\(\int \frac {e^{\frac {e^x+x^2}{x^3 \log (-2+e^{3/2}) \log (4+2 x)}} (-e^x x-x^3+(-2 x^2-x^3+e^x (-6-x+x^2)) \log (4+2 x))}{(2 x^4+x^5) \log (-2+e^{3/2}) \log ^2(4+2 x)} \, dx\) [2821]

Optimal result

Integrand size = 103, antiderivative size = 33 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {\frac {e^x}{x}+x}{x^2 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \] Output:

exp((exp(x)/x+x)/x^2/ln(4+2*x)/ln(exp(3/2)-2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 (2+x))}} \] Input:

Integrate[(E^((E^x + x^2)/(x^3*Log[-2 + E^(3/2)]*Log[4 + 2*x]))*(-(E^x*x) 
- x^3 + (-2*x^2 - x^3 + E^x*(-6 - x + x^2))*Log[4 + 2*x]))/((2*x^4 + x^5)* 
Log[-2 + E^(3/2)]*Log[4 + 2*x]^2),x]
 

Output:

E^((E^x + x^2)/(x^3*Log[-2 + E^(3/2)]*Log[2*(2 + x)]))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (e^{3/2}-2\right ) \log (2 x+4)}} \left (-x^3+\left (-x^3-2 x^2+e^x \left (x^2-x-6\right )\right ) \log (2 x+4)-e^x x\right )}{\left (x^5+2 x^4\right ) \log \left (e^{3/2}-2\right ) \log ^2(2 x+4)} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int -\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{\left (x^5+2 x^4\right ) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{\left (x^5+2 x^4\right ) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 2026

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x^3+e^x x+\left (x^3+2 x^2+e^x \left (-x^2+x+6\right )\right ) \log (2 x+4)\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7239

\(\displaystyle -\frac {\int \frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} \left (x \left (x^2+e^x\right )+(x+2) \left (x^2-e^x (x-3)\right ) \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}dx}{\log \left (e^{3/2}-2\right )}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {e^{\frac {x^2+e^x}{x^3 \log \left (-2+e^{3/2}\right ) \log (2 x+4)}} (\log (2 (x+2)) x+x+2 \log (2 (x+2)))}{x^2 (x+2) \log ^2(2 x+4)}+\frac {\exp \left (x+\frac {x^2+e^x}{\log \left (-2+e^{3/2}\right ) \log (2 x+4) x^3}\right ) \left (-\log (2 (x+2)) x^2+\log (2 (x+2)) x+x+6 \log (2 (x+2))\right )}{x^4 (x+2) \log ^2(2 x+4)}\right )dx}{\log \left (e^{3/2}-2\right )}\)

Input:

Int[(E^((E^x + x^2)/(x^3*Log[-2 + E^(3/2)]*Log[4 + 2*x]))*(-(E^x*x) - x^3 
+ (-2*x^2 - x^3 + E^x*(-6 - x + x^2))*Log[4 + 2*x]))/((2*x^4 + x^5)*Log[-2 
 + E^(3/2)]*Log[4 + 2*x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82

\[{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{x}}{x^{3} \ln \left (4+2 x \right ) \ln \left ({\mathrm e}^{\frac {3}{2}}-2\right )}}\]

Input:

int((((x^2-x-6)*exp(x)-x^3-2*x^2)*ln(4+2*x)-exp(x)*x-x^3)*exp((x^2+exp(x)) 
/x^3/ln(4+2*x)/ln(exp(3/2)-2))/(x^5+2*x^4)/ln(4+2*x)^2/ln(exp(3/2)-2),x)
 

Output:

exp((x^2+exp(x))/x^3/ln(4+2*x)/ln(exp(3/2)-2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\left (\frac {x^{2} + e^{x}}{x^{3} \log \left (2 \, x + 4\right ) \log \left (e^{\frac {3}{2}} - 2\right )}\right )} \] Input:

integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ 
exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 
/2)-2),x, algorithm="fricas")
 

Output:

e^((x^2 + e^x)/(x^3*log(2*x + 4)*log(e^(3/2) - 2)))
 

Sympy [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {x^{2} + e^{x}}{x^{3} \log {\left (-2 + e^{\frac {3}{2}} \right )} \log {\left (2 x + 4 \right )}}} \] Input:

integrate((((x**2-x-6)*exp(x)-x**3-2*x**2)*ln(4+2*x)-exp(x)*x-x**3)*exp((x 
**2+exp(x))/x**3/ln(4+2*x)/ln(exp(3/2)-2))/(x**5+2*x**4)/ln(4+2*x)**2/ln(e 
xp(3/2)-2),x)
 

Output:

exp((x**2 + exp(x))/(x**3*log(-2 + exp(3/2))*log(2*x + 4)))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ 
exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 
/2)-2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: In function CAR, the value of 
the first argument is  0which is not of the expected type LIST
 

Giac [F]

\[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=\int { -\frac {{\left (x^{3} + x e^{x} + {\left (x^{3} + 2 \, x^{2} - {\left (x^{2} - x - 6\right )} e^{x}\right )} \log \left (2 \, x + 4\right )\right )} e^{\left (\frac {x^{2} + e^{x}}{x^{3} \log \left (2 \, x + 4\right ) \log \left (e^{\frac {3}{2}} - 2\right )}\right )}}{{\left (x^{5} + 2 \, x^{4}\right )} \log \left (2 \, x + 4\right )^{2} \log \left (e^{\frac {3}{2}} - 2\right )} \,d x } \] Input:

integrate((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+ 
exp(x))/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3 
/2)-2),x, algorithm="giac")
 

Output:

integrate(-(x^3 + x*e^x + (x^3 + 2*x^2 - (x^2 - x - 6)*e^x)*log(2*x + 4))* 
e^((x^2 + e^x)/(x^3*log(2*x + 4)*log(e^(3/2) - 2)))/((x^5 + 2*x^4)*log(2*x 
 + 4)^2*log(e^(3/2) - 2)), x)
 

Mupad [B] (verification not implemented)

Time = 3.57 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^x+x^2}{x^3\,\ln \left ({\mathrm {e}}^{3/2}-2\right )\,\ln \left (2\,x+4\right )}} \] Input:

int(-(exp((exp(x) + x^2)/(x^3*log(exp(3/2) - 2)*log(2*x + 4)))*(x*exp(x) + 
 log(2*x + 4)*(exp(x)*(x - x^2 + 6) + 2*x^2 + x^3) + x^3))/(log(exp(3/2) - 
 2)*log(2*x + 4)^2*(2*x^4 + x^5)),x)
 

Output:

exp((exp(x) + x^2)/(x^3*log(exp(3/2) - 2)*log(2*x + 4)))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {e^{\frac {e^x+x^2}{x^3 \log \left (-2+e^{3/2}\right ) \log (4+2 x)}} \left (-e^x x-x^3+\left (-2 x^2-x^3+e^x \left (-6-x+x^2\right )\right ) \log (4+2 x)\right )}{\left (2 x^4+x^5\right ) \log \left (-2+e^{3/2}\right ) \log ^2(4+2 x)} \, dx=e^{\frac {e^{x}+x^{2}}{\mathrm {log}\left (\sqrt {e}\, e -2\right ) \mathrm {log}\left (2 x +4\right ) x^{3}}} \] Input:

int((((x^2-x-6)*exp(x)-x^3-2*x^2)*log(4+2*x)-exp(x)*x-x^3)*exp((x^2+exp(x) 
)/x^3/log(4+2*x)/log(exp(3/2)-2))/(x^5+2*x^4)/log(4+2*x)^2/log(exp(3/2)-2) 
,x)
 

Output:

e**((e**x + x**2)/(log(sqrt(e)*e - 2)*log(2*x + 4)*x**3))