\(\int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} (27+12 x+3 x^2)+e^{\frac {e^{6-x}}{3 x}} (-30 x+e^{6-x} (4+8 x+5 x^2+x^3))}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} (12+12 x+3 x^2)+e^{\frac {e^{6-x}}{3 x}} (-24 x-24 x^2-6 x^3)} \, dx\) [1560]

Optimal result

Integrand size = 152, antiderivative size = 34 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=5-\frac {5}{2+x}+\frac {x}{1-e^{-\frac {e^{6-x}}{3 x}} x} \] Output:

5+x/(1-x/exp(1/3*exp(6-x)/x))-5/(2+x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\frac {1}{3} \left (3 x+\frac {3 x^2}{e^{\frac {e^{6-x}}{3 x}}-x}-\frac {15}{2+x}\right ) \] Input:

Integrate[(15*x^2 + E^((2*E^(6 - x))/(3*x))*(27 + 12*x + 3*x^2) + E^(E^(6 
- x)/(3*x))*(-30*x + E^(6 - x)*(4 + 8*x + 5*x^2 + x^3)))/(12*x^2 + 12*x^3 
+ 3*x^4 + E^((2*E^(6 - x))/(3*x))*(12 + 12*x + 3*x^2) + E^(E^(6 - x)/(3*x) 
)*(-24*x - 24*x^2 - 6*x^3)),x]
 

Output:

(3*x + (3*x^2)/(E^(E^(6 - x)/(3*x)) - x) - 15/(2 + x))/3
 

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.

\(\displaystyle \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+27\right )+e^{\frac {e^{6-x}}{3 x}} \left (e^{6-x} \left (x^3+5 x^2+8 x+4\right )-30 x\right )}{3 x^4+12 x^3+12 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+12\right )+e^{\frac {e^{6-x}}{3 x}} \left (-6 x^3-24 x^2-24 x\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (3 x^2+12 x+27\right )+e^{\frac {e^{6-x}}{3 x}} \left (e^{6-x} \left (x^3+5 x^2+8 x+4\right )-30 x\right )}{3 \left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {15 x^2+3 e^{\frac {2 e^{6-x}}{3 x}} \left (x^2+4 x+9\right )-e^{\frac {e^{6-x}}{3 x}} \left (30 x-e^{6-x} \left (x^3+5 x^2+8 x+4\right )\right )}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} \int \left (\frac {15 x^2}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}-\frac {30 e^{\frac {e^{6-x}}{3 x}} x}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}+\frac {e^{-x+6+\frac {e^{6-x}}{3 x}} (x+1)}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}+\frac {3 e^{\frac {2 e^{6-x}}{3 x}} \left (x^2+4 x+9\right )}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (15 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+\int \frac {e^{-x+6+\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+3 \int \frac {e^{\frac {2 e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+\int \frac {e^{-x+6+\frac {e^{6-x}}{3 x}} x}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2}dx+60 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx+60 \int \frac {e^{\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx+15 \int \frac {e^{\frac {2 e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)^2}dx-60 \int \frac {1}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)}dx-30 \int \frac {e^{\frac {e^{6-x}}{3 x}}}{\left (e^{\frac {e^{6-x}}{3 x}}-x\right )^2 (x+2)}dx\right )\)

Input:

Int[(15*x^2 + E^((2*E^(6 - x))/(3*x))*(27 + 12*x + 3*x^2) + E^(E^(6 - x)/( 
3*x))*(-30*x + E^(6 - x)*(4 + 8*x + 5*x^2 + x^3)))/(12*x^2 + 12*x^3 + 3*x^ 
4 + E^((2*E^(6 - x))/(3*x))*(12 + 12*x + 3*x^2) + E^(E^(6 - x)/(3*x))*(-24 
*x - 24*x^2 - 6*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97

method	result	size
risch	\(x -\frac {5}{2+x}-\frac {x^{2}}{x -{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}\)	\(33\)
parallelrisch	\(-\frac {3 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x^{2}+6 x^{2}+27 x -27 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}{3 \left (x^{2}-{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x +2 x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}\right )}\)	\(81\)
norman	\(\frac {-5 x +5 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x -{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x^{2}}{x^{2}-{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}} x +2 x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{-x +6}}{3 x}}}\)	\(90\)

Input:

int(((3*x^2+12*x+27)*exp(1/3*exp(-x+6)/x)^2+((x^3+5*x^2+8*x+4)*exp(-x+6)-3 
0*x)*exp(1/3*exp(-x+6)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(-x+6)/x)^2+ 
(-6*x^3-24*x^2-24*x)*exp(1/3*exp(-x+6)/x)+3*x^4+12*x^3+12*x^2),x,method=_R 
ETURNVERBOSE)
 

Output:

x-5/(2+x)-x^2/(x-exp(1/3*exp(-x+6)/x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {{\left (x^{2} + 2 \, x - 5\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 5 \, x}{x^{2} - {\left (x + 2\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 2 \, x} \] Input:

integrate(((3*x^2+12*x+27)*exp(1/3*exp(6-x)/x)^2+((x^3+5*x^2+8*x+4)*exp(6- 
x)-30*x)*exp(1/3*exp(6-x)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(6-x)/x)^ 
2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(6-x)/x)+3*x^4+12*x^3+12*x^2),x, algorit 
hm="fricas")
 

Output:

-((x^2 + 2*x - 5)*e^(1/3*e^(-x + 6)/x) + 5*x)/(x^2 - (x + 2)*e^(1/3*e^(-x 
+ 6)/x) + 2*x)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.59 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\frac {x^{2}}{- x + e^{\frac {e^{6 - x}}{3 x}}} + x - \frac {5}{x + 2} \] Input:

integrate(((3*x**2+12*x+27)*exp(1/3*exp(6-x)/x)**2+((x**3+5*x**2+8*x+4)*ex 
p(6-x)-30*x)*exp(1/3*exp(6-x)/x)+15*x**2)/((3*x**2+12*x+12)*exp(1/3*exp(6- 
x)/x)**2+(-6*x**3-24*x**2-24*x)*exp(1/3*exp(6-x)/x)+3*x**4+12*x**3+12*x**2 
),x)
 

Output:

x**2/(-x + exp(exp(6 - x)/(3*x))) + x - 5/(x + 2)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {{\left (x^{2} + 2 \, x - 5\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 5 \, x}{x^{2} - {\left (x + 2\right )} e^{\left (\frac {e^{\left (-x + 6\right )}}{3 \, x}\right )} + 2 \, x} \] Input:

integrate(((3*x^2+12*x+27)*exp(1/3*exp(6-x)/x)^2+((x^3+5*x^2+8*x+4)*exp(6- 
x)-30*x)*exp(1/3*exp(6-x)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(6-x)/x)^ 
2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(6-x)/x)+3*x^4+12*x^3+12*x^2),x, algorit 
hm="maxima")
 

Output:

-((x^2 + 2*x - 5)*e^(1/3*e^(-x + 6)/x) + 5*x)/(x^2 - (x + 2)*e^(1/3*e^(-x 
+ 6)/x) + 2*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2098 vs. \(2 (30) = 60\).

Time = 0.19 (sec) , antiderivative size = 2098, normalized size of antiderivative = 61.71 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate(((3*x^2+12*x+27)*exp(1/3*exp(6-x)/x)^2+((x^3+5*x^2+8*x+4)*exp(6- 
x)-30*x)*exp(1/3*exp(6-x)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(6-x)/x)^ 
2+(-6*x^3-24*x^2-24*x)*exp(1/3*exp(6-x)/x)+3*x^4+12*x^3+12*x^2),x, algorit 
hm="giac")
 

Output:

-(9*x^5*e^(2*x + 1/3*e^(-x + 6)/x) + 3*x^5*e^(x + 1/3*(18*x + e^(-x + 6))/ 
x) + 3*x^5*e^(x + 1/3*e^(-x + 6)/x + 6) + x^5*e^(1/3*(18*x + e^(-x + 6))/x 
 + 6) + 5*x^4*e^12 + 45*x^4*e^(2*x) + 18*x^4*e^(2*x + 1/3*e^(-x + 6)/x) - 
9*x^4*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x) - 3*x^4* 
e^(x + 1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/x) + 9*x^4*e^(x + 1/3*(1 
8*x + e^(-x + 6))/x) + 9*x^4*e^(x + 1/3*e^(-x + 6)/x + 6) + 30*x^4*e^(x + 
6) - 3*x^4*e^(1/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) 
 - x^4*e^(1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) + 4*x^4*e^(1/3 
*(18*x + e^(-x + 6))/x + 6) + 10*x^3*e^12 - 45*x^3*e^(2*x + 1/3*e^(-x + 6) 
/x) - 18*x^3*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x) - 
 45*x^3*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x) - 9*x^3*e^(x + 1/3*(18*x 
+ e^(-x + 6))/x + 1/3*e^(-x + 6)/x) + 6*x^3*e^(x + 1/3*(18*x + e^(-x + 6)) 
/x) - 24*x^3*e^(x + 1/3*e^(-x + 6)/x + 6) + 30*x^3*e^(x + 6) - 9*x^3*e^(1/ 
2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1/3*e^(-x + 6)/x + 6) - 30*x^3*e^(1/2 
*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 6) - 5*x^3*e^(-1/2*x + 1/6*(3*x^2 + 2* 
e^(-x + 6))/x + 12) - 4*x^3*e^(1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6)/ 
x + 6) + 5*x^3*e^(1/3*(18*x + e^(-x + 6))/x + 6) - 5*x^3*e^(1/3*e^(-x + 6) 
/x + 12) + 5*x^2*e^12 + 45*x^2*e^(3/2*x + 1/6*(3*x^2 + 2*e^(-x + 6))/x + 1 
/3*e^(-x + 6)/x) - 6*x^2*e^(x + 1/3*(18*x + e^(-x + 6))/x + 1/3*e^(-x + 6) 
/x) - 30*x^2*e^(x + 1/3*e^(-x + 6)/x + 6) + 24*x^2*e^(1/2*x + 1/6*(3*x^...
 

Mupad [B] (verification not implemented)

Time = 3.71 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=-\frac {5\,x-5\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}+2\,x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}+x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}}{\left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}{3\,x}}\right )\,\left (x+2\right )} \] Input:

int((exp((2*exp(6 - x))/(3*x))*(12*x + 3*x^2 + 27) - exp(exp(6 - x)/(3*x)) 
*(30*x - exp(6 - x)*(8*x + 5*x^2 + x^3 + 4)) + 15*x^2)/(exp((2*exp(6 - x)) 
/(3*x))*(12*x + 3*x^2 + 12) - exp(exp(6 - x)/(3*x))*(24*x + 24*x^2 + 6*x^3 
) + 12*x^2 + 12*x^3 + 3*x^4),x)
 

Output:

-(5*x - 5*exp((exp(-x)*exp(6))/(3*x)) + 2*x*exp((exp(-x)*exp(6))/(3*x)) + 
x^2*exp((exp(-x)*exp(6))/(3*x)))/((x - exp((exp(-x)*exp(6))/(3*x)))*(x + 2 
))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.56 \[ \int \frac {15 x^2+e^{\frac {2 e^{6-x}}{3 x}} \left (27+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-30 x+e^{6-x} \left (4+8 x+5 x^2+x^3\right )\right )}{12 x^2+12 x^3+3 x^4+e^{\frac {2 e^{6-x}}{3 x}} \left (12+12 x+3 x^2\right )+e^{\frac {e^{6-x}}{3 x}} \left (-24 x-24 x^2-6 x^3\right )} \, dx=\frac {x \left (2 e^{\frac {e^{6}}{3 e^{x} x}} x +9 e^{\frac {e^{6}}{3 e^{x} x}}-5 x \right )}{2 e^{\frac {e^{6}}{3 e^{x} x}} x +4 e^{\frac {e^{6}}{3 e^{x} x}}-2 x^{2}-4 x} \] Input:

int(((3*x^2+12*x+27)*exp(1/3*exp(6-x)/x)^2+((x^3+5*x^2+8*x+4)*exp(6-x)-30* 
x)*exp(1/3*exp(6-x)/x)+15*x^2)/((3*x^2+12*x+12)*exp(1/3*exp(6-x)/x)^2+(-6* 
x^3-24*x^2-24*x)*exp(1/3*exp(6-x)/x)+3*x^4+12*x^3+12*x^2),x)
 

Output:

(x*(2*e**(e**6/(3*e**x*x))*x + 9*e**(e**6/(3*e**x*x)) - 5*x))/(2*(e**(e**6 
/(3*e**x*x))*x + 2*e**(e**6/(3*e**x*x)) - x**2 - 2*x))