What is the integral of 1/((a+bx)^3) ?

The integral of 1/((a+bx)^3) is x/((a+bx)^3)-3/(2b(bx+a)^2)+a/(b(bx+a)^3)+C

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integral of 1/((a+bx)^3)

\int \frac{1}{( a + b x ) ^{3}} d x

\frac{x}{( a + b x ) ^{3}} - \frac{3}{2 b ( b x + a ) ^{2}} + \frac{a}{b ( b x + a ) ^{3}} + C

Solution steps

\int \frac{1}{( a + b x ) ^{3}} d x

Apply Integration By Parts

: \frac{x}{( a + b x ) ^{3}} - \int - \frac{3 b x}{( a + b x ) ^{4}} d x

= \frac{x}{( a + b x ) ^{3}} - \int - \frac{3 b x}{( a + b x ) ^{4}} d x

\int - \frac{3 b x}{( a + b x ) ^{4}} d x = \frac{3}{2 b ( b x + a ) ^{2}} - \frac{a}{b ( b x + a ) ^{3}}

= \frac{x}{( a + b x ) ^{3}} - (\frac{3}{2 b ( b x + a ) ^{2}} - \frac{a}{b ( b x + a ) ^{3}})

Simplify

= \frac{x}{( a + b x ) ^{3}} - \frac{3}{2 b ( b x + a ) ^{2}} + \frac{a}{b ( b x + a ) ^{3}}

Add a constant to the solution

= \frac{x}{( a + b x ) ^{3}} - \frac{3}{2 b ( b x + a ) ^{2}} + \frac{a}{b ( b x + a ) ^{3}} + C

\int_{0}^{0.5} (300 x) d x

y^{^{''}} = 4 t^{3} + t

y = (x - 5) x (x + 5), y = 5 - x

\int_{0}^{0.6} 150 x d x

\frac{( 1 - e ^{- s} )}{s}

What is the integral of 1/((a+bx)^3) ?
The integral of 1/((a+bx)^3) is x/((a+bx)^3)-3/(2b(bx+a)^2)+a/(b(bx+a)^3)+C