Integrand size = 97, antiderivative size = 23 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=9 \left (1-4 e^{-\frac {25 e^{3-x}}{x}}+x\right )^2 \] Output:
3*(1-4/exp(25*exp(3-x)/x)+x)*(3-12/exp(25*exp(3-x)/x)+3*x)
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 0.78 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=18 \left (8 e^{-\frac {50 e^{3-x}}{x}}+e^{-\frac {25 e^{3-x}}{x}} (-4-4 x)+x+\frac {x^2}{2}\right ) \] Input:
Integrate[(E^(3 - x)*(7200 + 7200*x) + E^((50*E^(3 - x))/x)*(18*x^2 + 18*x ^3) + E^((25*E^(3 - x))/x)*(-72*x^2 + E^(3 - x)*(-1800 - 3600*x - 1800*x^2 )))/(E^((50*E^(3 - x))/x)*x^2),x]
Output:
18*(8/E^((50*E^(3 - x))/x) + (-4 - 4*x)/E^((25*E^(3 - x))/x) + x + x^2/2)
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.
\(\displaystyle \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{\frac {25 e^{3-x}}{x}} \left (e^{3-x} \left (-1800 x^2-3600 x-1800\right )-72 x^2\right )+e^{\frac {50 e^{3-x}}{x}} \left (18 x^3+18 x^2\right )+e^{3-x} (7200 x+7200)\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {18 e^{-x-\frac {50 e^{3-x}}{x}} \left (-e^{\frac {25 e^{3-x}}{x}} x-e^{\frac {25 e^{3-x}}{x}}+4\right ) \left (-e^{x+\frac {25 e^{3-x}}{x}} x^2+100 e^3 x+100 e^3\right )}{x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 18 \int \frac {e^{-x-\frac {50 e^{3-x}}{x}} \left (-e^{\frac {25 e^{3-x}}{x}} x-e^{\frac {25 e^{3-x}}{x}}+4\right ) \left (-e^{x+\frac {25 e^{3-x}}{x}} x^2+100 e^3 x+100 e^3\right )}{x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 18 \int \left (e^{-\frac {25 e^{3-x}}{x}} \left (e^{\frac {25 e^{3-x}}{x}} x+e^{\frac {25 e^{3-x}}{x}}-4\right )-\frac {100 e^{-x+3-\frac {50 e^{3-x}}{x}} (x+1) \left (e^{\frac {25 e^{3-x}}{x}} x+e^{\frac {25 e^{3-x}}{x}}-4\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 18 \left (400 \int \frac {e^{-x+3-\frac {50 e^{3-x}}{x}}}{x^2}dx-100 \int \frac {e^{-x+3-\frac {25 e^{3-x}}{x}}}{x^2}dx-100 \int e^{-x+3-\frac {25 e^{3-x}}{x}}dx-4 \int e^{-\frac {25 e^{3-x}}{x}}dx+400 \int \frac {e^{-x+3-\frac {50 e^{3-x}}{x}}}{x}dx-200 \int \frac {e^{-x+3-\frac {25 e^{3-x}}{x}}}{x}dx+\frac {x^2}{2}+x\right )\) |
Input:
Int[(E^(3 - x)*(7200 + 7200*x) + E^((50*E^(3 - x))/x)*(18*x^2 + 18*x^3) + E^((25*E^(3 - x))/x)*(-72*x^2 + E^(3 - x)*(-1800 - 3600*x - 1800*x^2)))/(E ^((50*E^(3 - x))/x)*x^2),x]
Output:
$Aborted
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83
method | result | size |
risch | \(9 x^{2}+18 x +\left (-72-72 x \right ) {\mathrm e}^{-\frac {25 \,{\mathrm e}^{-x +3}}{x}}+144 \,{\mathrm e}^{-\frac {50 \,{\mathrm e}^{-x +3}}{x}}\) | \(42\) |
parts | \(18 x +\frac {\left (-72 x^{2}-72 x \right ) {\mathrm e}^{-\frac {25 \,{\mathrm e}^{-x +3}}{x}}}{x}+9 x^{2}+144 \,{\mathrm e}^{-\frac {50 \,{\mathrm e}^{-x +3}}{x}}\) | \(53\) |
norman | \(\frac {\left (144 x -72 x \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{-x +3}}{x}}+9 \,{\mathrm e}^{\frac {50 \,{\mathrm e}^{-x +3}}{x}} x^{3}-72 \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{-x +3}}{x}} x^{2}+18 \,{\mathrm e}^{\frac {50 \,{\mathrm e}^{-x +3}}{x}} x^{2}\right ) {\mathrm e}^{-\frac {50 \,{\mathrm e}^{-x +3}}{x}}}{x}\) | \(93\) |
parallelrisch | \(-\frac {\left (-225 \,{\mathrm e}^{\frac {50 \,{\mathrm e}^{-x +3}}{x}} x^{3}-450 \,{\mathrm e}^{\frac {50 \,{\mathrm e}^{-x +3}}{x}} x^{2}+1800 \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{-x +3}}{x}} x^{2}+1800 x \,{\mathrm e}^{\frac {25 \,{\mathrm e}^{-x +3}}{x}}-3600 x \right ) {\mathrm e}^{-\frac {50 \,{\mathrm e}^{-x +3}}{x}}}{25 x}\) | \(94\) |
Input:
int(((18*x^3+18*x^2)*exp(25*exp(-x+3)/x)^2+((-1800*x^2-3600*x-1800)*exp(-x +3)-72*x^2)*exp(25*exp(-x+3)/x)+(7200*x+7200)*exp(-x+3))/x^2/exp(25*exp(-x +3)/x)^2,x,method=_RETURNVERBOSE)
Output:
9*x^2+18*x+(-72-72*x)*exp(-25*exp(-x+3)/x)+144*exp(-50*exp(-x+3)/x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.30 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=9 \, {\left ({\left (x^{2} + 2 \, x\right )} e^{\left (\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} - 8 \, {\left (x + 1\right )} e^{\left (\frac {25 \, e^{\left (-x + 3\right )}}{x}\right )} + 16\right )} e^{\left (-\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} \] Input:
integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*e xp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(7200*x+7200)*exp(3-x))/x^2/exp(25*exp( 3-x)/x)^2,x, algorithm="fricas")
Output:
9*((x^2 + 2*x)*e^(50*e^(-x + 3)/x) - 8*(x + 1)*e^(25*e^(-x + 3)/x) + 16)*e ^(-50*e^(-x + 3)/x)
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 3.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=9 x^{2} + 18 x + \left (- 72 x - 72\right ) e^{- \frac {25 e^{3 - x}}{x}} + 144 e^{- \frac {50 e^{3 - x}}{x}} \] Input:
integrate(((18*x**3+18*x**2)*exp(25*exp(3-x)/x)**2+((-1800*x**2-3600*x-180 0)*exp(3-x)-72*x**2)*exp(25*exp(3-x)/x)+(7200*x+7200)*exp(3-x))/x**2/exp(2 5*exp(3-x)/x)**2,x)
Output:
9*x**2 + 18*x + (-72*x - 72)*exp(-25*exp(3 - x)/x) + 144*exp(-50*exp(3 - x )/x)
\[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=\int { \frac {18 \, {\left (400 \, {\left (x + 1\right )} e^{\left (-x + 3\right )} + {\left (x^{3} + x^{2}\right )} e^{\left (\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )} - 4 \, {\left (x^{2} + 25 \, {\left (x^{2} + 2 \, x + 1\right )} e^{\left (-x + 3\right )}\right )} e^{\left (\frac {25 \, e^{\left (-x + 3\right )}}{x}\right )}\right )} e^{\left (-\frac {50 \, e^{\left (-x + 3\right )}}{x}\right )}}{x^{2}} \,d x } \] Input:
integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*e xp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(7200*x+7200)*exp(3-x))/x^2/exp(25*exp( 3-x)/x)^2,x, algorithm="maxima")
Output:
9*x^2 + 18*x + 144*e^(-50*e^(-x + 3)/x) - 18*integrate(4*(25*x^2*e^3 + x^2 *e^x + 50*x*e^3 + 25*e^3)*e^(-x - 25*e^(-x + 3)/x)/x^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.17 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=9 \, {\left (x^{2} e^{\left (-x + 3\right )} + 2 \, x e^{\left (-x + 3\right )} - 8 \, x e^{\left (-\frac {x^{2} - 3 \, x + 25 \, e^{\left (-x + 3\right )}}{x}\right )} + 16 \, e^{\left (-\frac {x^{2} - 3 \, x + 50 \, e^{\left (-x + 3\right )}}{x}\right )} - 8 \, e^{\left (-\frac {x^{2} - 3 \, x + 25 \, e^{\left (-x + 3\right )}}{x}\right )}\right )} e^{\left (x - 3\right )} \] Input:
integrate(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*e xp(3-x)-72*x^2)*exp(25*exp(3-x)/x)+(7200*x+7200)*exp(3-x))/x^2/exp(25*exp( 3-x)/x)^2,x, algorithm="giac")
Output:
9*(x^2*e^(-x + 3) + 2*x*e^(-x + 3) - 8*x*e^(-(x^2 - 3*x + 25*e^(-x + 3))/x ) + 16*e^(-(x^2 - 3*x + 50*e^(-x + 3))/x) - 8*e^(-(x^2 - 3*x + 25*e^(-x + 3))/x))*e^(x - 3)
Time = 2.67 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=18\,x+144\,{\mathrm {e}}^{-\frac {50\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}{x}}-{\mathrm {e}}^{-\frac {25\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^3}{x}}\,\left (72\,x+72\right )+9\,x^2 \] Input:
int((exp(-(50*exp(3 - x))/x)*(exp(3 - x)*(7200*x + 7200) + exp((50*exp(3 - x))/x)*(18*x^2 + 18*x^3) - exp((25*exp(3 - x))/x)*(exp(3 - x)*(3600*x + 1 800*x^2 + 1800) + 72*x^2)))/x^2,x)
Output:
18*x + 144*exp(-(50*exp(-x)*exp(3))/x) - exp(-(25*exp(-x)*exp(3))/x)*(72*x + 72) + 9*x^2
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.04 \[ \int \frac {e^{-\frac {50 e^{3-x}}{x}} \left (e^{3-x} (7200+7200 x)+e^{\frac {50 e^{3-x}}{x}} \left (18 x^2+18 x^3\right )+e^{\frac {25 e^{3-x}}{x}} \left (-72 x^2+e^{3-x} \left (-1800-3600 x-1800 x^2\right )\right )\right )}{x^2} \, dx=\frac {9 e^{\frac {50 e^{3}}{e^{x} x}} x^{2}+18 e^{\frac {50 e^{3}}{e^{x} x}} x -72 e^{\frac {25 e^{3}}{e^{x} x}} x -72 e^{\frac {25 e^{3}}{e^{x} x}}+144}{e^{\frac {50 e^{3}}{e^{x} x}}} \] Input:
int(((18*x^3+18*x^2)*exp(25*exp(3-x)/x)^2+((-1800*x^2-3600*x-1800)*exp(3-x )-72*x^2)*exp(25*exp(3-x)/x)+(7200*x+7200)*exp(3-x))/x^2/exp(25*exp(3-x)/x )^2,x)
Output:
(9*(e**((50*e**3)/(e**x*x))*x**2 + 2*e**((50*e**3)/(e**x*x))*x - 8*e**((25 *e**3)/(e**x*x))*x - 8*e**((25*e**3)/(e**x*x)) + 16))/e**((50*e**3)/(e**x* x))