\(\int \frac {-364+406 x+189 x^2+e^{2 x} (405+162 x^2)+e^x (270+594 x-108 x^2+162 x^3)}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} (54432-157464 x+118098 x^2)+e^x (-25920+108864 x-157464 x^2+78732 x^3)} \, dx\) [978]

Optimal result

Integrand size = 118, antiderivative size = 31 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\frac {-3-x-x^2}{3 \left (4+9 \left (2-3 \left (e^x+x\right )\right )^2\right )} \] Output:

1/3*(-x^2-x-3)/(4+3*(2-3*exp(x)-3*x)*(6-9*exp(x)-9*x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {3+x+x^2}{3 \left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )} \] Input:

Integrate[(-364 + 406*x + 189*x^2 + E^(2*x)*(405 + 162*x^2) + E^x*(270 + 5 
94*x - 108*x^2 + 162*x^3))/(4800 + 19683*E^(4*x) - 25920*x + 54432*x^2 - 5 
2488*x^3 + 19683*x^4 + E^(3*x)*(-52488 + 78732*x) + E^(2*x)*(54432 - 15746 
4*x + 118098*x^2) + E^x*(-25920 + 108864*x - 157464*x^2 + 78732*x^3)),x]
 

Output:

-1/3*(3 + x + x^2)/(40 + 81*E^(2*x) - 108*x + 81*x^2 + 54*E^x*(-2 + 3*x))
 

Rubi [F]

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.

\(\displaystyle \int \frac {189 x^2+e^{2 x} \left (162 x^2+405\right )+e^x \left (162 x^3-108 x^2+594 x+270\right )+406 x-364}{19683 x^4-52488 x^3+54432 x^2+e^{2 x} \left (118098 x^2-157464 x+54432\right )+e^x \left (78732 x^3-157464 x^2+108864 x-25920\right )-25920 x+19683 e^{4 x}+e^{3 x} (78732 x-52488)+4800} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {189 x^2+e^{2 x} \left (162 x^2+405\right )+e^x \left (162 x^3-108 x^2+594 x+270\right )+406 x-364}{3 \left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int -\frac {-189 x^2-406 x-81 e^{2 x} \left (2 x^2+5\right )-54 e^x \left (3 x^3-2 x^2+11 x+5\right )+364}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} \int \frac {-189 x^2-406 x-81 e^{2 x} \left (2 x^2+5\right )-54 e^x \left (3 x^3-2 x^2+11 x+5\right )+364}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{3} \int \left (\frac {2 \left (x^2+x+3\right ) \left (81 x^2+81 e^x x-189 x-135 e^x+94\right )}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}-\frac {2 x^2+5}{81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (-564 \int \frac {1}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx+810 \int \frac {e^x}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx+946 \int \frac {x}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx-216 \int \frac {e^x x}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx-296 \int \frac {x^2}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx+108 \int \frac {e^x x^2}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx+5 \int \frac {1}{81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40}dx+2 \int \frac {x^2}{81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40}dx-162 \int \frac {x^4}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx+216 \int \frac {x^3}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx-162 \int \frac {e^x x^3}{\left (81 x^2+162 e^x x-108 x-108 e^x+81 e^{2 x}+40\right )^2}dx\right )\)

Input:

Int[(-364 + 406*x + 189*x^2 + E^(2*x)*(405 + 162*x^2) + E^x*(270 + 594*x - 
 108*x^2 + 162*x^3))/(4800 + 19683*E^(4*x) - 25920*x + 54432*x^2 - 52488*x 
^3 + 19683*x^4 + E^(3*x)*(-52488 + 78732*x) + E^(2*x)*(54432 - 157464*x + 
118098*x^2) + E^x*(-25920 + 108864*x - 157464*x^2 + 78732*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16

method	result	size
risch	\(-\frac {x^{2}+x +3}{3 \left (81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40\right )}\)	\(36\)
parallelrisch	\(\frac {-81 x^{2}-81 x -243}{19683 \,{\mathrm e}^{2 x}+39366 \,{\mathrm e}^{x} x +19683 x^{2}-26244 \,{\mathrm e}^{x}-26244 x +9720}\)	\(40\)
norman	\(\frac {-\frac {7 x}{9}-\frac {4 \,{\mathrm e}^{x}}{9}+\frac {{\mathrm e}^{2 x}}{3}+\frac {2 \,{\mathrm e}^{x} x}{3}-\frac {203}{243}}{81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40}\)	\(49\)

Input:

int(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406 
*x-364)/(19683*exp(x)^4+(78732*x-52488)*exp(x)^3+(118098*x^2-157464*x+5443 
2)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-52488*x 
^3+54432*x^2-25920*x+4800),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(x^2+x+3)/(81*exp(2*x)+162*exp(x)*x+81*x^2-108*exp(x)-108*x+40)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \] Input:

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x 
^2+406*x-364)/(19683*exp(x)^4+(78732*x-52488)*exp(x)^3+(118098*x^2-157464* 
x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-5 
2488*x^3+54432*x^2-25920*x+4800),x, algorithm="fricas")
 

Output:

-1/3*(x^2 + x + 3)/(81*x^2 + 54*(3*x - 2)*e^x - 108*x + 81*e^(2*x) + 40)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\frac {- x^{2} - x - 3}{243 x^{2} - 324 x + \left (486 x - 324\right ) e^{x} + 243 e^{2 x} + 120} \] Input:

integrate(((162*x**2+405)*exp(x)**2+(162*x**3-108*x**2+594*x+270)*exp(x)+1 
89*x**2+406*x-364)/(19683*exp(x)**4+(78732*x-52488)*exp(x)**3+(118098*x**2 
-157464*x+54432)*exp(x)**2+(78732*x**3-157464*x**2+108864*x-25920)*exp(x)+ 
19683*x**4-52488*x**3+54432*x**2-25920*x+4800),x)
 

Output:

(-x**2 - x - 3)/(243*x**2 - 324*x + (486*x - 324)*exp(x) + 243*exp(2*x) + 
120)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \] Input:

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x 
^2+406*x-364)/(19683*exp(x)^4+(78732*x-52488)*exp(x)^3+(118098*x^2-157464* 
x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-5 
2488*x^3+54432*x^2-25920*x+4800),x, algorithm="maxima")
 

Output:

-1/3*(x^2 + x + 3)/(81*x^2 + 54*(3*x - 2)*e^x - 108*x + 81*e^(2*x) + 40)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 162 \, x e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} - 108 \, e^{x} + 40\right )}} \] Input:

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x 
^2+406*x-364)/(19683*exp(x)^4+(78732*x-52488)*exp(x)^3+(118098*x^2-157464* 
x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-5 
2488*x^3+54432*x^2-25920*x+4800),x, algorithm="giac")
 

Output:

-1/3*(x^2 + x + 3)/(81*x^2 + 162*x*e^x - 108*x + 81*e^(2*x) - 108*e^x + 40 
)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\int \frac {406\,x+{\mathrm {e}}^{2\,x}\,\left (162\,x^2+405\right )+189\,x^2+{\mathrm {e}}^x\,\left (162\,x^3-108\,x^2+594\,x+270\right )-364}{19683\,{\mathrm {e}}^{4\,x}-25920\,x+{\mathrm {e}}^{2\,x}\,\left (118098\,x^2-157464\,x+54432\right )+{\mathrm {e}}^{3\,x}\,\left (78732\,x-52488\right )+54432\,x^2-52488\,x^3+19683\,x^4+{\mathrm {e}}^x\,\left (78732\,x^3-157464\,x^2+108864\,x-25920\right )+4800} \,d x \] Input:

int((406*x + exp(2*x)*(162*x^2 + 405) + 189*x^2 + exp(x)*(594*x - 108*x^2 
+ 162*x^3 + 270) - 364)/(19683*exp(4*x) - 25920*x + exp(2*x)*(118098*x^2 - 
 157464*x + 54432) + exp(3*x)*(78732*x - 52488) + 54432*x^2 - 52488*x^3 + 
19683*x^4 + exp(x)*(108864*x - 157464*x^2 + 78732*x^3 - 25920) + 4800),x)
 

Output:

int((406*x + exp(2*x)*(162*x^2 + 405) + 189*x^2 + exp(x)*(594*x - 108*x^2 
+ 162*x^3 + 270) - 364)/(19683*exp(4*x) - 25920*x + exp(2*x)*(118098*x^2 - 
 157464*x + 54432) + exp(3*x)*(78732*x - 52488) + 54432*x^2 - 52488*x^3 + 
19683*x^4 + exp(x)*(108864*x - 157464*x^2 + 78732*x^3 - 25920) + 4800), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\frac {-x^{2}-x -3}{243 e^{2 x}+486 e^{x} x -324 e^{x}+243 x^{2}-324 x +120} \] Input:

int(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406 
*x-364)/(19683*exp(x)^4+(78732*x-52488)*exp(x)^3+(118098*x^2-157464*x+5443 
2)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-52488*x 
^3+54432*x^2-25920*x+4800),x)
 

Output:

( - x**2 - x - 3)/(3*(81*e**(2*x) + 162*e**x*x - 108*e**x + 81*x**2 - 108* 
x + 40))