What is the integral of ((x^2-5x)^2)/(sqrt(x)) ?

The integral of ((x^2-5x)^2)/(sqrt(x)) is 2/9 x^{9/2}-20/7 x^{7/2}+10x^{5/2}+C

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integral of ((x^2-5x)^2)/(sqrt(x))

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

\frac{2}{9} x^{\frac{9}{2}} - \frac{20}{7} x^{\frac{7}{2}} + 10 x^{\frac{5}{2}} + C

Solution steps

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

Simplify

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

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

Expand

x^{\frac{3}{2}} (x - 5)^{2} : x^{\frac{7}{2}} - 10 x^{\frac{5}{2}} + 25 x^{\frac{3}{2}}

= \int x^{\frac{7}{2}} - 10 x^{\frac{5}{2}} + 25 x^{\frac{3}{2}} 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 x^{\frac{7}{2}} d x - \int 10 x^{\frac{5}{2}} d x + \int 25 x^{\frac{3}{2}} d x

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 10 x^{\frac{5}{2}} d x = \frac{20}{7} x^{\frac{7}{2}}

\int 25 x^{\frac{3}{2}} d x = 10 x^{\frac{5}{2}}

= \frac{2}{9} x^{\frac{9}{2}} - \frac{20}{7} x^{\frac{7}{2}} + 10 x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{9} x^{\frac{9}{2}} - \frac{20}{7} x^{\frac{7}{2}} + 10 x^{\frac{5}{2}} + C

\int \frac{x ^{2} + 4}{3 x ^{3} + 4 x ^{2} - 4 x} d x

f (x) = 2 x + 45, at x = 2

49 + 23 sin (\frac{2 π}{52} (t - 10))

\int \frac{1}{x 1 - ln ( x ^{2} )} d x

y^{^{'}} = \frac{2 x}{y}

What is the integral of ((x^2-5x)^2)/(sqrt(x)) ?
The integral of ((x^2-5x)^2)/(sqrt(x)) is 2/9 x^{9/2}-20/7 x^{7/2}+10x^{5/2}+C