\(\int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8)}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx\) [730]

Optimal result

Integrand size = 132, antiderivative size = 24 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{x-x^3 \left (2+\frac {1}{x^2+(16+x)^2}\right )^2} \] Output:

exp(x-x^3*(2+1/(x^2+(x+16)^2))^2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{32-x-4 x^3-\frac {32 (16+x)}{\left (128+16 x+x^2\right )^2}+\frac {-16368-1025 x}{4 \left (128+16 x+x^2\right )}} \] Input:

Integrate[(E^((65536*x + 16384*x^2 - 261121*x^3 - 65536*x^4 - 8196*x^5 - 5 
12*x^6 - 16*x^7)/(65536 + 16384*x + 2048*x^2 + 128*x^3 + 4*x^4))*(8388608 
+ 3145728*x - 100467072*x^2 - 37765136*x^3 - 7081471*x^4 - 786624*x^5 - 55 
300*x^6 - 2304*x^7 - 48*x^8))/(8388608 + 3145728*x + 589824*x^2 + 65536*x^ 
3 + 4608*x^4 + 192*x^5 + 4*x^6),x]
 

Output:

E^(32 - x - 4*x^3 - (32*(16 + x))/(128 + 16*x + x^2)^2 + (-16368 - 1025*x) 
/(4*(128 + 16*x + x^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.

\(\displaystyle \int \frac {\left (-48 x^8-2304 x^7-55300 x^6-786624 x^5-7081471 x^4-37765136 x^3-100467072 x^2+3145728 x+8388608\right ) \exp \left (\frac {-16 x^7-512 x^6-8196 x^5-65536 x^4-261121 x^3+16384 x^2+65536 x}{4 x^4+128 x^3+2048 x^2+16384 x+65536}\right )}{4 x^6+192 x^5+4608 x^4+65536 x^3+589824 x^2+3145728 x+8388608} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {\left (-48 x^8-2304 x^7-55300 x^6-786624 x^5-7081471 x^4-37765136 x^3-100467072 x^2+3145728 x+8388608\right ) \exp \left (\frac {-16 x^7-512 x^6-8196 x^5-65536 x^4-261121 x^3+16384 x^2+65536 x}{4 x^4+128 x^3+2048 x^2+16384 x+65536}\right )}{4 \left (x^2+16 x+128\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int \frac {\exp \left (\frac {-16 x^7-512 x^6-8196 x^5-65536 x^4-261121 x^3+16384 x^2+65536 x}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) \left (-48 x^8-2304 x^7-55300 x^6-786624 x^5-7081471 x^4-37765136 x^3-100467072 x^2+3145728 x+8388608\right )}{\left (x^2+16 x+128\right )^3}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {1}{4} \int \frac {\exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) \left (-48 x^8-2304 x^7-55300 x^6-786624 x^5-7081471 x^4-37765136 x^3-100467072 x^2+3145728 x+8388608\right )}{\left (x^2+16 x+128\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{4} \int \left (-48 \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) x^2+\frac {4096 \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) x}{\left (x^2+16 x+128\right )^3}-4 \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )+\frac {1025 \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )}{x^2+16 x+128}+\frac {16 \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) (1021 x-8)}{\left (x^2+16 x+128\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (-4 \int \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )dx+(64+64 i) \int \frac {\exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )}{(-2 x-(16-16 i))^3}dx+(2050-2044 i) \int \frac {\exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )}{(-2 x-(16-16 i))^2}dx-48 \int \exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right ) x^2dx-(64-64 i) \int \frac {\exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )}{(2 x+(16+16 i))^3}dx+(2050+2044 i) \int \frac {\exp \left (\frac {x \left (-16 x^6-512 x^5-8196 x^4-65536 x^3-261121 x^2+16384 x+65536\right )}{4 \left (x^4+32 x^3+512 x^2+4096 x+16384\right )}\right )}{(2 x+(16+16 i))^2}dx\right )\)

Input:

Int[(E^((65536*x + 16384*x^2 - 261121*x^3 - 65536*x^4 - 8196*x^5 - 512*x^6 
 - 16*x^7)/(65536 + 16384*x + 2048*x^2 + 128*x^3 + 4*x^4))*(8388608 + 3145 
728*x - 100467072*x^2 - 37765136*x^3 - 7081471*x^4 - 786624*x^5 - 55300*x^ 
6 - 2304*x^7 - 48*x^8))/(8388608 + 3145728*x + 589824*x^2 + 65536*x^3 + 46 
08*x^4 + 192*x^5 + 4*x^6),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.72 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88

method	result	size
risch	\({\mathrm e}^{-\frac {x \left (4 x^{3}+62 x^{2}+481 x -256\right ) \left (4 x^{3}+66 x^{2}+545 x +256\right )}{4 \left (x^{2}+16 x +128\right )^{2}}}\)	\(45\)
gosper	\({\mathrm e}^{-\frac {x \left (16 x^{6}+512 x^{5}+8196 x^{4}+65536 x^{3}+261121 x^{2}-16384 x -65536\right )}{4 \left (x^{4}+32 x^{3}+512 x^{2}+4096 x +16384\right )}}\)	\(55\)
parallelrisch	\({\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}\)	\(58\)
orering	\(-\frac {4 \left (x^{2}+16 x +128\right )^{3} \left (-48 x^{8}-2304 x^{7}-55300 x^{6}-786624 x^{5}-7081471 x^{4}-37765136 x^{3}-100467072 x^{2}+3145728 x +8388608\right ) {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}}{\left (48 x^{8}+2304 x^{7}+55300 x^{6}+786624 x^{5}+7081471 x^{4}[phone] x^{3}[phone] x^{2}-3145728 x -8388608\right ) \left (4 x^{6}+192 x^{5}+4608 x^{4}+65536 x^{3}+589824 x^{2}+3145728 x +8388608\right )}\)	\(185\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+4096 x \,{\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+512 x^{2} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+32 x^{3} {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}+16384 \,{\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}}{\left (x^{2}+16 x +128\right )^{2}}\)	\(322\)

Input:

int((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467 
072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121* 
x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^6+192* 
x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x,method=_RETURNVERBO 
SE)
 

Output:

exp(-1/4*x*(4*x^3+62*x^2+481*x-256)*(4*x^3+66*x^2+545*x+256)/(x^2+16*x+128 
)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{\left (-\frac {16 \, x^{7} + 512 \, x^{6} + 8196 \, x^{5} + 65536 \, x^{4} + 261121 \, x^{3} - 16384 \, x^{2} - 65536 \, x}{4 \, {\left (x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384\right )}}\right )} \] Input:

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3- 
100467072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-2 
61121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^ 
6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="f 
ricas")
 

Output:

e^(-1/4*(16*x^7 + 512*x^6 + 8196*x^5 + 65536*x^4 + 261121*x^3 - 16384*x^2 
- 65536*x)/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{\frac {- 16 x^{7} - 512 x^{6} - 8196 x^{5} - 65536 x^{4} - 261121 x^{3} + 16384 x^{2} + 65536 x}{4 x^{4} + 128 x^{3} + 2048 x^{2} + 16384 x + 65536}} \] Input:

integrate((-48*x**8-2304*x**7-55300*x**6-786624*x**5-7081471*x**4-37765136 
*x**3-100467072*x**2+3145728*x+8388608)*exp((-16*x**7-512*x**6-8196*x**5-6 
5536*x**4-261121*x**3+16384*x**2+65536*x)/(4*x**4+128*x**3+2048*x**2+16384 
*x+65536))/(4*x**6+192*x**5+4608*x**4+65536*x**3+589824*x**2+3145728*x+838 
8608),x)
 

Output:

exp((-16*x**7 - 512*x**6 - 8196*x**5 - 65536*x**4 - 261121*x**3 + 16384*x* 
*2 + 65536*x)/(4*x**4 + 128*x**3 + 2048*x**2 + 16384*x + 65536))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (23) = 46\).

Time = 0.45 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{\left (-4 \, x^{3} - x - \frac {32 \, x}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {1025 \, x}{4 \, {\left (x^{2} + 16 \, x + 128\right )}} - \frac {512}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {4092}{x^{2} + 16 \, x + 128} + 32\right )} \] Input:

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3- 
100467072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-2 
61121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^ 
6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="m 
axima")
 

Output:

e^(-4*x^3 - x - 32*x/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 1025/4*x/ 
(x^2 + 16*x + 128) - 512/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 4092/ 
(x^2 + 16*x + 128) + 32)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (23) = 46\).

Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 7.29 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=e^{\left (-\frac {4 \, x^{7}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {128 \, x^{6}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {2049 \, x^{5}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {16384 \, x^{4}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} - \frac {261121 \, x^{3}}{4 \, {\left (x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384\right )}} + \frac {4096 \, x^{2}}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384} + \frac {16384 \, x}{x^{4} + 32 \, x^{3} + 512 \, x^{2} + 4096 \, x + 16384}\right )} \] Input:

integrate((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3- 
100467072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-2 
61121*x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^ 
6+192*x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x, algorithm="g 
iac")
 

Output:

e^(-4*x^7/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 128*x^6/(x^4 + 32*x^ 
3 + 512*x^2 + 4096*x + 16384) - 2049*x^5/(x^4 + 32*x^3 + 512*x^2 + 4096*x 
+ 16384) - 16384*x^4/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) - 261121/4* 
x^3/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 16384) + 4096*x^2/(x^4 + 32*x^3 + 5 
12*x^2 + 4096*x + 16384) + 16384*x/(x^4 + 32*x^3 + 512*x^2 + 4096*x + 1638 
4))
 

Mupad [B] (verification not implemented)

Time = 4.40 (sec) , antiderivative size = 183, normalized size of antiderivative = 7.62 \[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx={\mathrm {e}}^{\frac {16384\,x}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {4\,x^7}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {128\,x^6}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {2049\,x^5}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{\frac {4096\,x^2}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {16384\,x^4}{x^4+32\,x^3+512\,x^2+4096\,x+16384}}\,{\mathrm {e}}^{-\frac {261121\,x^3}{4\,x^4+128\,x^3+2048\,x^2+16384\,x+65536}} \] Input:

int(-(exp(-(261121*x^3 - 16384*x^2 - 65536*x + 65536*x^4 + 8196*x^5 + 512* 
x^6 + 16*x^7)/(16384*x + 2048*x^2 + 128*x^3 + 4*x^4 + 65536))*(100467072*x 
^2 - 3145728*x + 37765136*x^3 + 7081471*x^4 + 786624*x^5 + 55300*x^6 + 230 
4*x^7 + 48*x^8 - 8388608))/(3145728*x + 589824*x^2 + 65536*x^3 + 4608*x^4 
+ 192*x^5 + 4*x^6 + 8388608),x)
 

Output:

exp((16384*x)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(4*x^7)/(409 
6*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(128*x^6)/(4096*x + 512*x^2 + 
32*x^3 + x^4 + 16384))*exp(-(2049*x^5)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 
16384))*exp((4096*x^2)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(16 
384*x^4)/(4096*x + 512*x^2 + 32*x^3 + x^4 + 16384))*exp(-(261121*x^3)/(163 
84*x + 2048*x^2 + 128*x^3 + 4*x^4 + 65536))
 

Reduce [F]

\[ \int \frac {e^{\frac {65536 x+16384 x^2-261121 x^3-65536 x^4-8196 x^5-512 x^6-16 x^7}{65536+16384 x+2048 x^2+128 x^3+4 x^4}} \left (8388608+3145728 x-100467072 x^2-37765136 x^3-7081471 x^4-786624 x^5-55300 x^6-2304 x^7-48 x^8\right )}{8388608+3145728 x+589824 x^2+65536 x^3+4608 x^4+192 x^5+4 x^6} \, dx=\int \frac {\left (-48 x^{8}-2304 x^{7}-55300 x^{6}-786624 x^{5}-7081471 x^{4}-37765136 x^{3}-100467072 x^{2}+3145728 x +8388608\right ) {\mathrm e}^{\frac {-16 x^{7}-512 x^{6}-8196 x^{5}-65536 x^{4}-261121 x^{3}+16384 x^{2}+65536 x}{4 x^{4}+128 x^{3}+2048 x^{2}+16384 x +65536}}}{4 x^{6}+192 x^{5}+4608 x^{4}+65536 x^{3}+589824 x^{2}+3145728 x +8388608}d x \] Input:

int((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467 
072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121* 
x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^6+192* 
x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x)
 

Output:

int((-48*x^8-2304*x^7-55300*x^6-786624*x^5-7081471*x^4-37765136*x^3-100467 
072*x^2+3145728*x+8388608)*exp((-16*x^7-512*x^6-8196*x^5-65536*x^4-261121* 
x^3+16384*x^2+65536*x)/(4*x^4+128*x^3+2048*x^2+16384*x+65536))/(4*x^6+192* 
x^5+4608*x^4+65536*x^3+589824*x^2+3145728*x+8388608),x)