What is the integral of (y^{1/3})/((y^{4/3)+9)^2} ?

The integral of (y^{1/3})/((y^{4/3)+9)^2} is

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integral of (y^{1/3})/((y^{4/3)+9)^2}

\int \frac{y ^{\frac{1}{3}}}{( y ^{\frac{4}{3}} + 9 ) ^{2}} d y

\frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} + \frac{3 ( - 27 y ^{\frac{4}{3}} - 2 y ^{\frac{8}{3}} - 81 )}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2} ( 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 ) ( - 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 )} + C

Solution steps

\int \frac{y ^{\frac{1}{3}}}{( y ^{\frac{4}{3}} + 9 ) ^{2}} d y

Apply Integration By Parts

: \frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} - \int - \frac{2 y ^{\frac{5}{3}}}{( y ^{\frac{4}{3}} + 9 ) ^{3}} d y

= \frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} - \int - \frac{2 y ^{\frac{5}{3}}}{( y ^{\frac{4}{3}} + 9 ) ^{3}} d y

\int - \frac{2 y ^{\frac{5}{3}}}{( y ^{\frac{4}{3}} + 9 ) ^{3}} d y = - \frac{3 ( - 27 y ^{\frac{4}{3}} - 2 y ^{\frac{8}{3}} - 81 )}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2} ( 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 ) ( - 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 )}

= \frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} - - \frac{3 ( - 27 y ^{\frac{4}{3}} - 2 y ^{\frac{8}{3}} - 81 )}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2} ( 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 ) ( - 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 )}

Simplify

= \frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} + \frac{3 ( - 27 y ^{\frac{4}{3}} - 2 y ^{\frac{8}{3}} - 81 )}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2} ( 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 ) ( - 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 )}

Add a constant to the solution

= \frac{3 y ^{\frac{4}{3}}}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2}} + \frac{3 ( - 27 y ^{\frac{4}{3}} - 2 y ^{\frac{8}{3}} - 81 )}{4 ( y ^{\frac{4}{3}} + 9 ) ^{2} ( 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 ) ( - 6 ^{\frac{1}{2}} 3 y + y ^{\frac{2}{3}} + 3 )} + C

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What is the integral of (y^{1/3})/((y^{4/3)+9)^2} ?
The integral of (y^{1/3})/((y^{4/3)+9)^2} is