What is the integral of ((sqrt(x+5))^4)/(3sqrt(x)) ?

The integral of ((sqrt(x+5))^4)/(3sqrt(x)) is 1/45 (6x^2sqrt(x)+100xsqrt(x)+750sqrt(x))+C

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integral of ((sqrt(x+5))^4)/(3sqrt(x))

\int \frac{( x + 5 ) ^{4}}{3 x} d x

\frac{1}{45} (6 x^{2} x + 100 x x + 750 x) + C

Solution steps

\int \frac{( x + 5 ) ^{4}}{3 x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

= \frac{1}{3} (x (x + 5)^{2} - \int \frac{3 x ^{2} + 10 x - 25}{2 x} d x)

\int \frac{3 x ^{2} + 10 x - 25}{2 x} d x = \frac{1}{15} x (9 x^{2} + 50 x - 375)

= \frac{1}{3} (x (x + 5)^{2} - \frac{1}{15} x (9 x^{2} + 50 x - 375))

Simplify

\frac{1}{3} (x (x + 5)^{2} - \frac{1}{15} x (9 x^{2} + 50 x - 375)) : \frac{1}{45} (6 x^{2} x + 100 x x + 750 x)

= \frac{1}{45} (6 x^{2} x + 100 x x + 750 x)

Add a constant to the solution

= \frac{1}{45} (6 x^{2} x + 100 x x + 750 x) + C

What is the integral of ((sqrt(x+5))^4)/(3sqrt(x)) ?
The integral of ((sqrt(x+5))^4)/(3sqrt(x)) is 1/45 (6x^2sqrt(x)+100xsqrt(x)+750sqrt(x))+C