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

The integral of x^3sqrt(a^2x^2+b^2) is ((3a^2x^2-2b^2)(a^2x^2+b^2)^{3/2})/(15a^4)+C

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integral of x^3sqrt(a^2x^2+b^2)

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

\frac{( 3 a ^{2} x ^{2} - 2 b ^{2} ) ( a ^{2} x ^{2} + b ^{2} ) ^{\frac{3}{2}}}{15 a ^{4}} + C

Solution steps

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

Simplify

x^{3} : x^{2} x

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

Apply u-substitution

: \int \frac{u a ^{2} u + b ^{2}}{2} d u

= \int \frac{u a ^{2} u + b ^{2}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u a^{2} u + b^{2} d u

Apply u-substitution

: \int \frac{v ( v - b ^{2} )}{a ^{4}} d v

= \frac{1}{2} \cdot \int \frac{v ( v - b ^{2} )}{a ^{4}} d v

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{a ^{4}} \cdot \int v (v - b^{2}) d v

Expand

v (v - b^{2}) : v^{\frac{3}{2}} - b^{2} v

= \frac{1}{2} \cdot \frac{1}{a ^{4}} \cdot \int v^{\frac{3}{2}} - b^{2} v d v

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}{2} \cdot \frac{1}{a ^{4}} (\int v^{\frac{3}{2}} d v - \int b^{2} v d v)

\int v^{\frac{3}{2}} d v = \frac{2}{5} v^{\frac{5}{2}}

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

= \frac{1}{2} \cdot \frac{1}{a ^{4}} (\frac{2}{5} v^{\frac{5}{2}} - b^{2} \frac{2}{3} v^{\frac{3}{2}})

Substitute back

= \frac{1}{2} \cdot \frac{1}{a ^{4}} (\frac{2}{5} (a^{2} x^{2} + b^{2})^{\frac{5}{2}} - b^{2} \frac{2}{3} (a^{2} x^{2} + b^{2})^{\frac{3}{2}})

Simplify

\frac{1}{2} \cdot \frac{1}{a ^{4}} (\frac{2}{5} (a^{2} x^{2} + b^{2})^{\frac{5}{2}} - b^{2} \frac{2}{3} (a^{2} x^{2} + b^{2})^{\frac{3}{2}}) : \frac{( 3 a ^{2} x ^{2} - 2 b ^{2} ) ( a ^{2} x ^{2} + b ^{2} ) ^{\frac{3}{2}}}{15 a ^{4}}

= \frac{( 3 a ^{2} x ^{2} - 2 b ^{2} ) ( a ^{2} x ^{2} + b ^{2} ) ^{\frac{3}{2}}}{15 a ^{4}}

Add a constant to the solution

= \frac{( 3 a ^{2} x ^{2} - 2 b ^{2} ) ( a ^{2} x ^{2} + b ^{2} ) ^{\frac{3}{2}}}{15 a ^{4}} + C

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What is the integral of x^3sqrt(a^2x^2+b^2) ?
The integral of x^3sqrt(a^2x^2+b^2) is ((3a^2x^2-2b^2)(a^2x^2+b^2)^{3/2})/(15a^4)+C