Final answer to the problem
$-\frac{1}{3124}\ln\left(x\right)+\frac{-\frac{1}{138}}{\left(2x-5\right)^{2}}-\frac{1}{959}\left(\frac{1}{2}\arctan\left(\frac{x+1}{2}\right)+\frac{x+1}{4+\left(x+1\right)^2}\right)+\frac{-\frac{1}{928}}{4+\left(x+1\right)^2}+\frac{1}{9999}\ln\left(2x-5\right)+\frac{\frac{1}{603}}{2x-5}+\frac{1}{2503}\arctan\left(\frac{x+1}{2}\right)+\frac{1}{9091}\ln\left(4+\left(x+1\right)^2\right)+C_0$
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Step-by-step Solution
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1
Find the integral
$\int\frac{x^3+x+1}{x\left(2x-5\right)^3\left(x^2+2x+5\right)^2}dx$
Learn how to solve integral calculus problems step by step online.
$\int\frac{x^3+x+1}{x\left(2x-5\right)^3\left(x^2+2x+5\right)^2}dx$
Learn how to solve integral calculus problems step by step online. Find the integral of (x^3+x+1)/(x(2x-5)^3(x^2+2x+5)^2). Find the integral. Rewrite the fraction \frac{x^3+x+1}{x\left(2x-5\right)^3\left(x^2+2x+5\right)^2} in 6 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D, F, G, H, I. The first step is to multiply both sides of the equation from the previous step by x\left(2x-5\right)^3\left(x^2+2x+5\right)^2. Multiplying polynomials.
Final answer to the problem
$-\frac{1}{3124}\ln\left(x\right)+\frac{-\frac{1}{138}}{\left(2x-5\right)^{2}}-\frac{1}{959}\left(\frac{1}{2}\arctan\left(\frac{x+1}{2}\right)+\frac{x+1}{4+\left(x+1\right)^2}\right)+\frac{-\frac{1}{928}}{4+\left(x+1\right)^2}+\frac{1}{9999}\ln\left(2x-5\right)+\frac{\frac{1}{603}}{2x-5}+\frac{1}{2503}\arctan\left(\frac{x+1}{2}\right)+\frac{1}{9091}\ln\left(4+\left(x+1\right)^2\right)+C_0$