What is the integral of x/((3+5x)^{3/5)} ?

The integral of x/((3+5x)^{3/5)} is 1/70 (2(5x+3)^{7/5}-21(5x+3)^{2/5})+C

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integral of x/((3+5x)^{3/5)}

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

\frac{1}{70} (2 (5 x + 3)^{\frac{7}{5}} - 21 (5 x + 3)^{\frac{2}{5}}) + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 3}{25 u ^{\frac{3}{5}}} d u

= \int \frac{u - 3}{25 u ^{\frac{3}{5}}} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{u - 3}{u ^{\frac{3}{5}}} d u

Expand

\frac{u - 3}{u ^{\frac{3}{5}}} : u^{\frac{2}{5}} - \frac{3}{u ^{\frac{3}{5}}}

= \frac{1}{25} \cdot \int u^{\frac{2}{5}} - \frac{3}{u ^{\frac{3}{5}}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int u^{\frac{2}{5}} d u = \frac{5}{7} u^{\frac{7}{5}}

\int \frac{3}{u ^{\frac{3}{5}}} d u = \frac{15}{2} u^{\frac{2}{5}}

= \frac{1}{25} (\frac{5}{7} u^{\frac{7}{5}} - \frac{15}{2} u^{\frac{2}{5}})

Substitute back

u = 3 + 5 x

= \frac{1}{25} (\frac{5}{7} (3 + 5 x)^{\frac{7}{5}} - \frac{15}{2} (3 + 5 x)^{\frac{2}{5}})

Simplify

\frac{1}{25} (\frac{5}{7} (3 + 5 x)^{\frac{7}{5}} - \frac{15}{2} (3 + 5 x)^{\frac{2}{5}}) : \frac{1}{70} (2 (5 x + 3)^{\frac{7}{5}} - 21 (5 x + 3)^{\frac{2}{5}})

= \frac{1}{70} (2 (5 x + 3)^{\frac{7}{5}} - 21 (5 x + 3)^{\frac{2}{5}})

Add a constant to the solution

= \frac{1}{70} (2 (5 x + 3)^{\frac{7}{5}} - 21 (5 x + 3)^{\frac{2}{5}}) + C

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What is the integral of x/((3+5x)^{3/5)} ?
The integral of x/((3+5x)^{3/5)} is 1/70 (2(5x+3)^{7/5}-21(5x+3)^{2/5})+C