Crazy Integral

Integrating this expression takes quite a few steps.

$\int { \cfrac { x^{ 4 } }{ 2(1-x)^{ 5 } } } +\frac { 4x }{ x^{ 2 }+5x+6 } dx$

By setting $x= sin^{ 2 }(u)$ in the first fraction and breaking the second one apart using partial fractions, you get the following:

$\int { tan^{ 9 }(u)du } +\int { \frac { -8 }{ x+2 } +\frac { 12 }{ x+3 } dx }$

The right integral here comes out as $-8ln|x+2| +12ln|x+3|$ so for now we will omit this from the integration and focus on the left integral. By setting $y = tan(u)$ you get this integral

$\int { \frac { y^{ 9 } }{ y^{ 2 }+1 } } dy$

By dividing that out you arrive at

$\int { y^{ 7 }-y^{ 5 }+y^{ 3 }-y+\frac { y }{ y^{ 2 }+1 } } dy$

Most of it can be integrated rather easily, but the last part we'll inetegrate by setting $y=tan(w)$ which gives us(I'm omitting the other terms because integration of them is simple)

$\int { tan(w)dw\quad =\quad -ln|cos(w)| }$

When we put this all together, we get this final result

$y\quad =\quad tan(asin(\sqrt { x } ))\\ w\quad =\quad asin(\sqrt { x } )$

$\frac { y^{ 8 } }{ 8 } -\frac { y^{ 6 } }{ 6 } +\frac { y^{ 4 } }{ 4 } -\frac { y^{ 2 } }{ 2 } -ln|cos(w)|-8ln|x+2|+12ln|x+3|$

After getting that result, you just have to plug and chug to get the correct answer ( Personally, I used desmos graphing calculator: https://www.desmos.com/calculator/ckyrrlf7dg )