16}.$

4. Rewrite the Integral:

   The integral now is:

   $\int \frac{2x + 5}{5x^4 + 8x^2 + 16} \, dx.$

Solving the Integral

Now, let's split the integral into two parts:

$\int \frac{2x}{5x^4 + 8x^2 + 16} \, dx + \int \frac{5}{5x^4 + 8x^2 + 16} \, dx.$

For the first integral:

• Substitute $u = 5x^4 + 8x^2 + 16$, $du = (20x^3 + 16x) dx = 4x(5x^2 + 4) dx$, which simplifies this part.

For the second integral:

• A suitable trigonometric substitution or table integration may apply, depending on its specific form.

Would you like to proceed with solving these integrals, or do you have a preference for a particular approach?