Integrand size = 111, antiderivative size = 32 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=\frac {3 \left (-4+e^{e^x} x\right )}{-e^{x/5}+\frac {4 \log (25)}{3 x^2}} \] Output:
3*(x*exp(exp(x))-4)/(8/3*ln(5)/x^2-exp(1/5*x))
Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=-\frac {9 x^2 \left (-4+e^{e^x} x\right )}{3 e^{x/5} x^2-4 \log (25)} \] Input:
Integrate[(-108*E^(x/5)*x^4 - 1440*x*Log[25] + E^E^x*(E^(x/5)*(-135*x^4 + 27*x^5) + 540*x^2*Log[25] + E^x*(-135*E^(x/5)*x^5 + 180*x^3*Log[25])))/(45 *E^((2*x)/5)*x^4 - 120*E^(x/5)*x^2*Log[25] + 80*Log[25]^2),x]
Output:
(-9*x^2*(-4 + E^E^x*x))/(3*E^(x/5)*x^2 - 4*Log[25])
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.
| \(\displaystyle \int \frac {-108 e^{x/5} x^4+e^{e^x} \left (540 x^2 \log (25)+e^{x/5} \left (27 x^5-135 x^4\right )+e^x \left (180 x^3 \log (25)-135 e^{x/5} x^5\right )\right )-1440 x \log (25)}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-108 e^{x/5} x^4+e^{e^x} \left (540 x^2 \log (25)+e^{x/5} \left (27 x^5-135 x^4\right )+e^x \left (180 x^3 \log (25)-135 e^{x/5} x^5\right )\right )-1440 x \log (25)}{5 \left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {9 \left (12 e^{x/5} x^4+160 \log (25) x+e^{e^x} \left (-60 \log (25) x^2+3 e^{x/5} \left (5 x^4-x^5\right )+5 e^x \left (3 e^{x/5} x^5-4 x^3 \log (25)\right )\right )\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {9}{5} \int \frac {12 e^{x/5} x^4+160 \log (25) x+e^{e^x} \left (-60 \log (25) x^2+3 e^{x/5} \left (5 x^4-x^5\right )+5 e^x \left (3 e^{x/5} x^5-4 x^3 \log (25)\right )\right )}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {9}{5} \int \left (\frac {5}{3} e^{\frac {4 x}{5}+e^x} x-\frac {4 (x+10) \left (e^{e^x} x-4\right ) \log (25) x}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}+\frac {20 e^{\frac {3 x}{5}+e^x} \log (25)}{9 x}+\frac {80 e^{\frac {2 x}{5}+e^x} \log ^2(25)}{27 x^3}+\frac {320 e^{\frac {1}{5} \left (x+5 e^x\right )} \log ^3(25)}{81 x^5}-\frac {243 e^{e^x} x^{10}-1215 e^{e^x} x^9-972 x^9-5120 e^{e^x} \log ^5(25)}{243 \left (3 e^{x/5} x^2-4 \log (25)\right ) x^7}+\frac {1280 e^{e^x} \log ^4(25)}{243 x^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9}{5} \left (\frac {1280}{243} \log ^4(25) \int \frac {e^{e^x}}{x^7}dx+\frac {320}{81} \log ^3(25) \int \frac {e^{\frac {1}{5} \left (x+5 e^x\right )}}{x^5}dx+\frac {80}{27} \log ^2(25) \int \frac {e^{\frac {2 x}{5}+e^x}}{x^3}dx+160 \log (25) \int \frac {x}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx+16 \log (25) \int \frac {x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx-40 \log (25) \int \frac {e^{e^x} x^2}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx+4 \int \frac {x^2}{3 e^{x/5} x^2-4 \log (25)}dx+5 \int \frac {e^{e^x} x^2}{3 e^{x/5} x^2-4 \log (25)}dx+\frac {5120}{243} \log ^5(25) \int \frac {e^{e^x}}{x^7 \left (3 e^{x/5} x^2-4 \log (25)\right )}dx-4 \log (25) \int \frac {e^{e^x} x^3}{\left (3 e^{x/5} x^2-4 \log (25)\right )^2}dx-\int \frac {e^{e^x} x^3}{3 e^{x/5} x^2-4 \log (25)}dx+\frac {5}{3} \int e^{\frac {4 x}{5}+e^x} xdx+\frac {20}{9} \log (25) \int \frac {e^{\frac {3 x}{5}+e^x}}{x}dx\right )\) |
Input:
Int[(-108*E^(x/5)*x^4 - 1440*x*Log[25] + E^E^x*(E^(x/5)*(-135*x^4 + 27*x^5 ) + 540*x^2*Log[25] + E^x*(-135*E^(x/5)*x^5 + 180*x^3*Log[25])))/(45*E^((2 *x)/5)*x^4 - 120*E^(x/5)*x^2*Log[25] + 80*Log[25]^2),x]
Output:
$Aborted
Time = 10.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
| method | result | size |
| parallelrisch | \(-\frac {-135 x^{3} {\mathrm e}^{{\mathrm e}^{x}}+540 x^{2}}{15 \left (-3 x^{2} {\mathrm e}^{\frac {x}{5}}+8 \ln \left (5\right )\right )}\) | \(33\) |
Input:
int((((-135*x^5*exp(1/5*x)+360*x^3*ln(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5* x)+1080*x^2*ln(5))*exp(exp(x))-108*x^4*exp(1/5*x)-2880*x*ln(5))/(45*x^4*ex p(1/5*x)^2-240*x^2*ln(5)*exp(1/5*x)+320*ln(5)^2),x,method=_RETURNVERBOSE)
Output:
-1/15*(-135*x^3*exp(exp(x))+540*x^2)/(-3*x^2*exp(1/5*x)+8*ln(5))
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=-\frac {9 \, {\left (x^{3} e^{\left (e^{x}\right )} - 4 \, x^{2}\right )}}{3 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} - 8 \, \log \left (5\right )} \] Input:
integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*e xp(1/5*x)+1080*x^2*log(5))*exp(exp(x))-108*x^4*exp(1/5*x)-2880*x*log(5))/( 45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm=" fricas")
Output:
-9*(x^3*e^(e^x) - 4*x^2)/(3*x^2*e^(1/5*x) - 8*log(5))
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=- \frac {9 x^{3} e^{e^{x}}}{3 x^{2} e^{\frac {x}{5}} - 8 \log {\left (5 \right )}} + \frac {36 x^{2}}{3 x^{2} e^{\frac {x}{5}} - 8 \log {\left (5 \right )}} \] Input:
integrate((((-135*x**5*exp(1/5*x)+360*x**3*ln(5))*exp(x)+(27*x**5-135*x**4 )*exp(1/5*x)+1080*x**2*ln(5))*exp(exp(x))-108*x**4*exp(1/5*x)-2880*x*ln(5) )/(45*x**4*exp(1/5*x)**2-240*x**2*ln(5)*exp(1/5*x)+320*ln(5)**2),x)
Output:
-9*x**3*exp(exp(x))/(3*x**2*exp(x/5) - 8*log(5)) + 36*x**2/(3*x**2*exp(x/5 ) - 8*log(5))
\[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=\int { -\frac {9 \, {\left (12 \, x^{4} e^{\left (\frac {1}{5} \, x\right )} - {\left (120 \, x^{2} \log \left (5\right ) + 3 \, {\left (x^{5} - 5 \, x^{4}\right )} e^{\left (\frac {1}{5} \, x\right )} - 5 \, {\left (3 \, x^{5} e^{\left (\frac {1}{5} \, x\right )} - 8 \, x^{3} \log \left (5\right )\right )} e^{x}\right )} e^{\left (e^{x}\right )} + 320 \, x \log \left (5\right )\right )}}{5 \, {\left (9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} - 48 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + 64 \, \log \left (5\right )^{2}\right )}} \,d x } \] Input:
integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*e xp(1/5*x)+1080*x^2*log(5))*exp(exp(x))-108*x^4*exp(1/5*x)-2880*x*log(5))/( 45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm=" maxima")
Output:
-4096/27*integrate(e^(e^x)/x^7, x)*log(5)^4 - 512/9*integrate(e^(1/5*x + e ^x)/x^5, x)*log(5)^3 - 64/3*integrate(e^(2/5*x + e^x)/x^3, x)*log(5)^2 - 8 *integrate(e^(3/5*x + e^x)/x, x)*log(5) - 9*(x^3*e^(e^x) - 4*x^2)/(3*x^2*e ^(1/5*x) - 8*log(5)) - 3*integrate(x*e^(4/5*x + e^x), x) + 9/5*integrate(5 /243*(243*x^10*e^x - 32768*log(5)^5)*e^(e^x)/(3*x^9*e^(1/5*x) - 8*x^7*log( 5)), x)
\[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=\int { -\frac {9 \, {\left (12 \, x^{4} e^{\left (\frac {1}{5} \, x\right )} - {\left (120 \, x^{2} \log \left (5\right ) + 3 \, {\left (x^{5} - 5 \, x^{4}\right )} e^{\left (\frac {1}{5} \, x\right )} - 5 \, {\left (3 \, x^{5} e^{\left (\frac {1}{5} \, x\right )} - 8 \, x^{3} \log \left (5\right )\right )} e^{x}\right )} e^{\left (e^{x}\right )} + 320 \, x \log \left (5\right )\right )}}{5 \, {\left (9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} - 48 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + 64 \, \log \left (5\right )^{2}\right )}} \,d x } \] Input:
integrate((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*e xp(1/5*x)+1080*x^2*log(5))*exp(exp(x))-108*x^4*exp(1/5*x)-2880*x*log(5))/( 45*x^4*exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x, algorithm=" giac")
Output:
integrate(-9/5*(12*x^4*e^(1/5*x) - (120*x^2*log(5) + 3*(x^5 - 5*x^4)*e^(1/ 5*x) - 5*(3*x^5*e^(1/5*x) - 8*x^3*log(5))*e^x)*e^(e^x) + 320*x*log(5))/(9* x^4*e^(2/5*x) - 48*x^2*e^(1/5*x)*log(5) + 64*log(5)^2), x)
Time = 1.60 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=\frac {9\,x^2\,\left (x\,{\mathrm {e}}^{{\mathrm {e}}^x}-4\right )}{8\,\ln \left (5\right )-3\,x^2\,{\mathrm {e}}^{x/5}} \] Input:
int(-(2880*x*log(5) + 108*x^4*exp(x/5) + exp(exp(x))*(exp(x/5)*(135*x^4 - 27*x^5) - 1080*x^2*log(5) + exp(x)*(135*x^5*exp(x/5) - 360*x^3*log(5))))/( 45*x^4*exp((2*x)/5) + 320*log(5)^2 - 240*x^2*exp(x/5)*log(5)),x)
Output:
(9*x^2*(x*exp(exp(x)) - 4))/(8*log(5) - 3*x^2*exp(x/5))
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {-108 e^{x/5} x^4-1440 x \log (25)+e^{e^x} \left (e^{x/5} \left (-135 x^4+27 x^5\right )+540 x^2 \log (25)+e^x \left (-135 e^{x/5} x^5+180 x^3 \log (25)\right )\right )}{45 e^{2 x/5} x^4-120 e^{x/5} x^2 \log (25)+80 \log ^2(25)} \, dx=\frac {9 x^{2} \left (-e^{e^{x}} x +4\right )}{3 e^{\frac {x}{5}} x^{2}-8 \,\mathrm {log}\left (5\right )} \] Input:
int((((-135*x^5*exp(1/5*x)+360*x^3*log(5))*exp(x)+(27*x^5-135*x^4)*exp(1/5 *x)+1080*x^2*log(5))*exp(exp(x))-108*x^4*exp(1/5*x)-2880*x*log(5))/(45*x^4 *exp(1/5*x)^2-240*x^2*log(5)*exp(1/5*x)+320*log(5)^2),x)
Output:
(9*x**2*( - e**(e**x)*x + 4))/(3*e**(x/5)*x**2 - 8*log(5))