What is the integral of (x^9)/((x^5+1)^3) ?

The integral of (x^9)/((x^5+1)^3) is (x^{10})/(10(x^5+1)^3)+(-3x^{10}-3x^5-1)/(10(x^5+1)^3)+C

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integral of (x^9)/((x^5+1)^3)

\int \frac{x ^{9}}{( x ^{5} + 1 ) ^{3}} d x

\frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} + \frac{- 3 x ^{10} - 3 x ^{5} - 1}{10 ( x ^{5} + 1 ) ^{3}} + C

Solution steps

\int \frac{x ^{9}}{( x ^{5} + 1 ) ^{3}} d x

Apply Integration By Parts

: \frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} - \int - \frac{3 x ^{14}}{2 ( x ^{5} + 1 ) ^{4}} d x

= \frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} - \int - \frac{3 x ^{14}}{2 ( x ^{5} + 1 ) ^{4}} d x

\int - \frac{3 x ^{14}}{2 ( x ^{5} + 1 ) ^{4}} d x = - \frac{- 3 x ^{10} - 3 x ^{5} - 1}{10 ( x ^{5} + 1 ) ^{3}}

= \frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} - (- \frac{- 3 x ^{10} - 3 x ^{5} - 1}{10 ( x ^{5} + 1 ) ^{3}})

Simplify

= \frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} + \frac{- 3 x ^{10} - 3 x ^{5} - 1}{10 ( x ^{5} + 1 ) ^{3}}

Add a constant to the solution

= \frac{x ^{10}}{10 ( x ^{5} + 1 ) ^{3}} + \frac{- 3 x ^{10} - 3 x ^{5} - 1}{10 ( x ^{5} + 1 ) ^{3}} + C

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What is the integral of (x^9)/((x^5+1)^3) ?
The integral of (x^9)/((x^5+1)^3) is (x^{10})/(10(x^5+1)^3)+(-3x^{10}-3x^5-1)/(10(x^5+1)^3)+C