Integrand size = 243, antiderivative size = 33 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\frac {e^{\left .\frac {1}{5}\right /x}+x}{\log \left (\frac {3+\frac {2}{x}}{5+\log (-1+2 x)}\right )} \end {dmath*}
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\frac {e^{\left .\frac {1}{5}\right /x}+x}{\log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \end {dmath*}
Integrate[(-50*x^2 + 120*x^3 + 30*x^4 + E^(1/(5*x))*(-50*x + 120*x^2 + 30* x^3) + (-10*x^2 + 20*x^3 + E^(1/(5*x))*(-10*x + 20*x^2))*Log[-1 + 2*x] + ( -50*x^2 + 25*x^3 + 150*x^4 + E^(1/(5*x))*(10 - 5*x - 30*x^2) + (-10*x^2 + 5*x^3 + 30*x^4 + E^(1/(5*x))*(2 - x - 6*x^2))*Log[-1 + 2*x])*Log[(2 + 3*x) /(5*x + x*Log[-1 + 2*x])])/((-50*x^2 + 25*x^3 + 150*x^4 + (-10*x^2 + 5*x^3 + 30*x^4)*Log[-1 + 2*x])*Log[(2 + 3*x)/(5*x + x*Log[-1 + 2*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.
| \(\displaystyle \int \frac {30 x^4+120 x^3-50 x^2+e^{\left .\frac {1}{5}\right /x} \left (30 x^3+120 x^2-50 x\right )+\left (20 x^3-10 x^2+e^{\left .\frac {1}{5}\right /x} \left (20 x^2-10 x\right )\right ) \log (2 x-1)+\left (150 x^4+25 x^3-50 x^2+e^{\left .\frac {1}{5}\right /x} \left (-30 x^2-5 x+10\right )+\left (30 x^4+5 x^3-10 x^2+e^{\left .\frac {1}{5}\right /x} \left (-6 x^2-x+2\right )\right ) \log (2 x-1)\right ) \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )}{\left (150 x^4+25 x^3-50 x^2+\left (30 x^4+5 x^3-10 x^2\right ) \log (2 x-1)\right ) \log ^2\left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-30 x^4-120 x^3+50 x^2-e^{\left .\frac {1}{5}\right /x} \left (30 x^3+120 x^2-50 x\right )-\left (20 x^3-10 x^2+e^{\left .\frac {1}{5}\right /x} \left (20 x^2-10 x\right )\right ) \log (2 x-1)-\left (150 x^4+25 x^3-50 x^2+e^{\left .\frac {1}{5}\right /x} \left (-30 x^2-5 x+10\right )+\left (30 x^4+5 x^3-10 x^2+e^{\left .\frac {1}{5}\right /x} \left (-6 x^2-x+2\right )\right ) \log (2 x-1)\right ) \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )}{5 x^2 \left (-6 x^2-x+2\right ) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {-30 x^4-120 x^3+50 x^2+10 e^{\left .\frac {1}{5}\right /x} \left (-3 x^3-12 x^2+5 x\right )+10 \left (-2 x^3+x^2+e^{\left .\frac {1}{5}\right /x} \left (x-2 x^2\right )\right ) \log (2 x-1)+\left (-150 x^4-25 x^3+50 x^2-5 e^{\left .\frac {1}{5}\right /x} \left (-6 x^2-x+2\right )+\left (-30 x^4-5 x^3+10 x^2-e^{\left .\frac {1}{5}\right /x} \left (-6 x^2-x+2\right )\right ) \log (2 x-1)\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}{x^2 \left (-6 x^2-x+2\right ) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {30 \log (2 x-1) x^2}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {150 x^2}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {30 x^2}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1) x}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25 x}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {20 \log (2 x-1) x}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {120 x}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}-\frac {10 \log (2 x-1)}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}-\frac {50}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}-\frac {10 \log (2 x-1)}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}-\frac {50}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right ) \left (3 x^2+12 x-5\right )+5 \left (e^{\left .\frac {1}{5}\right /x}-5 x^2\right ) \left (6 x^2+x-2\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )-(2 x-1) \log (2 x-1) \left (10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right )+(3 x+2) \left (5 x^2-e^{\left .\frac {1}{5}\right /x}\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(1-2 x) x^2 (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {10 \left (3 x^2+12 x-5\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{x^2 (2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {10 \log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right ) \left (3 x^2+12 x-5\right )+5 \left (e^{\left .\frac {1}{5}\right /x}-5 x^2\right ) \left (6 x^2+x-2\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )-(2 x-1) \log (2 x-1) \left (10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right )+(3 x+2) \left (5 x^2-e^{\left .\frac {1}{5}\right /x}\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(1-2 x) x^2 (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {10 \left (3 x^2+12 x-5\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{x^2 (2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {10 \log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right ) \left (3 x^2+12 x-5\right )+5 \left (e^{\left .\frac {1}{5}\right /x}-5 x^2\right ) \left (6 x^2+x-2\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )-(2 x-1) \log (2 x-1) \left (10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right )+(3 x+2) \left (5 x^2-e^{\left .\frac {1}{5}\right /x}\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(1-2 x) x^2 (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {10 \left (3 x^2+12 x-5\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{x^2 (2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {10 \log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right ) \left (3 x^2+12 x-5\right )+5 \left (e^{\left .\frac {1}{5}\right /x}-5 x^2\right ) \left (6 x^2+x-2\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )-(2 x-1) \log (2 x-1) \left (10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right )+(3 x+2) \left (5 x^2-e^{\left .\frac {1}{5}\right /x}\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(1-2 x) x^2 (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {10 \left (3 x^2+12 x-5\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{x^2 (2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {10 \log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right ) \left (3 x^2+12 x-5\right )+5 \left (e^{\left .\frac {1}{5}\right /x}-5 x^2\right ) \left (6 x^2+x-2\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )-(2 x-1) \log (2 x-1) \left (10 x \left (x+e^{\left .\frac {1}{5}\right /x}\right )+(3 x+2) \left (5 x^2-e^{\left .\frac {1}{5}\right /x}\right ) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{(1-2 x) x^2 (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {10 \left (3 x^2+12 x-5\right )}{(2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {e^{\left .\frac {1}{5}\right /x} \left (30 x^3+20 \log (2 x-1) x^2-6 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2-30 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x^2+120 x^2-10 \log (2 x-1) x-\log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-5 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right ) x-50 x+2 \log (2 x-1) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )+10 \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )\right )}{x^2 (2 x-1) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {5 \log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {25}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}+\frac {10 \log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (5 \int \frac {1}{(\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+5 \int \frac {1}{(2 x-1) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+50 \int \frac {1}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+10 \int \frac {\log (2 x-1)}{(3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+25 \int \frac {1}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+5 \int \frac {\log (2 x-1)}{(\log (2 x-1)+5) \log \left (\frac {3 x+2}{\log (2 x-1) x+5 x}\right )}dx+\frac {5 e^{\left .\frac {1}{5}\right /x} \left (-6 x^2 \log (2 x-1) \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )-30 x^2 \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )-x \log (2 x-1) \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )-5 x \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )+2 \log (2 x-1) \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )+10 \log \left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )\right )}{(1-2 x) (3 x+2) (\log (2 x-1)+5) \log ^2\left (\frac {3 x+2}{5 x+x \log (2 x-1)}\right )}\right )\) |
Int[(-50*x^2 + 120*x^3 + 30*x^4 + E^(1/(5*x))*(-50*x + 120*x^2 + 30*x^3) + (-10*x^2 + 20*x^3 + E^(1/(5*x))*(-10*x + 20*x^2))*Log[-1 + 2*x] + (-50*x^ 2 + 25*x^3 + 150*x^4 + E^(1/(5*x))*(10 - 5*x - 30*x^2) + (-10*x^2 + 5*x^3 + 30*x^4 + E^(1/(5*x))*(2 - x - 6*x^2))*Log[-1 + 2*x])*Log[(2 + 3*x)/(5*x + x*Log[-1 + 2*x])])/((-50*x^2 + 25*x^3 + 150*x^4 + (-10*x^2 + 5*x^3 + 30* x^4)*Log[-1 + 2*x])*Log[(2 + 3*x)/(5*x + x*Log[-1 + 2*x])]^2),x]
3.5.47.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Time = 46.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12
| method | result | size |
| parallelrisch | \(\frac {13500 x +13500 \,{\mathrm e}^{\frac {1}{5 x}}}{13500 \ln \left (\frac {2+3 x}{x \left (\ln \left (-1+2 x \right )+5\right )}\right )}\) | \(37\) |
| risch | \(\frac {2 \,{\mathrm e}^{\frac {1}{5 x}}+2 x}{2 \ln \left (3\right )-2 \ln \left (x \right )+2 \ln \left (\frac {2}{3}+x \right )-2 \ln \left (\ln \left (-1+2 x \right )+5\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (-1+2 x \right )+5}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right )^{2}+i \pi \,\operatorname {csgn}\left (i \left (\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right )^{2}-i \pi \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right )^{3}-i \pi \,\operatorname {csgn}\left (i \left (\frac {2}{3}+x \right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (-1+2 x \right )+5}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right )-i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{x \left (\ln \left (-1+2 x \right )+5\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{x \left (\ln \left (-1+2 x \right )+5\right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{x \left (\ln \left (-1+2 x \right )+5\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{\ln \left (-1+2 x \right )+5}\right ) \operatorname {csgn}\left (\frac {i \left (\frac {2}{3}+x \right )}{x \left (\ln \left (-1+2 x \right )+5\right )}\right )^{2}}\) | \(321\) |
int(((((-6*x^2-x+2)*exp(1/5/x)+30*x^4+5*x^3-10*x^2)*ln(-1+2*x)+(-30*x^2-5* x+10)*exp(1/5/x)+150*x^4+25*x^3-50*x^2)*ln((2+3*x)/(x*ln(-1+2*x)+5*x))+((2 0*x^2-10*x)*exp(1/5/x)+20*x^3-10*x^2)*ln(-1+2*x)+(30*x^3+120*x^2-50*x)*exp (1/5/x)+30*x^4+120*x^3-50*x^2)/((30*x^4+5*x^3-10*x^2)*ln(-1+2*x)+150*x^4+2 5*x^3-50*x^2)/ln((2+3*x)/(x*ln(-1+2*x)+5*x))^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\frac {x + e^{\left (\frac {1}{5 \, x}\right )}}{\log \left (\frac {3 \, x + 2}{x \log \left (2 \, x - 1\right ) + 5 \, x}\right )} \end {dmath*}
integrate(((((-6*x^2-x+2)*exp(1/5/x)+30*x^4+5*x^3-10*x^2)*log(-1+2*x)+(-30 *x^2-5*x+10)*exp(1/5/x)+150*x^4+25*x^3-50*x^2)*log((2+3*x)/(x*log(-1+2*x)+ 5*x))+((20*x^2-10*x)*exp(1/5/x)+20*x^3-10*x^2)*log(-1+2*x)+(30*x^3+120*x^2 -50*x)*exp(1/5/x)+30*x^4+120*x^3-50*x^2)/((30*x^4+5*x^3-10*x^2)*log(-1+2*x )+150*x^4+25*x^3-50*x^2)/log((2+3*x)/(x*log(-1+2*x)+5*x))^2,x, algorithm=\
Time = 0.81 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\frac {x}{\log {\left (\frac {3 x + 2}{x \log {\left (2 x - 1 \right )} + 5 x} \right )}} + \frac {e^{\frac {1}{5 x}}}{\log {\left (\frac {3 x + 2}{x \log {\left (2 x - 1 \right )} + 5 x} \right )}} \end {dmath*}
integrate(((((-6*x**2-x+2)*exp(1/5/x)+30*x**4+5*x**3-10*x**2)*ln(-1+2*x)+( -30*x**2-5*x+10)*exp(1/5/x)+150*x**4+25*x**3-50*x**2)*ln((2+3*x)/(x*ln(-1+ 2*x)+5*x))+((20*x**2-10*x)*exp(1/5/x)+20*x**3-10*x**2)*ln(-1+2*x)+(30*x**3 +120*x**2-50*x)*exp(1/5/x)+30*x**4+120*x**3-50*x**2)/((30*x**4+5*x**3-10*x **2)*ln(-1+2*x)+150*x**4+25*x**3-50*x**2)/ln((2+3*x)/(x*ln(-1+2*x)+5*x))** 2,x)
x/log((3*x + 2)/(x*log(2*x - 1) + 5*x)) + exp(1/(5*x))/log((3*x + 2)/(x*lo g(2*x - 1) + 5*x))
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\frac {x + e^{\left (\frac {1}{5 \, x}\right )}}{\log \left (3 \, x + 2\right ) - \log \left (x\right ) - \log \left (\log \left (2 \, x - 1\right ) + 5\right )} \end {dmath*}
integrate(((((-6*x^2-x+2)*exp(1/5/x)+30*x^4+5*x^3-10*x^2)*log(-1+2*x)+(-30 *x^2-5*x+10)*exp(1/5/x)+150*x^4+25*x^3-50*x^2)*log((2+3*x)/(x*log(-1+2*x)+ 5*x))+((20*x^2-10*x)*exp(1/5/x)+20*x^3-10*x^2)*log(-1+2*x)+(30*x^3+120*x^2 -50*x)*exp(1/5/x)+30*x^4+120*x^3-50*x^2)/((30*x^4+5*x^3-10*x^2)*log(-1+2*x )+150*x^4+25*x^3-50*x^2)/log((2+3*x)/(x*log(-1+2*x)+5*x))^2,x, algorithm=\
Time = 0.66 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=-\frac {x + e^{\left (\frac {1}{5 \, x}\right )}}{\log \left (x \log \left (2 \, x - 1\right ) + 5 \, x\right ) - \log \left (3 \, x + 2\right )} \end {dmath*}
integrate(((((-6*x^2-x+2)*exp(1/5/x)+30*x^4+5*x^3-10*x^2)*log(-1+2*x)+(-30 *x^2-5*x+10)*exp(1/5/x)+150*x^4+25*x^3-50*x^2)*log((2+3*x)/(x*log(-1+2*x)+ 5*x))+((20*x^2-10*x)*exp(1/5/x)+20*x^3-10*x^2)*log(-1+2*x)+(30*x^3+120*x^2 -50*x)*exp(1/5/x)+30*x^4+120*x^3-50*x^2)/((30*x^4+5*x^3-10*x^2)*log(-1+2*x )+150*x^4+25*x^3-50*x^2)/log((2+3*x)/(x*log(-1+2*x)+5*x))^2,x, algorithm=\
Timed out. \begin {dmath*} \int \frac {-50 x^2+120 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (-50 x+120 x^2+30 x^3\right )+\left (-10 x^2+20 x^3+e^{\left .\frac {1}{5}\right /x} \left (-10 x+20 x^2\right )\right ) \log (-1+2 x)+\left (-50 x^2+25 x^3+150 x^4+e^{\left .\frac {1}{5}\right /x} \left (10-5 x-30 x^2\right )+\left (-10 x^2+5 x^3+30 x^4+e^{\left .\frac {1}{5}\right /x} \left (2-x-6 x^2\right )\right ) \log (-1+2 x)\right ) \log \left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )}{\left (-50 x^2+25 x^3+150 x^4+\left (-10 x^2+5 x^3+30 x^4\right ) \log (-1+2 x)\right ) \log ^2\left (\frac {2+3 x}{5 x+x \log (-1+2 x)}\right )} \, dx=\int -\frac {\ln \left (2\,x-1\right )\,\left ({\mathrm {e}}^{\frac {1}{5\,x}}\,\left (10\,x-20\,x^2\right )+10\,x^2-20\,x^3\right )-{\mathrm {e}}^{\frac {1}{5\,x}}\,\left (30\,x^3+120\,x^2-50\,x\right )+50\,x^2-120\,x^3-30\,x^4+\ln \left (\frac {3\,x+2}{5\,x+x\,\ln \left (2\,x-1\right )}\right )\,\left (\ln \left (2\,x-1\right )\,\left ({\mathrm {e}}^{\frac {1}{5\,x}}\,\left (6\,x^2+x-2\right )+10\,x^2-5\,x^3-30\,x^4\right )+{\mathrm {e}}^{\frac {1}{5\,x}}\,\left (30\,x^2+5\,x-10\right )+50\,x^2-25\,x^3-150\,x^4\right )}{{\ln \left (\frac {3\,x+2}{5\,x+x\,\ln \left (2\,x-1\right )}\right )}^2\,\left (\ln \left (2\,x-1\right )\,\left (30\,x^4+5\,x^3-10\,x^2\right )-50\,x^2+25\,x^3+150\,x^4\right )} \,d x \end {dmath*}
int(-(log(2*x - 1)*(exp(1/(5*x))*(10*x - 20*x^2) + 10*x^2 - 20*x^3) - exp( 1/(5*x))*(120*x^2 - 50*x + 30*x^3) + 50*x^2 - 120*x^3 - 30*x^4 + log((3*x + 2)/(5*x + x*log(2*x - 1)))*(log(2*x - 1)*(exp(1/(5*x))*(x + 6*x^2 - 2) + 10*x^2 - 5*x^3 - 30*x^4) + exp(1/(5*x))*(5*x + 30*x^2 - 10) + 50*x^2 - 25 *x^3 - 150*x^4))/(log((3*x + 2)/(5*x + x*log(2*x - 1)))^2*(log(2*x - 1)*(5 *x^3 - 10*x^2 + 30*x^4) - 50*x^2 + 25*x^3 + 150*x^4)),x)
int(-(log(2*x - 1)*(exp(1/(5*x))*(10*x - 20*x^2) + 10*x^2 - 20*x^3) - exp( 1/(5*x))*(120*x^2 - 50*x + 30*x^3) + 50*x^2 - 120*x^3 - 30*x^4 + log((3*x + 2)/(5*x + x*log(2*x - 1)))*(log(2*x - 1)*(exp(1/(5*x))*(x + 6*x^2 - 2) + 10*x^2 - 5*x^3 - 30*x^4) + exp(1/(5*x))*(5*x + 30*x^2 - 10) + 50*x^2 - 25 *x^3 - 150*x^4))/(log((3*x + 2)/(5*x + x*log(2*x - 1)))^2*(log(2*x - 1)*(5 *x^3 - 10*x^2 + 30*x^4) - 50*x^2 + 25*x^3 + 150*x^4)), x)