What is the integral of ((x^2+1)/(x^3+3x+1)+5) ?

The integral of ((x^2+1)/(x^3+3x+1)+5) is 1/3 ln|x^3+3x+1|+5x+C

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integral of ((x^2+1)/(x^3+3x+1)+5)

\int (\frac{x ^{2} + 1}{x ^{3} + 3 x + 1} + 5) d x

\frac{1}{3} ln x^{3} + 3 x + 1 + 5 x + C

Solution steps

\int \frac{x ^{2} + 1}{x ^{3} + 3 x + 1} + 5 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2} + 1}{x ^{3} + 3 x + 1} d x + \int 5 d x

\int \frac{x ^{2} + 1}{x ^{3} + 3 x + 1} d x = \frac{1}{3} ln x^{3} + 3 x + 1

\int 5 d x = 5 x

= \frac{1}{3} ln x^{3} + 3 x + 1 + 5 x

Add a constant to the solution

= \frac{1}{3} ln x^{3} + 3 x + 1 + 5 x + C

What is the integral of ((x^2+1)/(x^3+3x+1)+5) ?
The integral of ((x^2+1)/(x^3+3x+1)+5) is 1/3 ln|x^3+3x+1|+5x+C