Integrand size = 139, antiderivative size = 33 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \left (-e^2+x\right )}{2+2 x-\frac {5}{5+x}-\log \left (-5+\frac {x}{4}\right )} \] Output:
4*(x-exp(2))/(2*x-5/(5+x)-ln(1/4*x-5)+2)
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 (5+x) \left (-e^2+x\right )}{5+12 x+2 x^2-(5+x) \log \left (-5+\frac {x}{4}\right )} \] Input:
Integrate[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 + 8*x^3) + (2000 + 700*x + 40*x^2 - 4*x^3)*Log[(-20 + x)/4])/(-500 - 2375* x - 3160*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*x^ 3 - 4*x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/ 4]^2),x]
Output:
(4*(5 + x)*(-E^2 + x))/(5 + 12*x + 2*x^2 - (5 + x)*Log[-5 + x/4])
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.
| \(\displaystyle \int \frac {12 x^3-80 x^2+e^2 \left (8 x^3-84 x^2-1420 x-4500\right )+\left (-4 x^3+40 x^2+700 x+2000\right ) \log \left (\frac {x-20}{4}\right )-600 x-2000}{4 x^5-32 x^4-796 x^3-3160 x^2+\left (x^3-10 x^2-175 x-500\right ) \log ^2\left (\frac {x-20}{4}\right )+\left (-4 x^4+36 x^3+750 x^2+2550 x+1000\right ) \log \left (\frac {x-20}{4}\right )-2375 x-500} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-12 x^3+80 x^2-e^2 \left (8 x^3-84 x^2-1420 x-4500\right )-\left (-4 x^3+40 x^2+700 x+2000\right ) \log \left (\frac {x-20}{4}\right )+600 x+2000}{(20-x) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {80 x^2}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {600 x}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {4 (x+5)^2 \log \left (\frac {x}{4}-5\right )}{\left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {2000}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}+\frac {12 x^3}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}+\frac {4 e^2 \left (2 x^3-21 x^2-355 x-1125\right )}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 100 e^2 \int \frac {1}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+2500 \int \frac {1}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-2500 e^2 \int \frac {1}{(x-20) \left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+50000 \int \frac {1}{(x-20) \left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+76 e^2 \int \frac {x}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-100 \int \frac {x}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+8 e^2 \int \frac {x^2}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-76 \int \frac {x^2}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+20 \int \frac {1}{2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5}dx+4 \int \frac {x}{2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5}dx-8 \int \frac {x^3}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx\) |
Input:
Int[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 + 8*x^ 3) + (2000 + 700*x + 40*x^2 - 4*x^3)*Log[(-20 + x)/4])/(-500 - 2375*x - 31 60*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*x^3 - 4* x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/4]^2), x]
Output:
$Aborted
Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(-\frac {4 \left ({\mathrm e}^{2}-x \right ) \left (5+x \right )}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\) | \(41\) |
| parallelrisch | \(-\frac {4 \,{\mathrm e}^{2} x -4 x^{2}+20 \,{\mathrm e}^{2}-20 x}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\) | \(50\) |
| norman | \(\frac {\left (-4-4 \,{\mathrm e}^{2}\right ) x +10 \ln \left (\frac {x}{4}-5\right )-10+2 \ln \left (\frac {x}{4}-5\right ) x -20 \,{\mathrm e}^{2}}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\) | \(62\) |
| derivativedivides | \(-\frac {4 \left (16 \left (\frac {x}{4}-5\right )^{2}+\left (180-4 \,{\mathrm e}^{2}\right ) \left (\frac {x}{4}-5\right )+500-25 \,{\mathrm e}^{2}\right )}{4 \ln \left (\frac {x}{4}-5\right ) \left (\frac {x}{4}-5\right )-32 \left (\frac {x}{4}-5\right )^{2}+25 \ln \left (\frac {x}{4}-5\right )-92 x +795}\) | \(67\) |
| default | \(-\frac {4 \left (16 \left (\frac {x}{4}-5\right )^{2}+\left (180-4 \,{\mathrm e}^{2}\right ) \left (\frac {x}{4}-5\right )+500-25 \,{\mathrm e}^{2}\right )}{4 \ln \left (\frac {x}{4}-5\right ) \left (\frac {x}{4}-5\right )-32 \left (\frac {x}{4}-5\right )^{2}+25 \ln \left (\frac {x}{4}-5\right )-92 x +795}\) | \(67\) |
Input:
int(((-4*x^3+40*x^2+700*x+2000)*ln(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp (2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*ln(1/4*x-5)^2+(-4*x^ 4+36*x^3+750*x^2+2550*x+1000)*ln(1/4*x-5)+4*x^5-32*x^4-796*x^3-3160*x^2-23 75*x-500),x,method=_RETURNVERBOSE)
Output:
-4*(exp(2)-x)*(5+x)/(2*x^2-ln(1/4*x-5)*x+12*x-5*ln(1/4*x-5)+5)
Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - {\left (x + 5\right )} e^{2} + 5 \, x\right )}}{2 \, x^{2} - {\left (x + 5\right )} \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x + 5} \] Input:
integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 60*x^2-2375*x-500),x, algorithm="fricas")
Output:
4*(x^2 - (x + 5)*e^2 + 5*x)/(2*x^2 - (x + 5)*log(1/4*x - 5) + 12*x + 5)
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {- 4 x^{2} - 20 x + 4 x e^{2} + 20 e^{2}}{- 2 x^{2} - 12 x + \left (x + 5\right ) \log {\left (\frac {x}{4} - 5 \right )} - 5} \] Input:
integrate(((-4*x**3+40*x**2+700*x+2000)*ln(1/4*x-5)+(8*x**3-84*x**2-1420*x -4500)*exp(2)+12*x**3-80*x**2-600*x-2000)/((x**3-10*x**2-175*x-500)*ln(1/4 *x-5)**2+(-4*x**4+36*x**3+750*x**2+2550*x+1000)*ln(1/4*x-5)+4*x**5-32*x**4 -796*x**3-3160*x**2-2375*x-500),x)
Output:
(-4*x**2 - 20*x + 4*x*exp(2) + 20*exp(2))/(-2*x**2 - 12*x + (x + 5)*log(x/ 4 - 5) - 5)
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - x {\left (e^{2} - 5\right )} - 5 \, e^{2}\right )}}{2 \, x^{2} + 2 \, x {\left (\log \left (2\right ) + 6\right )} - {\left (x + 5\right )} \log \left (x - 20\right ) + 10 \, \log \left (2\right ) + 5} \] Input:
integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 60*x^2-2375*x-500),x, algorithm="maxima")
Output:
4*(x^2 - x*(e^2 - 5) - 5*e^2)/(2*x^2 + 2*x*(log(2) + 6) - (x + 5)*log(x - 20) + 10*log(2) + 5)
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - x e^{2} + 5 \, x - 5 \, e^{2}\right )}}{2 \, x^{2} - x \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x - 5 \, \log \left (\frac {1}{4} \, x - 5\right ) + 5} \] Input:
integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 60*x^2-2375*x-500),x, algorithm="giac")
Output:
4*(x^2 - x*e^2 + 5*x - 5*e^2)/(2*x^2 - x*log(1/4*x - 5) + 12*x - 5*log(1/4 *x - 5) + 5)
Time = 7.53 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4\,\left (112500\,x-112500\,{\mathrm {e}}^2-52375\,x\,{\mathrm {e}}^2-6300\,x^2\,{\mathrm {e}}^2+240\,x^3\,{\mathrm {e}}^2+51\,x^4\,{\mathrm {e}}^2-2\,x^5\,{\mathrm {e}}^2+52375\,x^2+6300\,x^3-240\,x^4-51\,x^5+2\,x^6\right )}{\left (12\,x+2\,x^2-\ln \left (\frac {x}{4}-5\right )\,\left (x+5\right )+5\right )\,\left (2\,x^4-61\,x^3+65\,x^2+5975\,x+22500\right )} \] Input:
int((600*x + exp(2)*(1420*x + 84*x^2 - 8*x^3 + 4500) - log(x/4 - 5)*(700*x + 40*x^2 - 4*x^3 + 2000) + 80*x^2 - 12*x^3 + 2000)/(2375*x + log(x/4 - 5) ^2*(175*x + 10*x^2 - x^3 + 500) + 3160*x^2 + 796*x^3 + 32*x^4 - 4*x^5 - lo g(x/4 - 5)*(2550*x + 750*x^2 + 36*x^3 - 4*x^4 + 1000) + 500),x)
Output:
(4*(112500*x - 112500*exp(2) - 52375*x*exp(2) - 6300*x^2*exp(2) + 240*x^3* exp(2) + 51*x^4*exp(2) - 2*x^5*exp(2) + 52375*x^2 + 6300*x^3 - 240*x^4 - 5 1*x^5 + 2*x^6))/((12*x + 2*x^2 - log(x/4 - 5)*(x + 5) + 5)*(5975*x + 65*x^ 2 - 61*x^3 + 2*x^4 + 22500))
Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.06 \[ \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {-4 \,\mathrm {log}\left (x -20\right ) \mathrm {log}\left (\frac {x}{4}-5\right ) e^{2} x -20 \,\mathrm {log}\left (x -20\right ) \mathrm {log}\left (\frac {x}{4}-5\right ) e^{2}+8 \,\mathrm {log}\left (x -20\right ) e^{2} x^{2}+48 \,\mathrm {log}\left (x -20\right ) e^{2} x +20 \,\mathrm {log}\left (x -20\right ) e^{2}+4 \mathrm {log}\left (\frac {x}{4}-5\right )^{2} e^{2} x +20 \mathrm {log}\left (\frac {x}{4}-5\right )^{2} e^{2}-8 \,\mathrm {log}\left (\frac {x}{4}-5\right ) e^{2} x^{2}-44 \,\mathrm {log}\left (\frac {x}{4}-5\right ) e^{2} x -8 e^{2} x^{2}-44 e^{2} x -4 x^{2}-20 x}{\mathrm {log}\left (\frac {x}{4}-5\right ) x +5 \,\mathrm {log}\left (\frac {x}{4}-5\right )-2 x^{2}-12 x -5} \] Input:
int(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*ex p(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^2+(-4* x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-3160*x^2 -2375*x-500),x)
Output:
(4*( - log(x - 20)*log((x - 20)/4)*e**2*x - 5*log(x - 20)*log((x - 20)/4)* e**2 + 2*log(x - 20)*e**2*x**2 + 12*log(x - 20)*e**2*x + 5*log(x - 20)*e** 2 + log((x - 20)/4)**2*e**2*x + 5*log((x - 20)/4)**2*e**2 - 2*log((x - 20) /4)*e**2*x**2 - 11*log((x - 20)/4)*e**2*x - 2*e**2*x**2 - 11*e**2*x - x**2 - 5*x))/(log((x - 20)/4)*x + 5*log((x - 20)/4) - 2*x**2 - 12*x - 5)