Integrand size = 134, antiderivative size = 30 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {2 e^6 (3+x)^2}{e^x+\frac {1}{15} (x-\log (5-x))} \] Output:
2*exp(6)/(1/15*x-1/15*ln(5-x)+exp(x))*(3+x)^2
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=-\frac {30 e^6 (3+x)^2}{-15 e^x-x+\log (5-x)} \] Input:
Integrate[(E^(6 + x)*(6750 + 7650*x + 450*x^2 - 450*x^3) + E^6*(1620 - 90* x - 120*x^2 + 30*x^3) + E^6*(900 + 120*x - 60*x^2)*Log[5 - x])/(-5*x^2 + x ^3 + E^(2*x)*(-1125 + 225*x) + E^x*(-150*x + 30*x^2) + (E^x*(150 - 30*x) + 10*x - 2*x^2)*Log[5 - x] + (-5 + x)*Log[5 - x]^2),x]
Output:
(-30*E^6*(3 + x)^2)/(-15*E^x - x + Log[5 - 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.
| \(\displaystyle \int \frac {e^6 \left (-60 x^2+120 x+900\right ) \log (5-x)+e^{x+6} \left (-450 x^3+450 x^2+7650 x+6750\right )+e^6 \left (30 x^3-120 x^2-90 x+1620\right )}{x^3-5 x^2+e^x \left (30 x^2-150 x\right )+\left (-2 x^2+10 x+e^x (150-30 x)\right ) \log (5-x)+e^{2 x} (225 x-1125)+(x-5) \log ^2(5-x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {30 e^6 (x+3) \left (-x^2+15 e^x \left (x^2-4 x-5\right )+7 x+2 (x-5) \log (5-x)-18\right )}{(5-x) \left (x+15 e^x-\log (5-x)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 30 e^6 \int -\frac {(x+3) \left (x^2-7 x+15 e^x \left (-x^2+4 x+5\right )+2 (5-x) \log (5-x)+18\right )}{(5-x) \left (x+15 e^x-\log (5-x)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -30 e^6 \int \frac {(x+3) \left (x^2-7 x+15 e^x \left (-x^2+4 x+5\right )+2 (5-x) \log (5-x)+18\right )}{(5-x) \left (x+15 e^x-\log (5-x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -30 e^6 \int \left (\frac {x^2+4 x+3}{x+15 e^x-\log (5-x)}-\frac {(x+3)^2 \left (x^2-\log (5-x) x-6 x+5 \log (5-x)+6\right )}{(x-5) \left (x+15 e^x-\log (5-x)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -30 e^6 \left (-\int \frac {x^3}{\left (x+15 e^x-\log (5-x)\right )^2}dx-5 \int \frac {x^2}{\left (x+15 e^x-\log (5-x)\right )^2}dx+\int \frac {x^2}{x+15 e^x-\log (5-x)}dx+\int \frac {x^2 \log (5-x)}{\left (x+15 e^x-\log (5-x)\right )^2}dx-2 \int \frac {1}{\left (x+15 e^x-\log (5-x)\right )^2}dx-64 \int \frac {1}{(x-5) \left (x+15 e^x-\log (5-x)\right )^2}dx-4 \int \frac {x}{\left (x+15 e^x-\log (5-x)\right )^2}dx+3 \int \frac {1}{x+15 e^x-\log (5-x)}dx+4 \int \frac {x}{x+15 e^x-\log (5-x)}dx+9 \int \frac {\log (5-x)}{\left (x+15 e^x-\log (5-x)\right )^2}dx+6 \int \frac {x \log (5-x)}{\left (x+15 e^x-\log (5-x)\right )^2}dx\right )\) |
Input:
Int[(E^(6 + x)*(6750 + 7650*x + 450*x^2 - 450*x^3) + E^6*(1620 - 90*x - 12 0*x^2 + 30*x^3) + E^6*(900 + 120*x - 60*x^2)*Log[5 - x])/(-5*x^2 + x^3 + E ^(2*x)*(-1125 + 225*x) + E^x*(-150*x + 30*x^2) + (E^x*(150 - 30*x) + 10*x - 2*x^2)*Log[5 - x] + (-5 + x)*Log[5 - x]^2),x]
Output:
$Aborted
Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {30 \left (x^{2}+6 x +9\right ) {\mathrm e}^{6}}{-\ln \left (5-x \right )+15 \,{\mathrm e}^{x}+x}\) | \(29\) |
| parallelrisch | \(\frac {30 x^{2} {\mathrm e}^{6}+180 x \,{\mathrm e}^{6}+270 \,{\mathrm e}^{6}}{-\ln \left (5-x \right )+15 \,{\mathrm e}^{x}+x}\) | \(35\) |
Input:
int(((-60*x^2+120*x+900)*exp(6)*ln(5-x)+(-450*x^3+450*x^2+7650*x+6750)*exp (6)*exp(x)+(30*x^3-120*x^2-90*x+1620)*exp(6))/((-5+x)*ln(5-x)^2+((-30*x+15 0)*exp(x)-2*x^2+10*x)*ln(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x)*exp(x)+ x^3-5*x^2),x,method=_RETURNVERBOSE)
Output:
30*(x^2+6*x+9)*exp(6)/(-ln(5-x)+15*exp(x)+x)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left (x^{2} + 6 \, x + 9\right )} e^{12}}{x e^{6} - e^{6} \log \left (-x + 5\right ) + 15 \, e^{\left (x + 6\right )}} \] Input:
integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+67 50)*exp(6)*exp(x)+(30*x^3-120*x^2-90*x+1620)*exp(6))/((-5+x)*log(5-x)^2+(( -30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x )*exp(x)+x^3-5*x^2),x, algorithm="fricas")
Output:
30*(x^2 + 6*x + 9)*e^12/(x*e^6 - e^6*log(-x + 5) + 15*e^(x + 6))
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {2 x^{2} e^{6} + 12 x e^{6} + 18 e^{6}}{\frac {x}{15} + e^{x} - \frac {\log {\left (5 - x \right )}}{15}} \] Input:
integrate(((-60*x**2+120*x+900)*exp(6)*ln(5-x)+(-450*x**3+450*x**2+7650*x+ 6750)*exp(6)*exp(x)+(30*x**3-120*x**2-90*x+1620)*exp(6))/((-5+x)*ln(5-x)** 2+((-30*x+150)*exp(x)-2*x**2+10*x)*ln(5-x)+(225*x-1125)*exp(x)**2+(30*x**2 -150*x)*exp(x)+x**3-5*x**2),x)
Output:
(2*x**2*exp(6) + 12*x*exp(6) + 18*exp(6))/(x/15 + exp(x) - log(5 - x)/15)
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left (x^{2} e^{6} + 6 \, x e^{6} + 9 \, e^{6}\right )}}{x + 15 \, e^{x} - \log \left (-x + 5\right )} \] Input:
integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+67 50)*exp(6)*exp(x)+(30*x^3-120*x^2-90*x+1620)*exp(6))/((-5+x)*log(5-x)^2+(( -30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x )*exp(x)+x^3-5*x^2),x, algorithm="maxima")
Output:
30*(x^2*e^6 + 6*x*e^6 + 9*e^6)/(x + 15*e^x - log(-x + 5))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.43 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 \, {\left ({\left (x - 5\right )}^{2} e^{\left (-x + 11\right )} + 16 \, {\left (x - 5\right )} e^{\left (-x + 11\right )} + 64 \, e^{\left (-x + 11\right )}\right )}}{{\left (x - 5\right )} e^{\left (-x + 5\right )} - e^{\left (-x + 5\right )} \log \left (-x + 5\right ) + 15 \, e^{5} + 5 \, e^{\left (-x + 5\right )}} \] Input:
integrate(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+67 50)*exp(6)*exp(x)+(30*x^3-120*x^2-90*x+1620)*exp(6))/((-5+x)*log(5-x)^2+(( -30*x+150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x )*exp(x)+x^3-5*x^2),x, algorithm="giac")
Output:
30*((x - 5)^2*e^(-x + 11) + 16*(x - 5)*e^(-x + 11) + 64*e^(-x + 11))/((x - 5)*e^(-x + 5) - e^(-x + 5)*log(-x + 5) + 15*e^5 + 5*e^(-x + 5))
Timed out. \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\int \frac {{\mathrm {e}}^6\,{\mathrm {e}}^x\,\left (-450\,x^3+450\,x^2+7650\,x+6750\right )-{\mathrm {e}}^6\,\left (-30\,x^3+120\,x^2+90\,x-1620\right )+{\mathrm {e}}^6\,\ln \left (5-x\right )\,\left (-60\,x^2+120\,x+900\right )}{{\ln \left (5-x\right )}^2\,\left (x-5\right )-{\mathrm {e}}^x\,\left (150\,x-30\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (225\,x-1125\right )-5\,x^2+x^3-\ln \left (5-x\right )\,\left ({\mathrm {e}}^x\,\left (30\,x-150\right )-10\,x+2\,x^2\right )} \,d x \] Input:
int((exp(6)*exp(x)*(7650*x + 450*x^2 - 450*x^3 + 6750) - exp(6)*(90*x + 12 0*x^2 - 30*x^3 - 1620) + exp(6)*log(5 - x)*(120*x - 60*x^2 + 900))/(log(5 - x)^2*(x - 5) - exp(x)*(150*x - 30*x^2) + exp(2*x)*(225*x - 1125) - 5*x^2 + x^3 - log(5 - x)*(exp(x)*(30*x - 150) - 10*x + 2*x^2)),x)
Output:
int((exp(6)*exp(x)*(7650*x + 450*x^2 - 450*x^3 + 6750) - exp(6)*(90*x + 12 0*x^2 - 30*x^3 - 1620) + exp(6)*log(5 - x)*(120*x - 60*x^2 + 900))/(log(5 - x)^2*(x - 5) - exp(x)*(150*x - 30*x^2) + exp(2*x)*(225*x - 1125) - 5*x^2 + x^3 - log(5 - x)*(exp(x)*(30*x - 150) - 10*x + 2*x^2)), x)
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{6+x} \left (6750+7650 x+450 x^2-450 x^3\right )+e^6 \left (1620-90 x-120 x^2+30 x^3\right )+e^6 \left (900+120 x-60 x^2\right ) \log (5-x)}{-5 x^2+x^3+e^{2 x} (-1125+225 x)+e^x \left (-150 x+30 x^2\right )+\left (e^x (150-30 x)+10 x-2 x^2\right ) \log (5-x)+(-5+x) \log ^2(5-x)} \, dx=\frac {30 e^{6} \left (x^{2}+6 x +9\right )}{15 e^{x}-\mathrm {log}\left (-x +5\right )+x} \] Input:
int(((-60*x^2+120*x+900)*exp(6)*log(5-x)+(-450*x^3+450*x^2+7650*x+6750)*ex p(6)*exp(x)+(30*x^3-120*x^2-90*x+1620)*exp(6))/((-5+x)*log(5-x)^2+((-30*x+ 150)*exp(x)-2*x^2+10*x)*log(5-x)+(225*x-1125)*exp(x)^2+(30*x^2-150*x)*exp( x)+x^3-5*x^2),x)
Output:
(30*e**6*(x**2 + 6*x + 9))/(15*e**x - log( - x + 5) + x)