<s>[INST] indefinite integral of $\int \frac{dx}{\sqrt{(x)(x+1)(x+2)}}$ I tried integration by parts considering $dx=du$
$\frac{1}{\sqrt{(x)(x+1)(x+2)}}=v$ but im not getting the answer.
My attempt....
$uv=\sqrt{\frac{x}{(x+1)(x+2)}}$ and-$\int{dv u}$=$\frac{I}{2}$+$\int{\frac{\sqrt{x}}{2\sqrt{(x+1)}(x+2)^{\frac{3}{2}}}}$+$\int{\frac{\sqrt{x}}{2\sqrt{(x+2)}(x+1)^{\frac{3}{2}}}}$
How do we calculate $\int{\frac{\sqrt{x}}{2\sqrt{(x+1)}(x+2)^{\frac{3}{2}}}}$ and
$\int{\frac{\sqrt{x}}{2\sqrt{(x+2)}(x+1)^{\frac{3}{2}}}}$
How do we do it?
Please help!
 [/INST] In general, an integral with square-root of a polynomial of degree $3$ or $4$ is an elliptic integral.
Maple evaluates this in terms of the elliptic integral of the first kind $F$:
$$
\int \!{\frac {1}{\sqrt {x \left( x+1 \right)  \left( x+2 \right) }}}
\,{\rm d}x=2\;{ F} \left( \frac{\sqrt {x+2}}{\sqrt{2}},\sqrt {2} \right)
$$
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