What is the integral of ((z^2+1)^2)/(sqrt(z)) ?

The integral of ((z^2+1)^2)/(sqrt(z)) is 2/9 z^4sqrt(z)+4/5 z^2sqrt(z)+2sqrt(z)+C

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integral of ((z^2+1)^2)/(sqrt(z))

\int \frac{( z ^{2} + 1 ) ^{2}}{z} d z

\frac{2}{9} z^{4} z + \frac{4}{5} z^{2} z + 2 z + C

Solution steps

\int \frac{( z ^{2} + 1 ) ^{2}}{z} d z

Apply Integration By Parts

= z (z^{2} + 1)^{2} - \int \frac{7 z ^{4} + 6 z ^{2} - 1}{2 z} d z

\int \frac{7 z ^{4} + 6 z ^{2} - 1}{2 z} d z = \frac{1}{45} z (35 z^{4} + 54 z^{2} - 45)

= z (z^{2} + 1)^{2} - \frac{1}{45} z (35 z^{4} + 54 z^{2} - 45)

Simplify

z (z^{2} + 1)^{2} - \frac{1}{45} z (35 z^{4} + 54 z^{2} - 45) : \frac{2}{9} z^{4} z + \frac{4}{5} z^{2} z + 2 z

= \frac{2}{9} z^{4} z + \frac{4}{5} z^{2} z + 2 z

Add a constant to the solution

= \frac{2}{9} z^{4} z + \frac{4}{5} z^{2} z + 2 z + C

What is the integral of ((z^2+1)^2)/(sqrt(z)) ?
The integral of ((z^2+1)^2)/(sqrt(z)) is 2/9 z^4sqrt(z)+4/5 z^2sqrt(z)+2sqrt(z)+C