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

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

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

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

\frac{1}{196} ln ∣ x - 1 ∣ + \frac{352}{240737} ln ∣ 2 x + 5 ∣ - \frac{10}{2023 ( 2 x + 5 )} - \frac{129}{19652} ln ∣ 3 x - 1 ∣ + \frac{3}{578 ( 3 x - 1 )} + C

Solution steps

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

Take the partial fraction of

\frac{x}{( x - 1 ) ( 2 x + 5 ) ^{2} ( 3 x - 1 ) ^{2}} : \frac{1}{196 ( x - 1 )} + \frac{704}{240737 ( 2 x + 5 )} + \frac{20}{2023 ( 2 x + 5 ) ^{2}} - \frac{387}{19652 ( 3 x - 1 )} - \frac{9}{578 ( 3 x - 1 ) ^{2}}

= \int \frac{1}{196 ( x - 1 )} + \frac{704}{240737 ( 2 x + 5 )} + \frac{20}{2023 ( 2 x + 5 ) ^{2}} - \frac{387}{19652 ( 3 x - 1 )} - \frac{9}{578 ( 3 x - 1 ) ^{2}} 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{1}{196 ( x - 1 )} d x + \int \frac{704}{240737 ( 2 x + 5 )} d x + \int \frac{20}{2023 ( 2 x + 5 ) ^{2}} d x - \int \frac{387}{19652 ( 3 x - 1 )} d x - \int \frac{9}{578 ( 3 x - 1 ) ^{2}} d x

\int \frac{1}{196 ( x - 1 )} d x = \frac{1}{196} ln ∣ x - 1 ∣

\int \frac{704}{240737 ( 2 x + 5 )} d x = \frac{352}{240737} ln ∣ 2 x + 5 ∣

\int \frac{20}{2023 ( 2 x + 5 ) ^{2}} d x = - \frac{10}{2023 ( 2 x + 5 )}

\int \frac{387}{19652 ( 3 x - 1 )} d x = \frac{129}{19652} ln ∣ 3 x - 1 ∣

\int \frac{9}{578 ( 3 x - 1 ) ^{2}} d x = - \frac{3}{578 ( 3 x - 1 )}

= \frac{1}{196} ln ∣ x - 1 ∣ + \frac{352}{240737} ln ∣ 2 x + 5 ∣ - \frac{10}{2023 ( 2 x + 5 )} - \frac{129}{19652} ln ∣ 3 x - 1 ∣ - (- \frac{3}{578 ( 3 x - 1 )})

Simplify

= \frac{1}{196} ln ∣ x - 1 ∣ + \frac{352}{240737} ln ∣ 2 x + 5 ∣ - \frac{10}{2023 ( 2 x + 5 )} - \frac{129}{19652} ln ∣ 3 x - 1 ∣ + \frac{3}{578 ( 3 x - 1 )}

Add a constant to the solution

= \frac{1}{196} ln ∣ x - 1 ∣ + \frac{352}{240737} ln ∣ 2 x + 5 ∣ - \frac{10}{2023 ( 2 x + 5 )} - \frac{129}{19652} ln ∣ 3 x - 1 ∣ + \frac{3}{578 ( 3 x - 1 )} + C

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