\( \int \frac { x ^ { 2 } + 3 } { ( x ^ { 3 } + 2 x ) ^ { \frac { 1 } { 2 } } } d x \)

To solve the integral \( \int \frac{x^2 + 3}{(x^3 + 2x)^{\frac{1}{2}}} \, dx \), we can start by simplifying the integrand. Notice that we can factor the denominator: \[ x^3 + 2x = x(x^2 + 2) \] Thus, the integral becomes: \[ \int \frac{x^2 + 3}{\sqrt{x(x^2 + 2)}} \, dx \] Now, to deal with the square root, we can make a substitution. Let \( u = x^3 + 2x \), then the derivative \( du = (3x^2 + 2) \, dx \). However, since this will complicate the limits and expressions due to the square root, let's see if substitution relating to trigonometric or hyperbolic functions might help, considering the structure of the integral. For now, we can rewrite \( (x^3 + 2x)^{\frac{1}{2}} \): \[ = \sqrt{x^3(1+\frac{2}{x^2})} = x^{\frac{3}{2}}\sqrt{1+\frac{2}{x^2}} \] Now the integral becomes: \[ \int \frac{x^2 + 3}{x^{\frac{3}{2}}\sqrt{1+\frac{2}{x^2}}} \, dx = \int \frac{x^{\frac{1}{2}}(x^2 + 3)}{\sqrt{1+\frac{2}{x^2}}} \, dx \] This looks complex but notice how we can separate terms. You can simplify and break it down term by term, or even try several substitutions involving \( x^2 \) or the radical. Continuing with the problem, applying integration techniques like partial fractions may not be directly useful here; instead, recognizing patterns and checking for standard integral forms is key. If a trigonometric substitution leads you into a cycle or complex derivatives, don’t hesitate to revert back and try something different like integration by parts or regrouping terms wisely. Finally, evaluating this integral would likely yield a special function or combination of terms that might have direct integrations in tables or known results. Each step needs careful review, so ensure you track changes in variables or integrate with respect to the original substitution directly. Happy integrating!