\(\int \frac {162 x^5+9 x^6+e^{2 e^x} (135 x^4-108 e^x x^5)}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} (162 x+27 x^2)+e^{2 e^x} (324 x^2+108 x^3+9 x^4)} \, dx\) [1335]

Optimal result

Integrand size = 105, antiderivative size = 31 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {x^3}{\left (\frac {x}{3}+\left (\frac {e^{2 e^x}}{x^2}+\frac {2}{x}\right ) x\right )^2} \] Output:

x^3/(x*(exp(exp(x))^2/x^2+2/x)+1/3*x)^2
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 x^5}{\left (3 e^{2 e^x}+6 x+x^2\right )^2} \] Input:

Integrate[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E 
^x) + 216*x^3 + 108*x^4 + 18*x^5 + x^6 + E^(4*E^x)*(162*x + 27*x^2) + E^(2 
*E^x)*(324*x^2 + 108*x^3 + 9*x^4)),x]
 

Output:

(9*x^5)/(3*E^(2*E^x) + 6*x + x^2)^2
 

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.

\(\displaystyle \int \frac {9 x^6+162 x^5+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{x^6+18 x^5+108 x^4+216 x^3+e^{4 e^x} \left (27 x^2+162 x\right )+e^{2 e^x} \left (9 x^4+108 x^3+324 x^2\right )+27 e^{6 e^x}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {9 x^4 \left (-12 e^{x+2 e^x} x+(x+18) x+15 e^{2 e^x}\right )}{\left (x (x+6)+3 e^{2 e^x}\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 9 \int \frac {x^4 \left (-12 e^{x+2 e^x} x+(x+18) x+15 e^{2 e^x}\right )}{\left (x (x+6)+3 e^{2 e^x}\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 9 \int \left (\frac {x^4 \left (x^2+18 x+15 e^{2 e^x}\right )}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}-\frac {12 e^{x+2 e^x} x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 9 \left (-4 \int \frac {x^6}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx-12 \int \frac {x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx-12 \int \frac {e^{x+2 e^x} x^5}{\left (x^2+6 x+3 e^{2 e^x}\right )^3}dx+5 \int \frac {x^4}{\left (x^2+6 x+3 e^{2 e^x}\right )^2}dx\right )\)

Input:

Int[(162*x^5 + 9*x^6 + E^(2*E^x)*(135*x^4 - 108*E^x*x^5))/(27*E^(6*E^x) + 
216*x^3 + 108*x^4 + 18*x^5 + x^6 + E^(4*E^x)*(162*x + 27*x^2) + E^(2*E^x)* 
(324*x^2 + 108*x^3 + 9*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {9 x^{5}}{\left (3 \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+x^{2}+6 x \right )^{2}}\)	\(22\)
parallelrisch	\(\frac {9 x^{5}}{x^{4}+6 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x^{2}+9 \,{\mathrm e}^{4 \,{\mathrm e}^{x}}+12 x^{3}+36 \,{\mathrm e}^{2 \,{\mathrm e}^{x}} x +36 x^{2}}\)	\(47\)

Input:

int(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x) 
)^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6 
+18*x^5+108*x^4+216*x^3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \] Input:

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( 
exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) 
^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="fricas")
 

Output:

9*x^5/(x^4 + 12*x^3 + 36*x^2 + 6*(x^2 + 6*x)*e^(2*e^x) + 9*e^(4*e^x))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {x^{5}}{\frac {x^{4}}{9} + \frac {4 x^{3}}{3} + 4 x^{2} + \left (\frac {2 x^{2}}{3} + 4 x\right ) e^{2 e^{x}} + e^{4 e^{x}}} \] Input:

integrate(((-108*x**5*exp(x)+135*x**4)*exp(exp(x))**2+9*x**6+162*x**5)/(27 
*exp(exp(x))**6+(27*x**2+162*x)*exp(exp(x))**4+(9*x**4+108*x**3+324*x**2)* 
exp(exp(x))**2+x**6+18*x**5+108*x**4+216*x**3),x)
 

Output:

x**5/(x**4/9 + 4*x**3/3 + 4*x**2 + (2*x**2/3 + 4*x)*exp(2*exp(x)) + exp(4* 
exp(x)))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 36 \, x^{2} + 6 \, {\left (x^{2} + 6 \, x\right )} e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \] Input:

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( 
exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) 
^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="maxima")
 

Output:

9*x^5/(x^4 + 12*x^3 + 36*x^2 + 6*(x^2 + 6*x)*e^(2*e^x) + 9*e^(4*e^x))
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\frac {9 \, x^{5}}{x^{4} + 12 \, x^{3} + 6 \, x^{2} e^{\left (2 \, e^{x}\right )} + 36 \, x^{2} + 36 \, x e^{\left (2 \, e^{x}\right )} + 9 \, e^{\left (4 \, e^{x}\right )}} \] Input:

integrate(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp( 
exp(x))^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x)) 
^2+x^6+18*x^5+108*x^4+216*x^3),x, algorithm="giac")
 

Output:

9*x^5/(x^4 + 12*x^3 + 6*x^2*e^(2*e^x) + 36*x^2 + 36*x*e^(2*e^x) + 9*e^(4*e 
^x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=\int \frac {162\,x^5-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (108\,x^5\,{\mathrm {e}}^x-135\,x^4\right )+9\,x^6}{27\,{\mathrm {e}}^{6\,{\mathrm {e}}^x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (9\,x^4+108\,x^3+324\,x^2\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^x}\,\left (27\,x^2+162\,x\right )+216\,x^3+108\,x^4+18\,x^5+x^6} \,d x \] Input:

int((162*x^5 - exp(2*exp(x))*(108*x^5*exp(x) - 135*x^4) + 9*x^6)/(27*exp(6 
*exp(x)) + exp(2*exp(x))*(324*x^2 + 108*x^3 + 9*x^4) + exp(4*exp(x))*(162* 
x + 27*x^2) + 216*x^3 + 108*x^4 + 18*x^5 + x^6),x)
 

Output:

int((162*x^5 - exp(2*exp(x))*(108*x^5*exp(x) - 135*x^4) + 9*x^6)/(27*exp(6 
*exp(x)) + exp(2*exp(x))*(324*x^2 + 108*x^3 + 9*x^4) + exp(4*exp(x))*(162* 
x + 27*x^2) + 216*x^3 + 108*x^4 + 18*x^5 + x^6), x)
 

Reduce [F]

\[ \int \frac {162 x^5+9 x^6+e^{2 e^x} \left (135 x^4-108 e^x x^5\right )}{27 e^{6 e^x}+216 x^3+108 x^4+18 x^5+x^6+e^{4 e^x} \left (162 x+27 x^2\right )+e^{2 e^x} \left (324 x^2+108 x^3+9 x^4\right )} \, dx=9 \left (\int \frac {x^{6}}{27 e^{6 e^{x}}+27 e^{4 e^{x}} x^{2}+162 e^{4 e^{x}} x +9 e^{2 e^{x}} x^{4}+108 e^{2 e^{x}} x^{3}+324 e^{2 e^{x}} x^{2}+x^{6}+18 x^{5}+108 x^{4}+216 x^{3}}d x \right )+162 \left (\int \frac {x^{5}}{27 e^{6 e^{x}}+27 e^{4 e^{x}} x^{2}+162 e^{4 e^{x}} x +9 e^{2 e^{x}} x^{4}+108 e^{2 e^{x}} x^{3}+324 e^{2 e^{x}} x^{2}+x^{6}+18 x^{5}+108 x^{4}+216 x^{3}}d x \right )-108 \left (\int \frac {e^{2 e^{x}+x} x^{5}}{27 e^{6 e^{x}}+27 e^{4 e^{x}} x^{2}+162 e^{4 e^{x}} x +9 e^{2 e^{x}} x^{4}+108 e^{2 e^{x}} x^{3}+324 e^{2 e^{x}} x^{2}+x^{6}+18 x^{5}+108 x^{4}+216 x^{3}}d x \right )+135 \left (\int \frac {e^{2 e^{x}} x^{4}}{27 e^{6 e^{x}}+27 e^{4 e^{x}} x^{2}+162 e^{4 e^{x}} x +9 e^{2 e^{x}} x^{4}+108 e^{2 e^{x}} x^{3}+324 e^{2 e^{x}} x^{2}+x^{6}+18 x^{5}+108 x^{4}+216 x^{3}}d x \right ) \] Input:

int(((-108*x^5*exp(x)+135*x^4)*exp(exp(x))^2+9*x^6+162*x^5)/(27*exp(exp(x) 
)^6+(27*x^2+162*x)*exp(exp(x))^4+(9*x^4+108*x^3+324*x^2)*exp(exp(x))^2+x^6 
+18*x^5+108*x^4+216*x^3),x)
 

Output:

9*(int(x**6/(27*e**(6*e**x) + 27*e**(4*e**x)*x**2 + 162*e**(4*e**x)*x + 9* 
e**(2*e**x)*x**4 + 108*e**(2*e**x)*x**3 + 324*e**(2*e**x)*x**2 + x**6 + 18 
*x**5 + 108*x**4 + 216*x**3),x) + 18*int(x**5/(27*e**(6*e**x) + 27*e**(4*e 
**x)*x**2 + 162*e**(4*e**x)*x + 9*e**(2*e**x)*x**4 + 108*e**(2*e**x)*x**3 
+ 324*e**(2*e**x)*x**2 + x**6 + 18*x**5 + 108*x**4 + 216*x**3),x) - 12*int 
((e**(2*e**x + x)*x**5)/(27*e**(6*e**x) + 27*e**(4*e**x)*x**2 + 162*e**(4* 
e**x)*x + 9*e**(2*e**x)*x**4 + 108*e**(2*e**x)*x**3 + 324*e**(2*e**x)*x**2 
 + x**6 + 18*x**5 + 108*x**4 + 216*x**3),x) + 15*int((e**(2*e**x)*x**4)/(2 
7*e**(6*e**x) + 27*e**(4*e**x)*x**2 + 162*e**(4*e**x)*x + 9*e**(2*e**x)*x* 
*4 + 108*e**(2*e**x)*x**3 + 324*e**(2*e**x)*x**2 + x**6 + 18*x**5 + 108*x* 
*4 + 216*x**3),x))