\(\int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} (68-68 x+17 x^2)+e^{-\frac {x}{-2+x}} (-128 x+160 x^2-32 x^3)+e^{-x^2+x^3} (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} (-32-128 x+320 x^2-224 x^3+48 x^4))}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} (64-64 x+16 x^2)+e^{-x^2+x^3} (-128 x^2+128 x^3-32 x^4)} \, dx\) [1400]

Optimal result

Integrand size = 195, antiderivative size = 36 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=-2+\frac {17 x}{16}+\frac {e^{\frac {x}{2-x}}}{-e^{(-1+x) x^2}+x^2} \] Output:

exp(x/(2-x))/(x^2-exp((-1+x)*x^2))-2+17/16*x
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.17 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {1}{16} \left (17 x-\frac {16 e^{-1-\frac {2}{-2+x}+x^2}}{e^{x^3}-e^{x^2} x^2}\right ) \] Input:

Integrate[(68*x^4 - 68*x^5 + 17*x^6 + E^(-2*x^2 + 2*x^3)*(68 - 68*x + 17*x 
^2) + (-128*x + 160*x^2 - 32*x^3)/E^(x/(-2 + x)) + E^(-x^2 + x^3)*(-136*x^ 
2 + 136*x^3 - 34*x^4 + (-32 - 128*x + 320*x^2 - 224*x^3 + 48*x^4)/E^(x/(-2 
 + x))))/(64*x^4 - 64*x^5 + 16*x^6 + E^(-2*x^2 + 2*x^3)*(64 - 64*x + 16*x^ 
2) + E^(-x^2 + x^3)*(-128*x^2 + 128*x^3 - 32*x^4)),x]
 

Output:

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

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.

\(\displaystyle \int \frac {17 x^6-68 x^5+68 x^4+e^{2 x^3-2 x^2} \left (17 x^2-68 x+68\right )+e^{-\frac {x}{x-2}} \left (-32 x^3+160 x^2-128 x\right )+e^{x^3-x^2} \left (-34 x^4+136 x^3-136 x^2+e^{-\frac {x}{x-2}} \left (48 x^4-224 x^3+320 x^2-128 x-32\right )\right )}{16 x^6-64 x^5+64 x^4+e^{2 x^3-2 x^2} \left (16 x^2-64 x+64\right )+e^{x^3-x^2} \left (-32 x^4+128 x^3-128 x^2\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{2 x^2} \left (17 x^6-68 x^5+68 x^4+e^{2 x^3-2 x^2} \left (17 x^2-68 x+68\right )+e^{-\frac {x}{x-2}} \left (-32 x^3+160 x^2-128 x\right )+e^{x^3-x^2} \left (-34 x^4+136 x^3-136 x^2+e^{-\frac {x}{x-2}} \left (48 x^4-224 x^3+320 x^2-128 x-32\right )\right )\right )}{16 (2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{16} \int \frac {e^{2 x^2} \left (17 x^6-68 x^5+68 x^4+17 e^{2 x^3-2 x^2} \left (x^2-4 x+4\right )-32 e^{\frac {x}{2-x}} \left (x^3-5 x^2+4 x\right )-2 e^{x^3-x^2} \left (17 x^4-68 x^3+68 x^2+8 e^{\frac {x}{2-x}} \left (-3 x^4+14 x^3-20 x^2+8 x+2\right )\right )\right )}{(2-x)^2 \left (e^{x^3}-e^{x^2} x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{16} \int \left (\frac {16 e^{2 x^2+\frac {x}{2-x}} x \left (3 x^3-2 x^2-2\right )}{\left (e^{x^2} x^2-e^{x^3}\right )^2}-\frac {16 e^{2 x^2-\frac {(x-1)^2 x}{x-2}} \left (3 x^4-14 x^3+20 x^2-8 x-2\right )}{(x-2)^2 \left (e^{x^2} x^2-e^{x^3}\right )}+17\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{16} \left (-32 \int \frac {e^{\frac {x \left (2 x^2-4 x-1\right )}{x-2}} x}{\left (e^{x^2} x^2-e^{x^3}\right )^2}dx-32 \int \frac {e^{\frac {x \left (2 x^2-4 x-1\right )}{x-2}} x^3}{\left (e^{x^2} x^2-e^{x^3}\right )^2}dx+32 \int \frac {e^{\frac {x \left (x^2-2 x-1\right )}{x-2}}}{(x-2)^2 \left (e^{x^2} x^2-e^{x^3}\right )}dx+32 \int \frac {e^{\frac {x \left (x^2-2 x-1\right )}{x-2}} x}{e^{x^2} x^2-e^{x^3}}dx-48 \int \frac {e^{\frac {x \left (x^2-2 x-1\right )}{x-2}} x^2}{e^{x^2} x^2-e^{x^3}}dx+48 \int \frac {e^{\frac {x \left (2 x^2-4 x-1\right )}{x-2}} x^4}{\left (e^{x^2} x^2-e^{x^3}\right )^2}dx+17 x\right )\)

Input:

Int[(68*x^4 - 68*x^5 + 17*x^6 + E^(-2*x^2 + 2*x^3)*(68 - 68*x + 17*x^2) + 
(-128*x + 160*x^2 - 32*x^3)/E^(x/(-2 + x)) + E^(-x^2 + x^3)*(-136*x^2 + 13 
6*x^3 - 34*x^4 + (-32 - 128*x + 320*x^2 - 224*x^3 + 48*x^4)/E^(x/(-2 + x)) 
))/(64*x^4 - 64*x^5 + 16*x^6 + E^(-2*x^2 + 2*x^3)*(64 - 64*x + 16*x^2) + E 
^(-x^2 + x^3)*(-128*x^2 + 128*x^3 - 32*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72

method	result	size
parallelrisch	\(\frac {136 x^{3}+408 x^{2}-136 \,{\mathrm e}^{\left (-1+x \right ) x^{2}} x -408 \,{\mathrm e}^{\left (-1+x \right ) x^{2}}+128 \,{\mathrm e}^{-\frac {x}{-2+x}}}{128 x^{2}-128 \,{\mathrm e}^{\left (-1+x \right ) x^{2}}}\)	\(62\)

Input:

int(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*ex 
p(-x/(-2+x))-34*x^4+136*x^3-136*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)* 
exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^2+(-32 
*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x,method=_RETURNV 
ERBOSE)
 

Output:

1/128*(136*x^3+408*x^2-136*exp((-1+x)*x^2)*x-408*exp((-1+x)*x^2)+128*exp(- 
x/(-2+x)))/(x^2-exp((-1+x)*x^2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 \, x^{3} - 17 \, x e^{\left (x^{3} - x^{2}\right )} + 16 \, e^{\left (-\frac {x}{x - 2}\right )}}{16 \, {\left (x^{2} - e^{\left (x^{3} - x^{2}\right )}\right )}} \] Input:

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x- 
32)*exp(-x/(-2+x))-34*x^4+136*x^3-136*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-1 
28*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^ 
2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorith 
m="fricas")
 

Output:

1/16*(17*x^3 - 17*x*e^(x^3 - x^2) + 16*e^(-x/(x - 2)))/(x^2 - e^(x^3 - x^2 
))
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 x}{16} - \frac {e^{- \frac {x}{x - 2}}}{- x^{2} + e^{x^{3} - x^{2}}} \] Input:

integrate(((17*x**2-68*x+68)*exp(x**3-x**2)**2+((48*x**4-224*x**3+320*x**2 
-128*x-32)*exp(-x/(-2+x))-34*x**4+136*x**3-136*x**2)*exp(x**3-x**2)+(-32*x 
**3+160*x**2-128*x)*exp(-x/(-2+x))+17*x**6-68*x**5+68*x**4)/((16*x**2-64*x 
+64)*exp(x**3-x**2)**2+(-32*x**4+128*x**3-128*x**2)*exp(x**3-x**2)+16*x**6 
-64*x**5+64*x**4),x)
 

Output:

17*x/16 - exp(-x/(x - 2))/(-x**2 + exp(x**3 - x**2))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 \, x^{3} e^{\left (x^{2} + 1\right )} - 17 \, x e^{\left (x^{3} + 1\right )} + 16 \, e^{\left (x^{2} - \frac {2}{x - 2}\right )}}{16 \, {\left (x^{2} e^{\left (x^{2} + 1\right )} - e^{\left (x^{3} + 1\right )}\right )}} \] Input:

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x- 
32)*exp(-x/(-2+x))-34*x^4+136*x^3-136*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-1 
28*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^ 
2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorith 
m="maxima")
 

Output:

1/16*(17*x^3*e^(x^2 + 1) - 17*x*e^(x^3 + 1) + 16*e^(x^2 - 2/(x - 2)))/(x^2 
*e^(x^2 + 1) - e^(x^3 + 1))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x- 
32)*exp(-x/(-2+x))-34*x^4+136*x^3-136*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-1 
28*x)*exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^ 
2+(-32*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x, algorith 
m="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{864,[0,15]%%%}+%%%{-10368,[0,14]%%%}+%%%{52992,[0,13]%%%}+ 
%%%{-1511
 

Mupad [B] (verification not implemented)

Time = 1.98 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17\,x}{16}-\frac {{\mathrm {e}}^{-\frac {x}{x-2}}}{{\mathrm {e}}^{x^3-x^2}-x^2} \] Input:

int(-(exp(-x/(x - 2))*(128*x - 160*x^2 + 32*x^3) - 68*x^4 + 68*x^5 - 17*x^ 
6 + exp(x^3 - x^2)*(exp(-x/(x - 2))*(128*x - 320*x^2 + 224*x^3 - 48*x^4 + 
32) + 136*x^2 - 136*x^3 + 34*x^4) - exp(2*x^3 - 2*x^2)*(17*x^2 - 68*x + 68 
))/(64*x^4 - exp(x^3 - x^2)*(128*x^2 - 128*x^3 + 32*x^4) - 64*x^5 + 16*x^6 
 + exp(2*x^3 - 2*x^2)*(16*x^2 - 64*x + 64)),x)
 

Output:

(17*x)/16 - exp(-x/(x - 2))/(exp(x^3 - x^2) - x^2)
 

Reduce [B] (verification not implemented)

Time = 1.80 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.44 \[ \int \frac {68 x^4-68 x^5+17 x^6+e^{-2 x^2+2 x^3} \left (68-68 x+17 x^2\right )+e^{-\frac {x}{-2+x}} \left (-128 x+160 x^2-32 x^3\right )+e^{-x^2+x^3} \left (-136 x^2+136 x^3-34 x^4+e^{-\frac {x}{-2+x}} \left (-32-128 x+320 x^2-224 x^3+48 x^4\right )\right )}{64 x^4-64 x^5+16 x^6+e^{-2 x^2+2 x^3} \left (64-64 x+16 x^2\right )+e^{-x^2+x^3} \left (-128 x^2+128 x^3-32 x^4\right )} \, dx=\frac {17 e^{\frac {x^{4}-2 x^{3}+2}{x -2}} e x -17 e^{\frac {x^{3}-2 x^{2}+2}{x -2}} e \,x^{3}-16 e^{x^{2}}}{16 e^{\frac {2}{x -2}} e \left (e^{x^{3}}-e^{x^{2}} x^{2}\right )} \] Input:

int(((17*x^2-68*x+68)*exp(x^3-x^2)^2+((48*x^4-224*x^3+320*x^2-128*x-32)*ex 
p(-x/(-2+x))-34*x^4+136*x^3-136*x^2)*exp(x^3-x^2)+(-32*x^3+160*x^2-128*x)* 
exp(-x/(-2+x))+17*x^6-68*x^5+68*x^4)/((16*x^2-64*x+64)*exp(x^3-x^2)^2+(-32 
*x^4+128*x^3-128*x^2)*exp(x^3-x^2)+16*x^6-64*x^5+64*x^4),x)
 

Output:

(17*e**((x**4 - 2*x**3 + 2)/(x - 2))*e*x - 17*e**((x**3 - 2*x**2 + 2)/(x - 
 2))*e*x**3 - 16*e**(x**2))/(16*e**(2/(x - 2))*e*(e**(x**3) - e**(x**2)*x* 
*2))