What is the integral of x(ax^2+b)^3 ?

The integral of x(ax^2+b)^3 is (a^3x^8)/8+(a^2bx^6)/2+(3ab^2x^4)/4+(b^3x^2)/2+C

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integral of x(ax^2+b)^3

\int x (a x^{2} + b)^{3} d x

\frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2} + C

Solution steps

\int x (a x^{2} + b)^{3} d x

Expand

x (a x^{2} + b)^{3} : a^{3} x^{7} + 3 a^{2} b x^{5} + 3 a b^{2} x^{3} + b^{3} x

= \int a^{3} x^{7} + 3 a^{2} b x^{5} + 3 a b^{2} x^{3} + b^{3} x 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 a^{3} x^{7} d x + \int 3 a^{2} b x^{5} d x + \int 3 a b^{2} x^{3} d x + \int b^{3} x d x

\int a^{3} x^{7} d x = \frac{a ^{3} x ^{8}}{8}

\int 3 a^{2} b x^{5} d x = \frac{a ^{2} b x ^{6}}{2}

\int 3 a b^{2} x^{3} d x = \frac{3 a b ^{2} x ^{4}}{4}

\int b^{3} x d x = \frac{b ^{3} x ^{2}}{2}

= \frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2}

Add a constant to the solution

= \frac{a ^{3} x ^{8}}{8} + \frac{a ^{2} b x ^{6}}{2} + \frac{3 a b ^{2} x ^{4}}{4} + \frac{b ^{3} x ^{2}}{2} + C

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\frac{d}{d y} (1)

{e^{t} sin (3 t)}

n = 1 \sum \infty \frac{ln ( 6 n )}{n}

What is the integral of x(ax^2+b)^3 ?
The integral of x(ax^2+b)^3 is (a^3x^8)/8+(a^2bx^6)/2+(3ab^2x^4)/4+(b^3x^2)/2+C