What is the integral of (6x+5)/((9x^2+15x)^5) ?

The integral of (6x+5)/((9x^2+15x)^5) is

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integral of (6x+5)/((9x^2+15x)^5)

\int \frac{6 x + 5}{( 9 x ^{2} + 15 x ) ^{5}} d x

\frac{1}{140625 x} - \frac{1}{168750 x ^{2}} + \frac{1}{253125 x ^{3}} - \frac{1}{607500 x ^{4}} - \frac{1}{46875 ( 3 x + 5 )} - \frac{1}{18750 ( 3 x + 5 ) ^{2}} - \frac{1}{9375 ( 3 x + 5 ) ^{3}} - \frac{1}{7500 ( 3 x + 5 ) ^{4}} + C

Solution steps

\int \frac{6 x + 5}{( 9 x ^{2} + 15 x ) ^{5}} d x

Take the partial fraction of

\frac{6 x + 5}{( 9 x ^{2} + 15 x ) ^{5}} : - \frac{1}{140625 x ^{2}} + \frac{1}{84375 x ^{3}} - \frac{1}{84375 x ^{4}} + \frac{1}{151875 x ^{5}} + \frac{1}{15625 ( 3 x + 5 ) ^{2}} + \frac{1}{3125 ( 3 x + 5 ) ^{3}} + \frac{3}{3125 ( 3 x + 5 ) ^{4}} + \frac{1}{625 ( 3 x + 5 ) ^{5}}

= \int - \frac{1}{140625 x ^{2}} + \frac{1}{84375 x ^{3}} - \frac{1}{84375 x ^{4}} + \frac{1}{151875 x ^{5}} + \frac{1}{15625 ( 3 x + 5 ) ^{2}} + \frac{1}{3125 ( 3 x + 5 ) ^{3}} + \frac{3}{3125 ( 3 x + 5 ) ^{4}} + \frac{1}{625 ( 3 x + 5 ) ^{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{1}{140625 x ^{2}} d x + \int \frac{1}{84375 x ^{3}} d x - \int \frac{1}{84375 x ^{4}} d x + \int \frac{1}{151875 x ^{5}} d x + \int \frac{1}{15625 ( 3 x + 5 ) ^{2}} d x + \int \frac{1}{3125 ( 3 x + 5 ) ^{3}} d x + \int \frac{3}{3125 ( 3 x + 5 ) ^{4}} d x + \int \frac{1}{625 ( 3 x + 5 ) ^{5}} d x

\int \frac{1}{140625 x ^{2}} d x = - \frac{1}{140625 x}

\int \frac{1}{84375 x ^{3}} d x = - \frac{1}{168750 x ^{2}}

\int \frac{1}{84375 x ^{4}} d x = - \frac{1}{253125 x ^{3}}

\int \frac{1}{151875 x ^{5}} d x = - \frac{1}{607500 x ^{4}}

\int \frac{1}{15625 ( 3 x + 5 ) ^{2}} d x = - \frac{1}{46875 ( 3 x + 5 )}

\int \frac{1}{3125 ( 3 x + 5 ) ^{3}} d x = - \frac{1}{18750 ( 3 x + 5 ) ^{2}}

\int \frac{3}{3125 ( 3 x + 5 ) ^{4}} d x = - \frac{1}{9375 ( 3 x + 5 ) ^{3}}

\int \frac{1}{625 ( 3 x + 5 ) ^{5}} d x = - \frac{1}{7500 ( 3 x + 5 ) ^{4}}

= - (- \frac{1}{140625 x}) - \frac{1}{168750 x ^{2}} - (- \frac{1}{253125 x ^{3}}) - \frac{1}{607500 x ^{4}} - \frac{1}{46875 ( 3 x + 5 )} - \frac{1}{18750 ( 3 x + 5 ) ^{2}} - \frac{1}{9375 ( 3 x + 5 ) ^{3}} - \frac{1}{7500 ( 3 x + 5 ) ^{4}}

Simplify

= \frac{1}{140625 x} - \frac{1}{168750 x ^{2}} + \frac{1}{253125 x ^{3}} - \frac{1}{607500 x ^{4}} - \frac{1}{46875 ( 3 x + 5 )} - \frac{1}{18750 ( 3 x + 5 ) ^{2}} - \frac{1}{9375 ( 3 x + 5 ) ^{3}} - \frac{1}{7500 ( 3 x + 5 ) ^{4}}

Add a constant to the solution

= \frac{1}{140625 x} - \frac{1}{168750 x ^{2}} + \frac{1}{253125 x ^{3}} - \frac{1}{607500 x ^{4}} - \frac{1}{46875 ( 3 x + 5 )} - \frac{1}{18750 ( 3 x + 5 ) ^{2}} - \frac{1}{9375 ( 3 x + 5 ) ^{3}} - \frac{1}{7500 ( 3 x + 5 ) ^{4}} + C

What is the integral of (6x+5)/((9x^2+15x)^5) ?
The integral of (6x+5)/((9x^2+15x)^5) is