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

The integral of ((x^2+3x)(5x+3))/x is (5x+3)((x^2)/2+3x)-5((x^3)/6+(3x^2)/2)+C

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

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

(5 x + 3) (\frac{x ^{2}}{2} + 3 x) - 5 (\frac{x ^{3}}{6} + \frac{3 x ^{2}}{2}) + C

Solution steps

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

Simplify

\frac{( x ^{2} + 3 x ) ( 5 x + 3 )}{x} : (x + 3) (5 x + 3)

= \int (x + 3) (5 x + 3) d x

Apply Integration By Parts

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

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

\int 5 (\frac{x ^{2}}{2} + 3 x) d x = 5 (\frac{x ^{3}}{6} + \frac{3 x ^{2}}{2})

= (5 x + 3) (\frac{x ^{2}}{2} + 3 x) - 5 (\frac{x ^{3}}{6} + \frac{3 x ^{2}}{2})

Add a constant to the solution

= (5 x + 3) (\frac{x ^{2}}{2} + 3 x) - 5 (\frac{x ^{3}}{6} + \frac{3 x ^{2}}{2}) + C

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What is the integral of ((x^2+3x)(5x+3))/x ?
The integral of ((x^2+3x)(5x+3))/x is (5x+3)((x^2)/2+3x)-5((x^3)/6+(3x^2)/2)+C