integration problem

[tmwong]

can anyone solve this equation?

∫x^3/(x^2+1)^(3/2) dx

[NeutronStar]

\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx =\frac{x^2+2}{\sqrt{x^2+1}}+C

[Galileo]

Hmm, maybe integration by parts would work?
\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}=x^2\f rac{x}{\left(x^2+1\right)^{\frac{3}{2}}}
The primitive of the factor on the right is
\frac{-1}{\sqrt{x^2+1}}
looks like it may lead somewhere.
Appears tedious though

[sally]

:surprised

[dextercioby]

\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx =
\frac{x^2+2}{\sqrt{x^2+1}}+C

Proof
\int\frac{x^3}{\left(x^2+1\right)^{\frac{3}{2}}}dx =
\int\frac{x^2}{2}\frac{2x}{(\sqrt{x^2+1})^3}dx=
-\frac{x^2}{\sqrt{x^2+1}}+\int\frac{2x}{\sqrt{x^2+1 }}dx=
-\frac{x^2}{\sqrt{x^2+1}}+2\sqrt{x^2+1}+C=
\frac{x^2+2}{\sqrt{x^2+1}}+C