Integrand size = 156, antiderivative size = 29 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=x \left (1+3 \left (-\frac {2}{x}-x+\log \left (4+e-\frac {1}{(5-x)^2}+x\right )\right )\right ) \]
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=x-3 x^2+3 x \log \left (\frac {99+e (-5+x)^2-15 x-6 x^2+x^3}{(-5+x)^2}\right ) \]
Integrate[(-495 + 2775*x - 804*x^2 - 146*x^3 + 70*x^4 - 6*x^5 + E*(-125 + 825*x - 465*x^2 + 91*x^3 - 6*x^4) + (-1485 + 522*x + 45*x^2 - 33*x^3 + 3*x ^4 + E*(-375 + 225*x - 45*x^2 + 3*x^3))*Log[(99 - 15*x - 6*x^2 + x^3 + E*( 25 - 10*x + x^2))/(25 - 10*x + x^2)])/(-495 + 174*x + 15*x^2 - 11*x^3 + x^ 4 + E*(-125 + 75*x - 15*x^2 + x^3)),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.
| \(\displaystyle \int \frac {-6 x^5+70 x^4-146 x^3-804 x^2+e \left (-6 x^4+91 x^3-465 x^2+825 x-125\right )+\left (3 x^4-33 x^3+45 x^2+e \left (3 x^3-45 x^2+225 x-375\right )+522 x-1485\right ) \log \left (\frac {x^3-6 x^2+e \left (x^2-10 x+25\right )-15 x+99}{x^2-10 x+25}\right )+2775 x-495}{x^4-11 x^3+15 x^2+e \left (x^3-15 x^2+75 x-125\right )+174 x-495} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {-6 x^5+70 x^4-146 x^3-804 x^2+e \left (-6 x^4+91 x^3-465 x^2+825 x-125\right )+\left (3 x^4-33 x^3+45 x^2+e \left (3 x^3-45 x^2+225 x-375\right )+522 x-1485\right ) \log \left (\frac {x^3-6 x^2+e \left (x^2-10 x+25\right )-15 x+99}{x^2-10 x+25}\right )+2775 x-495}{5-x}+\frac {(5-x) (-x-e-4) \left (-6 x^5+70 x^4-146 x^3-804 x^2+e \left (-6 x^4+91 x^3-465 x^2+825 x-125\right )+\left (3 x^4-33 x^3+45 x^2+e \left (3 x^3-45 x^2+225 x-375\right )+522 x-1485\right ) \log \left (\frac {x^3-6 x^2+e \left (x^2-10 x+25\right )-15 x+99}{x^2-10 x+25}\right )+2775 x-495\right )}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {146 x^3}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {804 x^2}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {2775 x}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {e (1-6 x) (5-x)^2}{x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}+\frac {495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+3 \log \left (\frac {x^3-(6-e) x^2-5 (3+2 e) x+25 e+99}{(x-5)^2}\right )+\frac {6 x^5}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}+\frac {70 x^4}{(x-5) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 x^5-70 x^4+146 x^3+804 x^2-3 (x-5) \left (x^3-6 x^2-15 x+e (x-5)^2+99\right ) \log \left (\frac {x^3-6 x^2-15 x+e (x-5)^2+99}{(x-5)^2}\right )-2775 x+e (x-5)^3 (6 x-1)+495}{(5-x) \left (x^3-(6-e) x^2-5 (3+2 e) x+25 e+99\right )}dx\) |
Int[(-495 + 2775*x - 804*x^2 - 146*x^3 + 70*x^4 - 6*x^5 + E*(-125 + 825*x - 465*x^2 + 91*x^3 - 6*x^4) + (-1485 + 522*x + 45*x^2 - 33*x^3 + 3*x^4 + E *(-375 + 225*x - 45*x^2 + 3*x^3))*Log[(99 - 15*x - 6*x^2 + x^3 + E*(25 - 1 0*x + x^2))/(25 - 10*x + x^2)])/(-495 + 174*x + 15*x^2 - 11*x^3 + x^4 + E* (-125 + 75*x - 15*x^2 + x^3)),x]
3.6.18.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62
| method | result | size |
| norman | \(x -3 x^{2}+3 x \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}+x^{3}-6 x^{2}-15 x +99}{x^{2}-10 x +25}\right )\) | \(47\) |
| risch | \(x -3 x^{2}+3 x \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}+x^{3}-6 x^{2}-15 x +99}{x^{2}-10 x +25}\right )\) | \(47\) |
| parallelrisch | \(3 \,{\mathrm e}^{2}+475-3 x^{2}+3 x \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}+x^{3}-6 x^{2}-15 x +99}{x^{2}-10 x +25}\right )-158 \,{\mathrm e}+x\) | \(58\) |
| default | \(x -3 x^{2}-3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left ({\mathrm e}-6\right ) \textit {\_Z}^{2}+\left (-10 \,{\mathrm e}-15\right ) \textit {\_Z} +25 \,{\mathrm e}+99\right )}{\sum }\frac {\left (297+\textit {\_R}^{2} {\mathrm e}-20 \textit {\_R} \,{\mathrm e}-6 \textit {\_R}^{2}+75 \,{\mathrm e}-30 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}-10 \,{\mathrm e}-12 \textit {\_R} -15}\right )+3 \ln \left (\frac {x^{2} {\mathrm e}+x^{3}-10 x \,{\mathrm e}-6 x^{2}+25 \,{\mathrm e}-15 x +99}{x^{2}-10 x +25}\right ) x -3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left ({\mathrm e}-6\right ) \textit {\_Z}^{2}+\left (-10 \,{\mathrm e}-15\right ) \textit {\_Z} +25 \,{\mathrm e}+99\right )}{\sum }\frac {\left (-\textit {\_R}^{2} {\mathrm e}+20 \textit {\_R} \,{\mathrm e}+6 \textit {\_R}^{2}-75 \,{\mathrm e}+30 \textit {\_R} -297\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}-10 \,{\mathrm e}-12 \textit {\_R} -15}\right )\) | \(220\) |
| parts | \(x -3 x^{2}-3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left ({\mathrm e}-6\right ) \textit {\_Z}^{2}+\left (-10 \,{\mathrm e}-15\right ) \textit {\_Z} +25 \,{\mathrm e}+99\right )}{\sum }\frac {\left (297+\textit {\_R}^{2} {\mathrm e}-20 \textit {\_R} \,{\mathrm e}-6 \textit {\_R}^{2}+75 \,{\mathrm e}-30 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}-10 \,{\mathrm e}-12 \textit {\_R} -15}\right )+3 \ln \left (\frac {x^{2} {\mathrm e}+x^{3}-10 x \,{\mathrm e}-6 x^{2}+25 \,{\mathrm e}-15 x +99}{x^{2}-10 x +25}\right ) x -3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left ({\mathrm e}-6\right ) \textit {\_Z}^{2}+\left (-10 \,{\mathrm e}-15\right ) \textit {\_Z} +25 \,{\mathrm e}+99\right )}{\sum }\frac {\left (-\textit {\_R}^{2} {\mathrm e}+20 \textit {\_R} \,{\mathrm e}+6 \textit {\_R}^{2}-75 \,{\mathrm e}+30 \textit {\_R} -297\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}+3 \textit {\_R}^{2}-10 \,{\mathrm e}-12 \textit {\_R} -15}\right )\) | \(220\) |
int((((3*x^3-45*x^2+225*x-375)*exp(1)+3*x^4-33*x^3+45*x^2+522*x-1485)*ln(( (x^2-10*x+25)*exp(1)+x^3-6*x^2-15*x+99)/(x^2-10*x+25))+(-6*x^4+91*x^3-465* x^2+825*x-125)*exp(1)-6*x^5+70*x^4-146*x^3-804*x^2+2775*x-495)/((x^3-15*x^ 2+75*x-125)*exp(1)+x^4-11*x^3+15*x^2+174*x-495),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=-3 \, x^{2} + 3 \, x \log \left (\frac {x^{3} - 6 \, x^{2} + {\left (x^{2} - 10 \, x + 25\right )} e - 15 \, x + 99}{x^{2} - 10 \, x + 25}\right ) + x \]
integrate((((3*x^3-45*x^2+225*x-375)*exp(1)+3*x^4-33*x^3+45*x^2+522*x-1485 )*log(((x^2-10*x+25)*exp(1)+x^3-6*x^2-15*x+99)/(x^2-10*x+25))+(-6*x^4+91*x ^3-465*x^2+825*x-125)*exp(1)-6*x^5+70*x^4-146*x^3-804*x^2+2775*x-495)/((x^ 3-15*x^2+75*x-125)*exp(1)+x^4-11*x^3+15*x^2+174*x-495),x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=- 3 x^{2} + 3 x \log {\left (\frac {x^{3} - 6 x^{2} - 15 x + e \left (x^{2} - 10 x + 25\right ) + 99}{x^{2} - 10 x + 25} \right )} + x \]
integrate((((3*x**3-45*x**2+225*x-375)*exp(1)+3*x**4-33*x**3+45*x**2+522*x -1485)*ln(((x**2-10*x+25)*exp(1)+x**3-6*x**2-15*x+99)/(x**2-10*x+25))+(-6* x**4+91*x**3-465*x**2+825*x-125)*exp(1)-6*x**5+70*x**4-146*x**3-804*x**2+2 775*x-495)/((x**3-15*x**2+75*x-125)*exp(1)+x**4-11*x**3+15*x**2+174*x-495) ,x)
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=-3 \, x^{2} + 3 \, x \log \left (x^{3} + x^{2} {\left (e - 6\right )} - 5 \, x {\left (2 \, e + 3\right )} + 25 \, e + 99\right ) - 6 \, x \log \left (x - 5\right ) + x \]
integrate((((3*x^3-45*x^2+225*x-375)*exp(1)+3*x^4-33*x^3+45*x^2+522*x-1485 )*log(((x^2-10*x+25)*exp(1)+x^3-6*x^2-15*x+99)/(x^2-10*x+25))+(-6*x^4+91*x ^3-465*x^2+825*x-125)*exp(1)-6*x^5+70*x^4-146*x^3-804*x^2+2775*x-495)/((x^ 3-15*x^2+75*x-125)*exp(1)+x^4-11*x^3+15*x^2+174*x-495),x, algorithm=\
Time = 0.42 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=-3 \, x^{2} + 3 \, x \log \left (\frac {x^{3} + x^{2} e - 6 \, x^{2} - 10 \, x e - 15 \, x + 25 \, e + 99}{x^{2} - 10 \, x + 25}\right ) + x \]
integrate((((3*x^3-45*x^2+225*x-375)*exp(1)+3*x^4-33*x^3+45*x^2+522*x-1485 )*log(((x^2-10*x+25)*exp(1)+x^3-6*x^2-15*x+99)/(x^2-10*x+25))+(-6*x^4+91*x ^3-465*x^2+825*x-125)*exp(1)-6*x^5+70*x^4-146*x^3-804*x^2+2775*x-495)/((x^ 3-15*x^2+75*x-125)*exp(1)+x^4-11*x^3+15*x^2+174*x-495),x, algorithm=\
Time = 43.92 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {-495+2775 x-804 x^2-146 x^3+70 x^4-6 x^5+e \left (-125+825 x-465 x^2+91 x^3-6 x^4\right )+\left (-1485+522 x+45 x^2-33 x^3+3 x^4+e \left (-375+225 x-45 x^2+3 x^3\right )\right ) \log \left (\frac {99-15 x-6 x^2+x^3+e \left (25-10 x+x^2\right )}{25-10 x+x^2}\right )}{-495+174 x+15 x^2-11 x^3+x^4+e \left (-125+75 x-15 x^2+x^3\right )} \, dx=x+3\,x\,\ln \left (\frac {\mathrm {e}\,\left (x^2-10\,x+25\right )-15\,x-6\,x^2+x^3+99}{x^2-10\,x+25}\right )-3\,x^2 \]
int(-(exp(1)*(465*x^2 - 825*x - 91*x^3 + 6*x^4 + 125) - 2775*x + 804*x^2 + 146*x^3 - 70*x^4 + 6*x^5 - log((exp(1)*(x^2 - 10*x + 25) - 15*x - 6*x^2 + x^3 + 99)/(x^2 - 10*x + 25))*(522*x + exp(1)*(225*x - 45*x^2 + 3*x^3 - 37 5) + 45*x^2 - 33*x^3 + 3*x^4 - 1485) + 495)/(174*x + exp(1)*(75*x - 15*x^2 + x^3 - 125) + 15*x^2 - 11*x^3 + x^4 - 495),x)