Integrand size = 214, antiderivative size = 30 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=x+\frac {4}{3+\frac {3 e x}{\left (e^{\frac {x}{e^5}}-x\right ) (2+x)}} \] Output:
4/(3+3/(2+x)*x*exp(1)/(exp(x/exp(5))-x))+x
Time = 7.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\frac {1}{3} x \left (3-\frac {4 e}{e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)}\right ) \] Input:
Integrate[(E^(5 + (2*x)/E^5)*(12 + 12*x + 3*x^2) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 + 3*x^4 + E*(-16*x^2 - 6*x^3)) + E^(x/E^5)*(E*(8*x + 4*x^2) + E^ 5*(-24*x - 24*x^2 - 6*x^3 + E*(-8 + 12*x + 6*x^2))))/(E^(5 + (2*x)/E^5)*(1 2 + 12*x + 3*x^2) + E^(5 + x/E^5)*(-24*x - 24*x^2 - 6*x^3 + E*(12*x + 6*x^ 2)) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 + 3*x^4 + E*(-12*x^2 - 6*x^3))),x]
Output:
(x*(3 - (4*E)/(E*x + E^(x/E^5)*(2 + x) - x*(2 + x))))/3
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.
| \(\displaystyle \int \frac {e^{\frac {2 x}{e^5}+5} \left (3 x^2+12 x+12\right )+e^{\frac {x}{e^5}} \left (e \left (4 x^2+8 x\right )+e^5 \left (-6 x^3-24 x^2+e \left (6 x^2+12 x-8\right )-24 x\right )\right )+e^5 \left (3 x^4+12 x^3+3 e^2 x^2+12 x^2+e \left (-6 x^3-16 x^2\right )\right )}{e^{\frac {2 x}{e^5}+5} \left (3 x^2+12 x+12\right )+e^{\frac {x}{e^5}+5} \left (-6 x^3-24 x^2+e \left (6 x^2+12 x\right )-24 x\right )+e^5 \left (3 x^4+12 x^3+3 e^2 x^2+12 x^2+e \left (-6 x^3-12 x^2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {2 x}{e^5}+5} \left (3 x^2+12 x+12\right )+e^{\frac {x}{e^5}} \left (e \left (4 x^2+8 x\right )+e^5 \left (-6 x^3-24 x^2+e \left (6 x^2+12 x-8\right )-24 x\right )\right )+e^5 \left (3 x^4+12 x^3+3 e^2 x^2+12 x^2+e \left (-6 x^3-16 x^2\right )\right )}{3 e^5 \left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 e^{\frac {2 x}{e^5}+5} \left (x^2+4 x+4\right )+e^5 \left (3 x^4+12 x^3+3 e^2 x^2+12 x^2-2 e \left (3 x^3+8 x^2\right )\right )+2 e^{\frac {x}{e^5}} \left (2 e \left (x^2+2 x\right )-e^5 \left (3 x^3+12 x^2+12 x+e \left (-3 x^2-6 x+4\right )\right )\right )}{\left (-x^2+e^{\frac {x}{e^5}} x-(2-e) x+2 e^{\frac {x}{e^5}}\right )^2}dx}{3 e^5}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {4 e \left (x^2+2 x-2 e^5\right )}{(x+2) \left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )}+\frac {4 e x \left (x^3+\left (4-e-e^5\right ) x^2+2 \left (2-e-2 e^5\right ) x-2 (2-e) e^5\right )}{(x+2) \left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}+3 e^5\right )dx}{3 e^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8 e^7 \int \frac {1}{\left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx+16 e^7 \int \frac {1}{(-x-2) \left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx-8 e^6 \int \frac {x}{\left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx+4 e \left (2-e-e^5\right ) \int \frac {x^2}{\left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx+8 e^6 \int \frac {1}{(-x-2) \left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )}dx+4 e \int \frac {x}{-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}}dx+4 e \int \frac {x^3}{\left (-x^2+e^{\frac {x}{e^5}} x-2 \left (1-\frac {e}{2}\right ) x+2 e^{\frac {x}{e^5}}\right )^2}dx+3 e^5 x}{3 e^5}\) |
Input:
Int[(E^(5 + (2*x)/E^5)*(12 + 12*x + 3*x^2) + E^5*(12*x^2 + 3*E^2*x^2 + 12* x^3 + 3*x^4 + E*(-16*x^2 - 6*x^3)) + E^(x/E^5)*(E*(8*x + 4*x^2) + E^5*(-24 *x - 24*x^2 - 6*x^3 + E*(-8 + 12*x + 6*x^2))))/(E^(5 + (2*x)/E^5)*(12 + 12 *x + 3*x^2) + E^(5 + x/E^5)*(-24*x - 24*x^2 - 6*x^3 + E*(12*x + 6*x^2)) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 + 3*x^4 + E*(-12*x^2 - 6*x^3))),x]
Output:
$Aborted
Time = 2.57 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23
| method | result | size |
| risch | \(x -\frac {4 \,{\mathrm e} x}{3 \left (x \,{\mathrm e}-x^{2}+{\mathrm e}^{x \,{\mathrm e}^{-5}} x -2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}\right )}\) | \(37\) |
| norman | \(\frac {\left (2 \,{\mathrm e}-4\right ) {\mathrm e}^{x \,{\mathrm e}^{-5}}+\left ({\mathrm e}^{2}-\frac {16 \,{\mathrm e}}{3}+4\right ) x +{\mathrm e}^{x \,{\mathrm e}^{-5}} x^{2}+{\mathrm e} \,{\mathrm e}^{x \,{\mathrm e}^{-5}} x -x^{3}}{x \,{\mathrm e}-x^{2}+{\mathrm e}^{x \,{\mathrm e}^{-5}} x -2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}}\) | \(89\) |
| parallelrisch | \(\frac {\left (3 \,{\mathrm e} \,{\mathrm e}^{5} x^{2}-3 x^{3} {\mathrm e}^{5}+3 \,{\mathrm e}^{5} {\mathrm e}^{x \,{\mathrm e}^{-5}} x^{2}-10 \,{\mathrm e} \,{\mathrm e}^{5} x +12 x \,{\mathrm e}^{5}-12 \,{\mathrm e}^{5} {\mathrm e}^{x \,{\mathrm e}^{-5}}\right ) {\mathrm e}^{-5}}{3 x \,{\mathrm e}-3 x^{2}+3 \,{\mathrm e}^{x \,{\mathrm e}^{-5}} x -6 x +6 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}}\) | \(94\) |
Input:
int(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3- 24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6* x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp(5)*ex p(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5)) +(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x,met hod=_RETURNVERBOSE)
Output:
x-4/3*exp(1)*x/(x*exp(1)-x^2+exp(x*exp(-5))*x-2*x+2*exp(x*exp(-5)))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (28) = 56\).
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.63 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\frac {{\left (3 \, x^{2} - 4 \, x\right )} e^{6} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{5} + 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}}{3 \, {\left (x e^{6} - {\left (x^{2} + 2 \, x\right )} e^{5} + {\left (x + 2\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}\right )}} \] Input:
integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)- 6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^ 2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp (5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/e xp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)) ,x, algorithm="fricas")
Output:
1/3*((3*x^2 - 4*x)*e^6 - 3*(x^3 + 2*x^2)*e^5 + 3*(x^2 + 2*x)*e^((x + 5*e^5 )*e^(-5)))/(x*e^6 - (x^2 + 2*x)*e^5 + (x + 2)*e^((x + 5*e^5)*e^(-5)))
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=x - \frac {4 e x}{- 3 x^{2} - 6 x + 3 e x + \left (3 x + 6\right ) e^{\frac {x}{e^{5}}}} \] Input:
integrate(((3*x**2+12*x+12)*exp(5)*exp(x/exp(5))**2+(((6*x**2+12*x-8)*exp( 1)-6*x**3-24*x**2-24*x)*exp(5)+(4*x**2+8*x)*exp(1))*exp(x/exp(5))+(3*x**2* exp(1)**2+(-6*x**3-16*x**2)*exp(1)+3*x**4+12*x**3+12*x**2)*exp(5))/((3*x** 2+12*x+12)*exp(5)*exp(x/exp(5))**2+((6*x**2+12*x)*exp(1)-6*x**3-24*x**2-24 *x)*exp(5)*exp(x/exp(5))+(3*x**2*exp(1)**2+(-6*x**3-12*x**2)*exp(1)+3*x**4 +12*x**3+12*x**2)*exp(5)),x)
Output:
x - 4*E*x/(-3*x**2 - 6*x + 3*E*x + (3*x + 6)*exp(x*exp(-5)))
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (28) = 56\).
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\frac {3 \, x^{3} - 3 \, x^{2} {\left (e - 2\right )} + 4 \, x e - 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x e^{\left (-5\right )}\right )}}{3 \, {\left (x^{2} - x {\left (e - 2\right )} - {\left (x + 2\right )} e^{\left (x e^{\left (-5\right )}\right )}\right )}} \] Input:
integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)- 6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^ 2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp (5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/e xp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)) ,x, algorithm="maxima")
Output:
1/3*(3*x^3 - 3*x^2*(e - 2) + 4*x*e - 3*(x^2 + 2*x)*e^(x*e^(-5)))/(x^2 - x* (e - 2) - (x + 2)*e^(x*e^(-5)))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (28) = 56\).
Time = 0.49 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\frac {{\left (3 \, x^{3} e^{\left (-5\right )} - 3 \, x^{2} e^{\left (-4\right )} + 6 \, x^{2} e^{\left (-5\right )} - 3 \, x^{2} e^{\left (x e^{\left (-5\right )} - 5\right )} + 4 \, x e^{\left (-4\right )} - 6 \, x e^{\left (x e^{\left (-5\right )} - 5\right )}\right )} e^{5}}{3 \, {\left (x^{2} - x e - x e^{\left (x e^{\left (-5\right )}\right )} + 2 \, x - 2 \, e^{\left (x e^{\left (-5\right )}\right )}\right )}} \] Input:
integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)- 6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^ 2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp (5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/e xp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)) ,x, algorithm="giac")
Output:
1/3*(3*x^3*e^(-5) - 3*x^2*e^(-4) + 6*x^2*e^(-5) - 3*x^2*e^(x*e^(-5) - 5) + 4*x*e^(-4) - 6*x*e^(x*e^(-5) - 5))*e^5/(x^2 - x*e - x*e^(x*e^(-5)) + 2*x - 2*e^(x*e^(-5)))
Timed out. \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}}\,\left (\mathrm {e}\,\left (4\,x^2+8\,x\right )-{\mathrm {e}}^5\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x-8\right )+24\,x^2+6\,x^3\right )\right )+{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+16\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}+5}\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x\right )+24\,x^2+6\,x^3\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+12\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )} \,d x \] Input:
int((exp(x*exp(-5))*(exp(1)*(8*x + 4*x^2) - exp(5)*(24*x - exp(1)*(12*x + 6*x^2 - 8) + 24*x^2 + 6*x^3)) + exp(5)*(3*x^2*exp(2) - exp(1)*(16*x^2 + 6* x^3) + 12*x^2 + 12*x^3 + 3*x^4) + exp(5)*exp(2*x*exp(-5))*(12*x + 3*x^2 + 12))/(exp(5)*(3*x^2*exp(2) - exp(1)*(12*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3 *x^4) - exp(5)*exp(x*exp(-5))*(24*x - exp(1)*(12*x + 6*x^2) + 24*x^2 + 6*x ^3) + exp(5)*exp(2*x*exp(-5))*(12*x + 3*x^2 + 12)),x)
Output:
int((exp(x*exp(-5))*(exp(1)*(8*x + 4*x^2) - exp(5)*(24*x - exp(1)*(12*x + 6*x^2 - 8) + 24*x^2 + 6*x^3)) + exp(2*x*exp(-5) + 5)*(12*x + 3*x^2 + 12) + exp(5)*(3*x^2*exp(2) - exp(1)*(16*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^4) )/(exp(2*x*exp(-5) + 5)*(12*x + 3*x^2 + 12) - exp(x*exp(-5) + 5)*(24*x - e xp(1)*(12*x + 6*x^2) + 24*x^2 + 6*x^3) + exp(5)*(3*x^2*exp(2) - exp(1)*(12 *x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^4)), x)
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx=\frac {x \left (3 e^{\frac {x}{e^{5}}} x +6 e^{\frac {x}{e^{5}}}+3 e x -4 e -3 x^{2}-6 x \right )}{3 e^{\frac {x}{e^{5}}} x +6 e^{\frac {x}{e^{5}}}+3 e x -3 x^{2}-6 x} \] Input:
int(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3- 24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6* x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp(5)*ex p(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5)) +(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x)
Output:
(x*(3*e**(x/e**5)*x + 6*e**(x/e**5) + 3*e*x - 4*e - 3*x**2 - 6*x))/(3*(e** (x/e**5)*x + 2*e**(x/e**5) + e*x - x**2 - 2*x))