Integrand size = 89, antiderivative size = 31 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\frac {2 \left (-e^5+\left (-3+e^{e^{-x} x}-\frac {x}{25}\right )^2 x\right )}{x} \] Output:
2/x*((exp(x/exp(x))-3-1/25*x)^2*x-exp(5))
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=2 e^{2 e^{-x} x}-\frac {2 e^5}{x}-\frac {4}{25} e^{e^{-x} x} (75+x)+\frac {2}{625} x (150+x) \] Input:
Integrate[(E^((2*x)/E^x)*(2500*x^2 - 2500*x^3) + E^x*(1250*E^5 + 300*x^2 + 4*x^3) + E^(x/E^x)*(-7500*x^2 - 100*E^x*x^2 + 7400*x^3 + 100*x^4))/(625*E ^x*x^2),x]
Output:
2*E^((2*x)/E^x) - (2*E^5)/x - (4*E^(x/E^x)*(75 + x))/25 + (2*x*(150 + x))/ 625
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.
| \(\displaystyle \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (4 x^3+300 x^2+1250 e^5\right )+e^{e^{-x} x} \left (100 x^4+7400 x^3-100 e^x x^2-7500 x^2\right )\right )}{625 x^2} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{625} \int \frac {2 e^{-x} \left (1250 e^{2 e^{-x} x} \left (x^2-x^3\right )+e^x \left (2 x^3+150 x^2+625 e^5\right )-50 e^{e^{-x} x} \left (-x^4-74 x^3+e^x x^2+75 x^2\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{625} \int \frac {e^{-x} \left (1250 e^{2 e^{-x} x} \left (x^2-x^3\right )+e^x \left (2 x^3+150 x^2+625 e^5\right )-50 e^{e^{-x} x} \left (-x^4-74 x^3+e^x x^2+75 x^2\right )\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{625} \int \left (-1250 e^{2 e^{-x} x-x} (x-1)-50 e^{e^{-x} x-x} \left (-x^2-74 x+e^x+75\right )+\frac {2 x^3+150 x^2+625 e^5}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{625} \left (50 \int e^{-e^{-x} \left (-1+e^x\right ) x} x^2dx-50 \int e^{e^{-x} x}dx+1250 \int e^{-e^{-x} \left (-2+e^x\right ) x}dx-3750 \int e^{-e^{-x} \left (-1+e^x\right ) x}dx-1250 \int e^{-e^{-x} \left (-2+e^x\right ) x} xdx+3700 \int e^{-e^{-x} \left (-1+e^x\right ) x} xdx+x^2+150 x-\frac {625 e^5}{x}\right )\) |
Input:
Int[(E^((2*x)/E^x)*(2500*x^2 - 2500*x^3) + E^x*(1250*E^5 + 300*x^2 + 4*x^3 ) + E^(x/E^x)*(-7500*x^2 - 100*E^x*x^2 + 7400*x^3 + 100*x^4))/(625*E^x*x^2 ),x]
Output:
$Aborted
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(\frac {2 x^{2}}{625}+\frac {12 x}{25}-\frac {2 \,{\mathrm e}^{5}}{x}+2 \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}}+\frac {\left (-100 x -7500\right ) {\mathrm e}^{x \,{\mathrm e}^{-x}}}{625}\) | \(41\) |
| parallelrisch | \(\frac {2 x^{3}-100 \,{\mathrm e}^{x \,{\mathrm e}^{-x}} x^{2}+1250 \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}} x +300 x^{2}-7500 \,{\mathrm e}^{x \,{\mathrm e}^{-x}} x -1250 \,{\mathrm e}^{5}}{625 x}\) | \(55\) |
| norman | \(\frac {\left (-2 \,{\mathrm e}^{5} {\mathrm e}^{x}+\frac {12 \,{\mathrm e}^{x} x^{2}}{25}+\frac {2 \,{\mathrm e}^{x} x^{3}}{625}+2 \,{\mathrm e}^{x} x \,{\mathrm e}^{2 x \,{\mathrm e}^{-x}}-12 \,{\mathrm e}^{x \,{\mathrm e}^{-x}} {\mathrm e}^{x} x -\frac {4 \,{\mathrm e}^{x \,{\mathrm e}^{-x}} {\mathrm e}^{x} x^{2}}{25}\right ) {\mathrm e}^{-x}}{x}\) | \(70\) |
Input:
int(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7 400*x^3-7500*x^2)*exp(x/exp(x))+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x) /x^2,x,method=_RETURNVERBOSE)
Output:
2/625*x^2+12/25*x-2*exp(5)/x+2*exp(2*x*exp(-x))+1/625*(-100*x-7500)*exp(x* exp(-x))
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\frac {2 \, {\left (x^{3} + 150 \, x^{2} + 625 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - 50 \, {\left (x^{2} + 75 \, x\right )} e^{\left (x e^{\left (-x\right )}\right )} - 625 \, e^{5}\right )}}{625 \, x} \] Input:
integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100 *x^4+7400*x^3-7500*x^2)*exp(x/exp(x))+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/ exp(x)/x^2,x, algorithm="fricas")
Output:
2/625*(x^3 + 150*x^2 + 625*x*e^(2*x*e^(-x)) - 50*(x^2 + 75*x)*e^(x*e^(-x)) - 625*e^5)/x
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\frac {2 x^{2}}{625} + \frac {12 x}{25} + \frac {\left (- 4 x - 300\right ) e^{x e^{- x}}}{25} + 2 e^{2 x e^{- x}} - \frac {2 e^{5}}{x} \] Input:
integrate(1/625*((-2500*x**3+2500*x**2)*exp(x/exp(x))**2+(-100*exp(x)*x**2 +100*x**4+7400*x**3-7500*x**2)*exp(x/exp(x))+(1250*exp(5)+4*x**3+300*x**2) *exp(x))/exp(x)/x**2,x)
Output:
2*x**2/625 + 12*x/25 + (-4*x - 300)*exp(x*exp(-x))/25 + 2*exp(2*x*exp(-x)) - 2*exp(5)/x
\[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\int { -\frac {2 \, {\left (1250 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} - 50 \, {\left (x^{4} + 74 \, x^{3} - x^{2} e^{x} - 75 \, x^{2}\right )} e^{\left (x e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} + 150 \, x^{2} + 625 \, e^{5}\right )} e^{x}\right )} e^{\left (-x\right )}}{625 \, x^{2}} \,d x } \] Input:
integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100 *x^4+7400*x^3-7500*x^2)*exp(x/exp(x))+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/ exp(x)/x^2,x, algorithm="maxima")
Output:
2/625*x^2 + 12/25*x - 2*e^5/x + 2*e^(2*x*e^(-x)) - 2/625*integrate(-50*(x^ 2 + 74*x - e^x - 75)*e^(x*e^(-x) - x), x)
\[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\int { -\frac {2 \, {\left (1250 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} - 50 \, {\left (x^{4} + 74 \, x^{3} - x^{2} e^{x} - 75 \, x^{2}\right )} e^{\left (x e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} + 150 \, x^{2} + 625 \, e^{5}\right )} e^{x}\right )} e^{\left (-x\right )}}{625 \, x^{2}} \,d x } \] Input:
integrate(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100 *x^4+7400*x^3-7500*x^2)*exp(x/exp(x))+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/ exp(x)/x^2,x, algorithm="giac")
Output:
integrate(-2/625*(1250*(x^3 - x^2)*e^(2*x*e^(-x)) - 50*(x^4 + 74*x^3 - x^2 *e^x - 75*x^2)*e^(x*e^(-x)) - (2*x^3 + 150*x^2 + 625*e^5)*e^x)*e^(-x)/x^2, x)
Time = 8.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}+\frac {2\,\left (x^3+150\,x^2-625\,{\mathrm {e}}^5\right )}{625\,x}-\frac {4\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-x}}\,\left (x+75\right )}{25} \] Input:
int((exp(-x)*((exp(2*x*exp(-x))*(2500*x^2 - 2500*x^3))/625 + (exp(x)*(1250 *exp(5) + 300*x^2 + 4*x^3))/625 - (exp(x*exp(-x))*(100*x^2*exp(x) + 7500*x ^2 - 7400*x^3 - 100*x^4))/625))/x^2,x)
Output:
2*exp(2*x*exp(-x)) + (2*(150*x^2 - 625*exp(5) + x^3))/(625*x) - (4*exp(x*e xp(-x))*(x + 75))/25
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {e^{-x} \left (e^{2 e^{-x} x} \left (2500 x^2-2500 x^3\right )+e^x \left (1250 e^5+300 x^2+4 x^3\right )+e^{e^{-x} x} \left (-7500 x^2-100 e^x x^2+7400 x^3+100 x^4\right )\right )}{625 x^2} \, dx=\frac {2 e^{\frac {2 x}{e^{x}}} x -\frac {4 e^{\frac {x}{e^{x}}} x^{2}}{25}-12 e^{\frac {x}{e^{x}}} x -2 e^{5}+\frac {2 x^{3}}{625}+\frac {12 x^{2}}{25}}{x} \] Input:
int(1/625*((-2500*x^3+2500*x^2)*exp(x/exp(x))^2+(-100*exp(x)*x^2+100*x^4+7 400*x^3-7500*x^2)*exp(x/exp(x))+(1250*exp(5)+4*x^3+300*x^2)*exp(x))/exp(x) /x^2,x)
Output:
(2*(625*e**((2*x)/e**x)*x - 50*e**(x/e**x)*x**2 - 3750*e**(x/e**x)*x - 625 *e**5 + x**3 + 150*x**2))/(625*x)