Integrand size = 261, antiderivative size = 29 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=e^{\frac {x}{\left (\left (x+x \left (4+x+x^2\right )\right )^2\right )^x+\frac {x^2}{\log (x)}}} \]
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \]
Integrate[(x^(x/(x^2 + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x])) *(5*x^2 + x^3 + x^4 + (-5*x^2 - x^3 - x^4)*Log[x] + (25*x^2 + 10*x^3 + 11* x^4 + 2*x^5 + x^6)^x*((5 - 9*x - 3*x^2 - 6*x^3)*Log[x]^2 + (-5*x - x^2 - x ^3)*Log[x]^2*Log[25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6])))/(5*x^4 + x^5 + x^6 + (10*x^2 + 2*x^3 + 2*x^4)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x *Log[x] + (5 + x + x^2)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^(2*x)*Log [x]^2),x]
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.
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^6+2 x^5+11 x^4+10 x^3+25 x^2\right )^x \log (x)}} \left (x^4+x^3+5 x^2+\left (-x^4-x^3-5 x^2\right ) \log (x)+\left (x^6+2 x^5+11 x^4+10 x^3+25 x^2\right )^x \left (\left (-6 x^3-3 x^2-9 x+5\right ) \log ^2(x)+\left (-x^3-x^2-5 x\right ) \log \left (x^6+2 x^5+11 x^4+10 x^3+25 x^2\right ) \log ^2(x)\right )\right )}{x^6+x^5+5 x^4+\left (x^2+x+5\right ) \left (x^6+2 x^5+11 x^4+10 x^3+25 x^2\right )^{2 x} \log ^2(x)+\left (2 x^4+2 x^3+10 x^2\right ) \left (x^6+2 x^5+11 x^4+10 x^3+25 x^2\right )^x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \left (x^2 \left (x^2+x+5\right )-x^2 \left (x^2+x+5\right ) \log (x)+\left (x^2 \left (x^2+x+5\right )^2\right )^x \left (-\log ^2(x)\right ) \left (6 x^3+3 x^2+\left (x^2+x+5\right ) x \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}+2} \left (6 x^3 \log (x)+x^2+2 x^2 \log (x)+x^2 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (x^2+x+5\right )^2\right )+x+8 x \log (x)-10 \log (x)+5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )^2}-\frac {x^{\frac {x}{x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)}} \log (x) \left (6 x^3+3 x^2+x^2 \log \left (x^2 \left (x^2+x+5\right )^2\right )+5 x \log \left (x^2 \left (x^2+x+5\right )^2\right )+x^3 \log \left (x^2 \left (x^2+x+5\right )^2\right )+9 x-5\right )}{\left (x^2+x+5\right ) \left (x^2+\left (x^2 \left (x^2+x+5\right )^2\right )^x \log (x)\right )}\right )dx\) |
Int[(x^(x/(x^2 + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x]))*(5*x^ 2 + x^3 + x^4 + (-5*x^2 - x^3 - x^4)*Log[x] + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*((5 - 9*x - 3*x^2 - 6*x^3)*Log[x]^2 + (-5*x - x^2 - x^3)*Lo g[x]^2*Log[25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6])))/(5*x^4 + x^5 + x^6 + (10*x^2 + 2*x^3 + 2*x^4)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x ] + (5 + x + x^2)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^(2*x)*Log[x]^2) ,x]
3.7.98.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.28
\[x^{\frac {x}{\ln \left (x \right ) x^{2 x} \left (x^{2}+x +5\right )^{2 x} {\mathrm e}^{-\frac {i \pi x \left (\operatorname {csgn}\left (i \left (x^{2}+x +5\right )^{2}\right )-2 \,\operatorname {csgn}\left (i \left (x^{2}+x +5\right )\right )+\operatorname {csgn}\left (i \left (x^{2}+x +5\right )^{2}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} \left (x^{2}+x +5\right )^{2}\right )+\operatorname {csgn}\left (i x^{2}\right )-2 \,\operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2} \left (x^{2}+x +5\right )^{2}\right )\right )}{2}}+x^{2}}}\]
int((((-x^3-x^2-5*x)*ln(x)^2*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(-6*x^3-3* x^2-9*x+5)*ln(x)^2)*exp(x*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(-x^4-x^3-5* x^2)*ln(x)+x^4+x^3+5*x^2)*exp(x*ln(x)/(ln(x)*exp(x*ln(x^6+2*x^5+11*x^4+10* x^3+25*x^2))+x^2))/((x^2+x+5)*ln(x)^2*exp(x*ln(x^6+2*x^5+11*x^4+10*x^3+25* x^2))^2+(2*x^4+2*x^3+10*x^2)*ln(x)*exp(x*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2 ))+x^6+x^5+5*x^4),x)
x^(x/(ln(x)*x^(2*x)*(x^2+x+5)^(2*x)*exp(-1/2*I*Pi*x*(csgn(I*(x^2+x+5)^2)-2 *csgn(I*(x^2+x+5))+csgn(I*(x^2+x+5)^2)*csgn(I*x^2)*csgn(I*x^2*(x^2+x+5)^2) +csgn(I*x^2)-2*csgn(I*x)+csgn(I*x^2*(x^2+x+5)^2)))+x^2))
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^{2} + {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )}} \]
integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(- 6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(- x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+2* x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+1 1*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11 *x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm=\
Timed out. \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=\text {Timed out} \]
integrate((((-x**3-x**2-5*x)*ln(x)**2*ln(x**6+2*x**5+11*x**4+10*x**3+25*x* *2)+(-6*x**3-3*x**2-9*x+5)*ln(x)**2)*exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+ 25*x**2))+(-x**4-x**3-5*x**2)*ln(x)+x**4+x**3+5*x**2)*exp(x*ln(x)/(ln(x)*e xp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))+x**2))/((x**2+x+5)*ln(x)**2* exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))**2+(2*x**4+2*x**3+10*x**2)* ln(x)*exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))+x**6+x**5+5*x**4),x)
Time = 0.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^{2} + e^{\left (2 \, x \log \left (x^{2} + x + 5\right ) + 2 \, x \log \left (x\right )\right )} \log \left (x\right )}} \]
integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(- 6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(- x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+2* x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+1 1*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11 *x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm=\
\[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=\int { \frac {{\left (x^{4} + x^{3} - {\left ({\left (x^{3} + x^{2} + 5 \, x\right )} \log \left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right ) \log \left (x\right )^{2} + {\left (6 \, x^{3} + 3 \, x^{2} + 9 \, x - 5\right )} \log \left (x\right )^{2}\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} + 5 \, x^{2} - {\left (x^{4} + x^{3} + 5 \, x^{2}\right )} \log \left (x\right )\right )} x^{\frac {x}{x^{2} + {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )}}}{x^{6} + x^{5} + 5 \, x^{4} + {\left (x^{2} + x + 5\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{2 \, x} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + x^{3} + 5 \, x^{2}\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )} \,d x } \]
integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(- 6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(- x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+2* x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+1 1*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11 *x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm=\
integrate((x^4 + x^3 - ((x^3 + x^2 + 5*x)*log(x^6 + 2*x^5 + 11*x^4 + 10*x^ 3 + 25*x^2)*log(x)^2 + (6*x^3 + 3*x^2 + 9*x - 5)*log(x)^2)*(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x + 5*x^2 - (x^4 + x^3 + 5*x^2)*log(x))*x^(x/(x^ 2 + (x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x*log(x)))/(x^6 + x^5 + 5*x^4 + (x^2 + x + 5)*(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^(2*x)*log(x)^2 + 2*(x^4 + x^3 + 5*x^2)*(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x*log(x)), x)
Time = 8.73 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{\ln \left (x\right )\,{\left (x^6+2\,x^5+11\,x^4+10\,x^3+25\,x^2\right )}^x+x^2}} \]
int((exp((x*log(x))/(x^2 + exp(x*log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^ 6))*log(x)))*(5*x^2 - exp(x*log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6))*( log(x)^2*(9*x + 3*x^2 + 6*x^3 - 5) + log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)*log(x)^2*(5*x + x^2 + x^3)) - log(x)*(5*x^2 + x^3 + x^4) + x^3 + x^ 4))/(5*x^4 + x^5 + x^6 + exp(x*log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6) )*log(x)*(10*x^2 + 2*x^3 + 2*x^4) + exp(2*x*log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6))*log(x)^2*(x + x^2 + 5)),x)