Integrand size = 518, antiderivative size = 29 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5}{\left (1+\frac {x \left (5+e^x+x\right )}{5-x}-\log (5)-\log (x)\right )^2} \] Output:
5/(-ln(x)+1+(exp(x)+5+x)/(5-x)*x-ln(5))^2
Time = 0.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 (-5+x)^2}{\left (5+x^2-5 \log (5)+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^2} \] Input:
Integrate[(1250 - 2000*x - 100*x^2 + 140*x^3 - 10*x^4 + E^x*(-250*x - 200* x^2 + 100*x^3 - 10*x^4))/(125*x + 300*x^2 + 315*x^3 + 184*x^4 + E^(3*x)*x^ 4 + 63*x^5 + 12*x^6 + x^7 + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x^6)*Log[5] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5)*Log[5]^2 + (- 125*x + 75*x^2 - 15*x^3 + x^4)*Log[5]^3 + E^(2*x)*(15*x^3 + 12*x^4 + 3*x^5 + (-15*x^3 + 3*x^4)*Log[5]) + E^x*(75*x^2 + 120*x^3 + 78*x^4 + 24*x^5 + 3 *x^6 + (-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5)*Log[5] + (75*x^2 - 30*x^3 + 3*x ^4)*Log[5]^2) + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x^6 + E^( 2*x)*(-15*x^3 + 3*x^4) + (750*x + 300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)*Log[5 ] + (-375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5]^2 + E^x*(-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5 + (150*x^2 - 60*x^3 + 6*x^4)*Log[5]))*Log[x] + (375*x + 15 0*x^2 - 30*x^3 - 18*x^4 + 3*x^5 + E^x*(75*x^2 - 30*x^3 + 3*x^4) + (-375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5])*Log[x]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log[x]^3),x]
Output:
(5*(-5 + x)^2)/(5 + x^2 - 5*Log[5] + x*(4 + E^x + Log[5]) + (-5 + x)*Log[x ])^2
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.
| \(\displaystyle \int \frac {-10 x^4+140 x^3-100 x^2+e^x \left (-10 x^4+100 x^3-200 x^2-250 x\right )-2000 x+1250}{x^7+12 x^6+63 x^5+e^{3 x} x^4+184 x^4+315 x^3+300 x^2+e^{2 x} \left (3 x^5+12 x^4+15 x^3+\left (3 x^4-15 x^3\right ) \log (5)\right )+\left (x^4-15 x^3+75 x^2-125 x\right ) \log ^3(x)+\left (x^4-15 x^3+75 x^2-125 x\right ) \log ^3(5)+\left (3 x^5-18 x^4-30 x^3+150 x^2+e^x \left (3 x^4-30 x^3+75 x^2\right )+\left (3 x^4-45 x^3+225 x^2-375 x\right ) \log (5)+375 x\right ) \log ^2(x)+\left (3 x^5-18 x^4-30 x^3+150 x^2+375 x\right ) \log ^2(5)+e^x \left (3 x^6+24 x^5+78 x^4+120 x^3+75 x^2+\left (3 x^4-30 x^3+75 x^2\right ) \log ^2(5)+\left (6 x^5-6 x^4-90 x^3-150 x^2\right ) \log (5)\right )+\left (3 x^6+9 x^5-42 x^4-270 x^3-525 x^2+e^{2 x} \left (3 x^4-15 x^3\right )+\left (3 x^4-45 x^3+225 x^2-375 x\right ) \log ^2(5)+e^x \left (6 x^5-6 x^4-90 x^3-150 x^2+\left (6 x^4-60 x^3+150 x^2\right ) \log (5)\right )+\left (6 x^5-36 x^4-60 x^3+300 x^2+750 x\right ) \log (5)-375 x\right ) \log (x)+\left (3 x^6+9 x^5-42 x^4-270 x^3-525 x^2-375 x\right ) \log (5)+125 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {10 (5-x) \left (\left (e^x+1\right ) x^3-\left (5 e^x+9\right ) x^2-5 \left (e^x+7\right ) x+25\right )}{x \left (x^2+x \left (e^x+4+\log (5)\right )+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x-(5-x) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 10 \int \left (\frac {\left (x^3+\log (x) x^2+3 \left (1+\frac {\log (5)}{3}\right ) x^2-5 \log (x) x+4 \left (1-\frac {5 \log (5)}{4}\right ) x-5 \log (x)+10 \left (1-\frac {\log (5)}{2}\right )\right ) (5-x)^2}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^3}+\frac {-x^3+10 x^2-20 x-25}{x \left (x^2+e^x x+\log (x) x+4 \left (1+\frac {\log (5)}{4}\right ) x-5 \log (x)+5 (1-\log (5))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 10 \int \frac {(5-x) \left (\left (1+e^x\right ) x^3-\left (9+5 e^x\right ) x^2-5 \left (7+e^x\right ) x+25\right )}{x \left (x^2+\left (4+e^x+\log (5)\right ) x+(x-5) \log (x)+5 (1-\log (5))\right )^3}dx\) |
Input:
Int[(1250 - 2000*x - 100*x^2 + 140*x^3 - 10*x^4 + E^x*(-250*x - 200*x^2 + 100*x^3 - 10*x^4))/(125*x + 300*x^2 + 315*x^3 + 184*x^4 + E^(3*x)*x^4 + 63 *x^5 + 12*x^6 + x^7 + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x^6 )*Log[5] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5)*Log[5]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log[5]^3 + E^(2*x)*(15*x^3 + 12*x^4 + 3*x^5 + (-1 5*x^3 + 3*x^4)*Log[5]) + E^x*(75*x^2 + 120*x^3 + 78*x^4 + 24*x^5 + 3*x^6 + (-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5)*Log[5] + (75*x^2 - 30*x^3 + 3*x^4)*Lo g[5]^2) + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x^6 + E^(2*x)*( -15*x^3 + 3*x^4) + (750*x + 300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)*Log[5] + (- 375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5]^2 + E^x*(-150*x^2 - 90*x^3 - 6*x^ 4 + 6*x^5 + (150*x^2 - 60*x^3 + 6*x^4)*Log[5]))*Log[x] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5 + E^x*(75*x^2 - 30*x^3 + 3*x^4) + (-375*x + 225* x^2 - 45*x^3 + 3*x^4)*Log[5])*Log[x]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)* Log[x]^3),x]
Output:
$Aborted
Time = 4.43 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {5 \left (-5+x \right )^{2}}{\left (x \ln \left (x \right )+{\mathrm e}^{x} x +x \ln \left (5\right )+x^{2}-5 \ln \left (x \right )-5 \ln \left (5\right )+4 x +5\right )^{2}}\) | \(38\) |
| parallelrisch | \(\frac {5 x^{2}-50 x +125}{25+40 x -10 x \ln \left (5\right )^{2}+2 x^{3} \ln \left (5\right )-30 x \ln \left (5\right )+{\mathrm e}^{2 x} x^{2}+50 \ln \left (5\right ) \ln \left (x \right )-10 x \ln \left (x \right )^{2}+8 \,{\mathrm e}^{x} x^{2}-2 x^{2} \ln \left (5\right )+2 \,{\mathrm e}^{x} x^{3}+x^{2} \ln \left (x \right )^{2}+2 x^{2} {\mathrm e}^{x} \ln \left (x \right )+x^{2} \ln \left (5\right )^{2}-10 x \,{\mathrm e}^{x} \ln \left (5\right )-10 x \,{\mathrm e}^{x} \ln \left (x \right )+25 \ln \left (5\right )^{2}+2 x^{3} \ln \left (x \right )-30 x \ln \left (x \right )+10 \,{\mathrm e}^{x} x -2 x^{2} \ln \left (x \right )-50 \ln \left (5\right )-50 \ln \left (x \right )+25 \ln \left (x \right )^{2}+26 x^{2}+8 x^{3}+x^{4}+2 x^{2} \ln \left (5\right ) {\mathrm e}^{x}-20 x \ln \left (5\right ) \ln \left (x \right )+2 \ln \left (x \right ) \ln \left (5\right ) x^{2}}\) | \(201\) |
Input:
int(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+ 1250)/((x^4-15*x^3+75*x^2-125*x)*ln(x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3* x^4-45*x^3+225*x^2-375*x)*ln(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x)*ln(x)^2 +((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*ln(5)+6*x^5-6*x^4-90*x^3 -150*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*ln(5)^2+(6*x^5-36*x^4-60*x^3 +300*x^2+750*x)*ln(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*ln(x)+x^4* exp(x)^3+((3*x^4-15*x^3)*ln(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x^4-30*x^ 3+75*x^2)*ln(5)^2+(6*x^5-6*x^4-90*x^3-150*x^2)*ln(5)+3*x^6+24*x^5+78*x^4+1 20*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*ln(5)^3+(3*x^5-18*x^4-30*x ^3+150*x^2+375*x)*ln(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*ln(5) +x^7+12*x^6+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x,method=_RETURNVERBOSE)
Output:
5*(-5+x)^2/(x*ln(x)+exp(x)*x+x*ln(5)+x^2-5*ln(x)-5*ln(5)+4*x+5)^2
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.00 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 8 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{2} - 10 \, x + 25\right )} \log \left (5\right )^{2} + {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{2} + 26 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2} + {\left (x^{2} - 5 \, x\right )} \log \left (5\right ) + 5 \, x\right )} e^{x} + 2 \, {\left (x^{3} - x^{2} - 15 \, x - 25\right )} \log \left (5\right ) + 2 \, {\left (x^{3} - x^{2} + {\left (x^{2} - 5 \, x\right )} e^{x} + {\left (x^{2} - 10 \, x + 25\right )} \log \left (5\right ) - 15 \, x - 25\right )} \log \left (x\right ) + 40 \, x + 25} \] Input:
integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2 000*x+1250)/((x^4-15*x^3+75*x^2-125*x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp (x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x) *log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6* x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36 *x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x )*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2 +((3*x^4-30*x^3+75*x^2)*log(5)^2+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6 +24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(5)^3+( 3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-52 5*x^2-375*x)*log(5)+x^7+12*x^6+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, al gorithm="fricas")
Output:
5*(x^2 - 10*x + 25)/(x^4 + 8*x^3 + x^2*e^(2*x) + (x^2 - 10*x + 25)*log(5)^ 2 + (x^2 - 10*x + 25)*log(x)^2 + 26*x^2 + 2*(x^3 + 4*x^2 + (x^2 - 5*x)*log (5) + 5*x)*e^x + 2*(x^3 - x^2 - 15*x - 25)*log(5) + 2*(x^3 - x^2 + (x^2 - 5*x)*e^x + (x^2 - 10*x + 25)*log(5) - 15*x - 25)*log(x) + 40*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (22) = 44\).
Time = 0.61 (sec) , antiderivative size = 219, normalized size of antiderivative = 7.55 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 x^{2} - 50 x + 125}{x^{4} + 2 x^{3} \log {\left (x \right )} + 2 x^{3} \log {\left (5 \right )} + 8 x^{3} + x^{2} e^{2 x} + x^{2} \log {\left (x \right )}^{2} - 2 x^{2} \log {\left (x \right )} + 2 x^{2} \log {\left (5 \right )} \log {\left (x \right )} - 2 x^{2} \log {\left (5 \right )} + x^{2} \log {\left (5 \right )}^{2} + 26 x^{2} - 10 x \log {\left (x \right )}^{2} - 20 x \log {\left (5 \right )} \log {\left (x \right )} - 30 x \log {\left (x \right )} - 30 x \log {\left (5 \right )} - 10 x \log {\left (5 \right )}^{2} + 40 x + \left (2 x^{3} + 2 x^{2} \log {\left (x \right )} + 2 x^{2} \log {\left (5 \right )} + 8 x^{2} - 10 x \log {\left (x \right )} - 10 x \log {\left (5 \right )} + 10 x\right ) e^{x} + 25 \log {\left (x \right )}^{2} - 50 \log {\left (x \right )} + 50 \log {\left (5 \right )} \log {\left (x \right )} - 50 \log {\left (5 \right )} + 25 + 25 \log {\left (5 \right )}^{2}} \] Input:
integrate(((-10*x**4+100*x**3-200*x**2-250*x)*exp(x)-10*x**4+140*x**3-100* x**2-2000*x+1250)/((x**4-15*x**3+75*x**2-125*x)*ln(x)**3+((3*x**4-30*x**3+ 75*x**2)*exp(x)+(3*x**4-45*x**3+225*x**2-375*x)*ln(5)+3*x**5-18*x**4-30*x* *3+150*x**2+375*x)*ln(x)**2+((3*x**4-15*x**3)*exp(x)**2+((6*x**4-60*x**3+1 50*x**2)*ln(5)+6*x**5-6*x**4-90*x**3-150*x**2)*exp(x)+(3*x**4-45*x**3+225* x**2-375*x)*ln(5)**2+(6*x**5-36*x**4-60*x**3+300*x**2+750*x)*ln(5)+3*x**6+ 9*x**5-42*x**4-270*x**3-525*x**2-375*x)*ln(x)+x**4*exp(x)**3+((3*x**4-15*x **3)*ln(5)+3*x**5+12*x**4+15*x**3)*exp(x)**2+((3*x**4-30*x**3+75*x**2)*ln( 5)**2+(6*x**5-6*x**4-90*x**3-150*x**2)*ln(5)+3*x**6+24*x**5+78*x**4+120*x* *3+75*x**2)*exp(x)+(x**4-15*x**3+75*x**2-125*x)*ln(5)**3+(3*x**5-18*x**4-3 0*x**3+150*x**2+375*x)*ln(5)**2+(3*x**6+9*x**5-42*x**4-270*x**3-525*x**2-3 75*x)*ln(5)+x**7+12*x**6+63*x**5+184*x**4+315*x**3+300*x**2+125*x),x)
Output:
(5*x**2 - 50*x + 125)/(x**4 + 2*x**3*log(x) + 2*x**3*log(5) + 8*x**3 + x** 2*exp(2*x) + x**2*log(x)**2 - 2*x**2*log(x) + 2*x**2*log(5)*log(x) - 2*x** 2*log(5) + x**2*log(5)**2 + 26*x**2 - 10*x*log(x)**2 - 20*x*log(5)*log(x) - 30*x*log(x) - 30*x*log(5) - 10*x*log(5)**2 + 40*x + (2*x**3 + 2*x**2*log (x) + 2*x**2*log(5) + 8*x**2 - 10*x*log(x) - 10*x*log(5) + 10*x)*exp(x) + 25*log(x)**2 - 50*log(x) + 50*log(5)*log(x) - 50*log(5) + 25 + 25*log(5)** 2)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (22) = 44\).
Time = 1.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 5.07 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 2 \, x^{3} {\left (\log \left (5\right ) + 4\right )} + {\left (\log \left (5\right )^{2} - 2 \, \log \left (5\right ) + 26\right )} x^{2} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{2} - 10 \, x + 25\right )} \log \left (x\right )^{2} - 10 \, {\left (\log \left (5\right )^{2} + 3 \, \log \left (5\right ) - 4\right )} x + 2 \, {\left (x^{3} + x^{2} {\left (\log \left (5\right ) + 4\right )} - 5 \, x {\left (\log \left (5\right ) - 1\right )} + {\left (x^{2} - 5 \, x\right )} \log \left (x\right )\right )} e^{x} + 25 \, \log \left (5\right )^{2} + 2 \, {\left (x^{3} + x^{2} {\left (\log \left (5\right ) - 1\right )} - 5 \, x {\left (2 \, \log \left (5\right ) + 3\right )} + 25 \, \log \left (5\right ) - 25\right )} \log \left (x\right ) - 50 \, \log \left (5\right ) + 25} \] Input:
integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2 000*x+1250)/((x^4-15*x^3+75*x^2-125*x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp (x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x) *log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6* x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36 *x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x )*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2 +((3*x^4-30*x^3+75*x^2)*log(5)^2+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6 +24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(5)^3+( 3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-52 5*x^2-375*x)*log(5)+x^7+12*x^6+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, al gorithm="maxima")
Output:
5*(x^2 - 10*x + 25)/(x^4 + 2*x^3*(log(5) + 4) + (log(5)^2 - 2*log(5) + 26) *x^2 + x^2*e^(2*x) + (x^2 - 10*x + 25)*log(x)^2 - 10*(log(5)^2 + 3*log(5) - 4)*x + 2*(x^3 + x^2*(log(5) + 4) - 5*x*(log(5) - 1) + (x^2 - 5*x)*log(x) )*e^x + 25*log(5)^2 + 2*(x^3 + x^2*(log(5) - 1) - 5*x*(2*log(5) + 3) + 25* log(5) - 25)*log(x) - 50*log(5) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (22) = 44\).
Time = 0.55 (sec) , antiderivative size = 199, normalized size of antiderivative = 6.86 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 2 \, x^{3} e^{x} + 2 \, x^{3} \log \left (5\right ) + 2 \, x^{2} e^{x} \log \left (5\right ) + x^{2} \log \left (5\right )^{2} + 2 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{x} \log \left (x\right ) + 2 \, x^{2} \log \left (5\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 8 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 2 \, x^{2} \log \left (5\right ) - 10 \, x e^{x} \log \left (5\right ) - 10 \, x \log \left (5\right )^{2} - 2 \, x^{2} \log \left (x\right ) - 10 \, x e^{x} \log \left (x\right ) - 20 \, x \log \left (5\right ) \log \left (x\right ) - 10 \, x \log \left (x\right )^{2} + 26 \, x^{2} + 10 \, x e^{x} - 30 \, x \log \left (5\right ) + 25 \, \log \left (5\right )^{2} - 30 \, x \log \left (x\right ) + 50 \, \log \left (5\right ) \log \left (x\right ) + 25 \, \log \left (x\right )^{2} + 40 \, x - 50 \, \log \left (5\right ) - 50 \, \log \left (x\right ) + 25} \] Input:
integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2 000*x+1250)/((x^4-15*x^3+75*x^2-125*x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp (x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x) *log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6* x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36 *x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x )*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2 +((3*x^4-30*x^3+75*x^2)*log(5)^2+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6 +24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(5)^3+( 3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-52 5*x^2-375*x)*log(5)+x^7+12*x^6+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, al gorithm="giac")
Output:
5*(x^2 - 10*x + 25)/(x^4 + 2*x^3*e^x + 2*x^3*log(5) + 2*x^2*e^x*log(5) + x ^2*log(5)^2 + 2*x^3*log(x) + 2*x^2*e^x*log(x) + 2*x^2*log(5)*log(x) + x^2* log(x)^2 + 8*x^3 + x^2*e^(2*x) + 8*x^2*e^x - 2*x^2*log(5) - 10*x*e^x*log(5 ) - 10*x*log(5)^2 - 2*x^2*log(x) - 10*x*e^x*log(x) - 20*x*log(5)*log(x) - 10*x*log(x)^2 + 26*x^2 + 10*x*e^x - 30*x*log(5) + 25*log(5)^2 - 30*x*log(x ) + 50*log(5)*log(x) + 25*log(x)^2 + 40*x - 50*log(5) - 50*log(x) + 25)
Timed out. \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\int -\frac {2000\,x+{\mathrm {e}}^x\,\left (10\,x^4-100\,x^3+200\,x^2+250\,x\right )+100\,x^2-140\,x^3+10\,x^4-1250}{125\,x-{\ln \left (5\right )}^3\,\left (-x^4+15\,x^3-75\,x^2+125\,x\right )-\ln \left (5\right )\,\left (-3\,x^6-9\,x^5+42\,x^4+270\,x^3+525\,x^2+375\,x\right )-\ln \left (x\right )\,\left (375\,x+{\ln \left (5\right )}^2\,\left (-3\,x^4+45\,x^3-225\,x^2+375\,x\right )+{\mathrm {e}}^x\,\left (150\,x^2-\ln \left (5\right )\,\left (6\,x^4-60\,x^3+150\,x^2\right )+90\,x^3+6\,x^4-6\,x^5\right )+{\mathrm {e}}^{2\,x}\,\left (15\,x^3-3\,x^4\right )+525\,x^2+270\,x^3+42\,x^4-9\,x^5-3\,x^6-\ln \left (5\right )\,\left (6\,x^5-36\,x^4-60\,x^3+300\,x^2+750\,x\right )\right )+{\mathrm {e}}^{2\,x}\,\left (15\,x^3-\ln \left (5\right )\,\left (15\,x^3-3\,x^4\right )+12\,x^4+3\,x^5\right )+{\ln \left (5\right )}^2\,\left (3\,x^5-18\,x^4-30\,x^3+150\,x^2+375\,x\right )+x^4\,{\mathrm {e}}^{3\,x}-{\ln \left (x\right )}^3\,\left (-x^4+15\,x^3-75\,x^2+125\,x\right )+{\ln \left (x\right )}^2\,\left (375\,x+{\mathrm {e}}^x\,\left (3\,x^4-30\,x^3+75\,x^2\right )-\ln \left (5\right )\,\left (-3\,x^4+45\,x^3-225\,x^2+375\,x\right )+150\,x^2-30\,x^3-18\,x^4+3\,x^5\right )+300\,x^2+315\,x^3+184\,x^4+63\,x^5+12\,x^6+x^7+{\mathrm {e}}^x\,\left ({\ln \left (5\right )}^2\,\left (3\,x^4-30\,x^3+75\,x^2\right )-\ln \left (5\right )\,\left (-6\,x^5+6\,x^4+90\,x^3+150\,x^2\right )+75\,x^2+120\,x^3+78\,x^4+24\,x^5+3\,x^6\right )} \,d x \] Input:
int(-(2000*x + exp(x)*(250*x + 200*x^2 - 100*x^3 + 10*x^4) + 100*x^2 - 140 *x^3 + 10*x^4 - 1250)/(125*x - log(5)^3*(125*x - 75*x^2 + 15*x^3 - x^4) - log(5)*(375*x + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6) - log(x)*(375* x + log(5)^2*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + exp(x)*(150*x^2 - log(5) *(150*x^2 - 60*x^3 + 6*x^4) + 90*x^3 + 6*x^4 - 6*x^5) + exp(2*x)*(15*x^3 - 3*x^4) + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6 - log(5)*(750*x + 300 *x^2 - 60*x^3 - 36*x^4 + 6*x^5)) + exp(2*x)*(15*x^3 - log(5)*(15*x^3 - 3*x ^4) + 12*x^4 + 3*x^5) + log(5)^2*(375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^ 5) + x^4*exp(3*x) - log(x)^3*(125*x - 75*x^2 + 15*x^3 - x^4) + log(x)^2*(3 75*x + exp(x)*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5) + 300*x^2 + 315*x^3 + 184*x ^4 + 63*x^5 + 12*x^6 + x^7 + exp(x)*(log(5)^2*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(150*x^2 + 90*x^3 + 6*x^4 - 6*x^5) + 75*x^2 + 120*x^3 + 78*x^4 + 24 *x^5 + 3*x^6)),x)
Output:
int(-(2000*x + exp(x)*(250*x + 200*x^2 - 100*x^3 + 10*x^4) + 100*x^2 - 140 *x^3 + 10*x^4 - 1250)/(125*x - log(5)^3*(125*x - 75*x^2 + 15*x^3 - x^4) - log(5)*(375*x + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6) - log(x)*(375* x + log(5)^2*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + exp(x)*(150*x^2 - log(5) *(150*x^2 - 60*x^3 + 6*x^4) + 90*x^3 + 6*x^4 - 6*x^5) + exp(2*x)*(15*x^3 - 3*x^4) + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6 - log(5)*(750*x + 300 *x^2 - 60*x^3 - 36*x^4 + 6*x^5)) + exp(2*x)*(15*x^3 - log(5)*(15*x^3 - 3*x ^4) + 12*x^4 + 3*x^5) + log(5)^2*(375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^ 5) + x^4*exp(3*x) - log(x)^3*(125*x - 75*x^2 + 15*x^3 - x^4) + log(x)^2*(3 75*x + exp(x)*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5) + 300*x^2 + 315*x^3 + 184*x ^4 + 63*x^5 + 12*x^6 + x^7 + exp(x)*(log(5)^2*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(150*x^2 + 90*x^3 + 6*x^4 - 6*x^5) + 75*x^2 + 120*x^3 + 78*x^4 + 24 *x^5 + 3*x^6)), x)
Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 7.14 \[ \int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {5 x^{2}-50 x +125}{25+40 x +2 e^{x} \mathrm {log}\left (x \right ) x^{2}+2 e^{x} \mathrm {log}\left (5\right ) x^{2}+2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x^{2}-20 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x +e^{2 x} x^{2}+2 e^{x} x^{3}-2 \,\mathrm {log}\left (5\right ) x^{2}+\mathrm {log}\left (x \right )^{2} x^{2}-30 \,\mathrm {log}\left (5\right ) x +2 \,\mathrm {log}\left (5\right ) x^{3}+50 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right )+26 x^{2}+8 x^{3}-10 \mathrm {log}\left (5\right )^{2} x +8 e^{x} x^{2}-10 e^{x} \mathrm {log}\left (5\right ) x -50 \,\mathrm {log}\left (x \right )-2 \,\mathrm {log}\left (x \right ) x^{2}+25 \mathrm {log}\left (5\right )^{2}+\mathrm {log}\left (5\right )^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) x^{3}+x^{4}+25 \mathrm {log}\left (x \right )^{2}-10 e^{x} \mathrm {log}\left (x \right ) x +10 e^{x} x -30 \,\mathrm {log}\left (x \right ) x -10 \mathrm {log}\left (x \right )^{2} x -50 \,\mathrm {log}\left (5\right )} \] Input:
int(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+ 1250)/((x^4-15*x^3+75*x^2-125*x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3 *x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(x )^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6*x^4-90 *x^3-150*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36*x^4-6 0*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*log( x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x ^4-30*x^3+75*x^2)*log(5)^2+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6+24*x^ 5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(5)^3+(3*x^5- 18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2- 375*x)*log(5)+x^7+12*x^6+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x)
Output:
(5*(x**2 - 10*x + 25))/(e**(2*x)*x**2 + 2*e**x*log(x)*x**2 - 10*e**x*log(x )*x + 2*e**x*log(5)*x**2 - 10*e**x*log(5)*x + 2*e**x*x**3 + 8*e**x*x**2 + 10*e**x*x + log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + 2*log(x)*log( 5)*x**2 - 20*log(x)*log(5)*x + 50*log(x)*log(5) + 2*log(x)*x**3 - 2*log(x) *x**2 - 30*log(x)*x - 50*log(x) + log(5)**2*x**2 - 10*log(5)**2*x + 25*log (5)**2 + 2*log(5)*x**3 - 2*log(5)*x**2 - 30*log(5)*x - 50*log(5) + x**4 + 8*x**3 + 26*x**2 + 40*x + 25)