Primitive of Root of x squared plus a squared cubed over x cubed

Theorem

$\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^3} \ \mathrm d x = \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$

Proof

Let:

 $\displaystyle z$ $=$ $\displaystyle x^2$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d z} {\mathrm d x}$ $=$ $\displaystyle 2 x$ Power Rule for Derivatives $\displaystyle \implies \ \$ $\displaystyle \int \frac{\left({\sqrt {x^2 + a^2} }\right)^3} {x^3} \ \mathrm d x$ $=$ $\displaystyle \int \frac{\left({\sqrt {z + a^2} }\right)^3} {2 z^2} \ \mathrm d z$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac 1 2 \left({\frac {-\left({\sqrt {z + a^2} }\right)^3} z + \frac 3 2 \int \frac{\sqrt {z + a^2} } z \ \mathrm d z}\right)$ Primitive of $\dfrac {\left({a x + b}\right)^m} {\left({p x + q}\right)^n}$: Formulation 3 $\displaystyle$ $=$ $\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac 3 2 \int \frac {\sqrt {x^2 + a^2} } x \ \mathrm d x$ substituting for $z$ and simplifying $\displaystyle$ $=$ $\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac 3 2 \left({\sqrt {x^2 + a^2} - a \ln \left({\frac {a + \sqrt {x^2 + a^2} } a}\right)}\right) + C$ Primitive of $\dfrac {\sqrt {x^2 + a^2} } x$ $\displaystyle$ $=$ $\displaystyle \frac {-\left({\sqrt {x^2 + a^2} }\right)^3} {2 x^2} + \frac {3 \sqrt {x^2 + a^2} } 2 - \frac {3 a} 2 \ln \left({\frac {a + \sqrt {x^2 + a^2} } x}\right) + C$ simplifying

$\blacksquare$