\(\int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} (576+1728 x-216 x^2)+e^{-6 e^5+6 x} (96+576 x-72 x^2)}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} (1944 x^2+12 x^3)+e^{-3 e^5+3 x} (3888 x^2+72 x^3)} \, dx\) [2940]

Optimal result

Integrand size = 149, antiderivative size = 32 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {-8+x}{\left (3 \left (3+e^{-3 \left (e^5-x\right )}\right )^2+\frac {x}{2}\right ) x} \] Output:

(-8+x)/(1/2*x+3*(exp(-3*exp(5)+3*x)+3)^2)/x
 

Mathematica [A] (verified)

Time = 3.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=-\frac {2 e^{6 e^5} (8-x)}{x \left (6 e^{6 x}+36 e^{3 \left (e^5+x\right )}+e^{6 e^5} (54+x)\right )} \] Input:

Integrate[(864 + 32*x - 2*x^2 + E^(-3*E^5 + 3*x)*(576 + 1728*x - 216*x^2) 
+ E^(-6*E^5 + 6*x)*(96 + 576*x - 72*x^2))/(2916*x^2 + 432*E^(-9*E^5 + 9*x) 
*x^2 + 36*E^(-12*E^5 + 12*x)*x^2 + 108*x^3 + x^4 + E^(-6*E^5 + 6*x)*(1944* 
x^2 + 12*x^3) + E^(-3*E^5 + 3*x)*(3888*x^2 + 72*x^3)),x]
 

Output:

(-2*E^(6*E^5)*(8 - x))/(x*(6*E^(6*x) + 36*E^(3*(E^5 + x)) + E^(6*E^5)*(54 
+ 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.

\(\displaystyle \int \frac {-2 x^2+e^{3 x-3 e^5} \left (-216 x^2+1728 x+576\right )+e^{6 x-6 e^5} \left (-72 x^2+576 x+96\right )+32 x+864}{x^4+108 x^3+432 e^{9 x-9 e^5} x^2+36 e^{12 x-12 e^5} x^2+2916 x^2+e^{6 x-6 e^5} \left (12 x^3+1944 x^2\right )+e^{3 x-3 e^5} \left (72 x^3+3888 x^2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{12 e^5} \left (-2 x^2+e^{3 x-3 e^5} \left (-216 x^2+1728 x+576\right )+e^{6 x-6 e^5} \left (-72 x^2+576 x+96\right )+32 x+864\right )}{x^2 \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle e^{12 e^5} \int \frac {2 \left (-x^2+16 x+12 e^{6 x-6 e^5} \left (-3 x^2+24 x+4\right )+36 e^{3 x-3 e^5} \left (-3 x^2+24 x+8\right )+432\right )}{x^2 \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^{12 e^5} \int \frac {-x^2+16 x+12 e^{6 x-6 e^5} \left (-3 x^2+24 x+4\right )+36 e^{3 x-3 e^5} \left (-3 x^2+24 x+8\right )+432}{x^2 \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 e^{12 e^5} \int \left (\frac {e^{-3 e^5} (x-8) \left (6 e^{3 e^5} x+108 e^{3 x}+323 e^{3 e^5}\right )}{x \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}-\frac {2 e^{-6 e^5} \left (3 x^2-24 x-4\right )}{x^2 \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e^{12 e^5} \left (8 e^{-6 e^5} \int \frac {1}{x^2 \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )}dx+275 \int \frac {1}{\left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx+108 e^{-3 e^5} \int \frac {e^{3 x}}{\left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx-2584 \int \frac {1}{x \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx-864 e^{-3 e^5} \int \frac {e^{3 x}}{x \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx+6 \int \frac {x}{\left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )^2}dx-6 e^{-6 e^5} \int \frac {1}{e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}}dx+48 e^{-6 e^5} \int \frac {1}{x \left (e^{6 e^5} x+6 e^{6 x}+36 e^{3 x+3 e^5}+54 e^{6 e^5}\right )}dx\right )\)

Input:

Int[(864 + 32*x - 2*x^2 + E^(-3*E^5 + 3*x)*(576 + 1728*x - 216*x^2) + E^(- 
6*E^5 + 6*x)*(96 + 576*x - 72*x^2))/(2916*x^2 + 432*E^(-9*E^5 + 9*x)*x^2 + 
 36*E^(-12*E^5 + 12*x)*x^2 + 108*x^3 + x^4 + E^(-6*E^5 + 6*x)*(1944*x^2 + 
12*x^3) + E^(-3*E^5 + 3*x)*(3888*x^2 + 72*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12

method	result	size
risch	\(\frac {2 x -16}{x \left (6 \,{\mathrm e}^{-6 \,{\mathrm e}^{5}+6 x}+36 \,{\mathrm e}^{-3 \,{\mathrm e}^{5}+3 x}+x +54\right )}\)	\(36\)
norman	\(\frac {2 x -16}{x \left (6 \,{\mathrm e}^{-6 \,{\mathrm e}^{5}+6 x}+36 \,{\mathrm e}^{-3 \,{\mathrm e}^{5}+3 x}+x +54\right )}\)	\(39\)
parallelrisch	\(\frac {-96+12 x}{6 x \left (6 \,{\mathrm e}^{-6 \,{\mathrm e}^{5}+6 x}+36 \,{\mathrm e}^{-3 \,{\mathrm e}^{5}+3 x}+x +54\right )}\)	\(40\)

Input:

int(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3* 
exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*ex 
p(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2)*exp(- 
3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x,method=_RETURNVERBOSE)
 

Output:

2*(-8+x)/x/(6*exp(-6*exp(5)+6*x)+36*exp(-3*exp(5)+3*x)+x+54)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {2 \, {\left (x - 8\right )}}{x^{2} + 6 \, x e^{\left (6 \, x - 6 \, e^{5}\right )} + 36 \, x e^{\left (3 \, x - 3 \, e^{5}\right )} + 54 \, x} \] Input:

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*e 
xp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp 
(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2) 
*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="fricas")
 

Output:

2*(x - 8)/(x^2 + 6*x*e^(6*x - 6*e^5) + 36*x*e^(3*x - 3*e^5) + 54*x)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {2 x - 16}{x^{2} + 36 x e^{3 x - 3 e^{5}} + 6 x e^{6 x - 6 e^{5}} + 54 x} \] Input:

integrate(((-72*x**2+576*x+96)*exp(-3*exp(5)+3*x)**2+(-216*x**2+1728*x+576 
)*exp(-3*exp(5)+3*x)-2*x**2+32*x+864)/(36*x**2*exp(-3*exp(5)+3*x)**4+432*x 
**2*exp(-3*exp(5)+3*x)**3+(12*x**3+1944*x**2)*exp(-3*exp(5)+3*x)**2+(72*x* 
*3+3888*x**2)*exp(-3*exp(5)+3*x)+x**4+108*x**3+2916*x**2),x)
 

Output:

(2*x - 16)/(x**2 + 36*x*exp(3*x - 3*exp(5)) + 6*x*exp(6*x - 6*exp(5)) + 54 
*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {2 \, {\left (x e^{\left (6 \, e^{5}\right )} - 8 \, e^{\left (6 \, e^{5}\right )}\right )}}{x^{2} e^{\left (6 \, e^{5}\right )} + 6 \, x e^{\left (6 \, x\right )} + 36 \, x e^{\left (3 \, x + 3 \, e^{5}\right )} + 54 \, x e^{\left (6 \, e^{5}\right )}} \] Input:

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*e 
xp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp 
(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2) 
*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="maxima")
 

Output:

2*(x*e^(6*e^5) - 8*e^(6*e^5))/(x^2*e^(6*e^5) + 6*x*e^(6*x) + 36*x*e^(3*x + 
 3*e^5) + 54*x*e^(6*e^5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (27) = 54\).

Time = 0.99 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {4 \, {\left (x - 8\right )}}{{\left (x - e^{5}\right )}^{2} + 2 \, {\left (x - e^{5}\right )} e^{5} + 6 \, {\left (x - e^{5}\right )} e^{\left (6 \, x - 6 \, e^{5}\right )} + 36 \, {\left (x - e^{5}\right )} e^{\left (3 \, x - 3 \, e^{5}\right )} + 54 \, x + e^{10} + 6 \, e^{\left (6 \, x - 6 \, e^{5} + 5\right )} + 36 \, e^{\left (3 \, x - 3 \, e^{5} + 5\right )}} \] Input:

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*e 
xp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp 
(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2) 
*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="giac")
 

Output:

4*(x - 8)/((x - e^5)^2 + 2*(x - e^5)*e^5 + 6*(x - e^5)*e^(6*x - 6*e^5) + 3 
6*(x - e^5)*e^(3*x - 3*e^5) + 54*x + e^10 + 6*e^(6*x - 6*e^5 + 5) + 36*e^( 
3*x - 3*e^5 + 5))
 

Mupad [B] (verification not implemented)

Time = 3.80 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=-\frac {2\,\left (-36\,x^4-1644\,x^3+15455\,x^2+8\,x\right )}{x^2\,\left (36\,x^2+1932\,x+1\right )\,\left (x+36\,{\mathrm {e}}^{3\,x-3\,{\mathrm {e}}^5}+6\,{\mathrm {e}}^{6\,x-6\,{\mathrm {e}}^5}+54\right )} \] Input:

int((32*x + exp(6*x - 6*exp(5))*(576*x - 72*x^2 + 96) + exp(3*x - 3*exp(5) 
)*(1728*x - 216*x^2 + 576) - 2*x^2 + 864)/(exp(6*x - 6*exp(5))*(1944*x^2 + 
 12*x^3) + exp(3*x - 3*exp(5))*(3888*x^2 + 72*x^3) + 432*x^2*exp(9*x - 9*e 
xp(5)) + 36*x^2*exp(12*x - 12*exp(5)) + 2916*x^2 + 108*x^3 + x^4),x)
 

Output:

-(2*(8*x + 15455*x^2 - 1644*x^3 - 36*x^4))/(x^2*(1932*x + 36*x^2 + 1)*(x + 
 36*exp(3*x - 3*exp(5)) + 6*exp(6*x - 6*exp(5)) + 54))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81 \[ \int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx=\frac {-e^{6 e^{5}} x^{2}-432 e^{6 e^{5}}-36 e^{3 e^{5}+3 x} x -6 e^{6 x} x}{27 x \left (e^{6 e^{5}} x +54 e^{6 e^{5}}+36 e^{3 e^{5}+3 x}+6 e^{6 x}\right )} \] Input:

int(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3* 
exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*ex 
p(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2)*exp(- 
3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x)
 

Output:

( - e**(6*e**5)*x**2 - 432*e**(6*e**5) - 36*e**(3*e**5 + 3*x)*x - 6*e**(6* 
x)*x)/(27*x*(e**(6*e**5)*x + 54*e**(6*e**5) + 36*e**(3*e**5 + 3*x) + 6*e** 
(6*x)))