\(\int \frac {e^{\frac {8}{9} e^{-2 x} x^2} (-72 e^{2+2 x}+e^2 (64 x^2-64 x^3)+e^{2 x} (-144 e^{2+2 x}+e^2 (64 x-64 x^2)))}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx\) [1621]

Optimal result

Integrand size = 107, antiderivative size = 27 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \] Output:

4/(exp(2*x)+x)^2*exp(1)^2*exp(4/9*x^2/exp(x)^2)^2
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \] Input:

Integrate[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3 
) + E^(2*x)*(-144*E^(2 + 2*x) + E^2*(64*x - 64*x^2))))/(9*E^(8*x) + 27*E^( 
6*x)*x + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),x]
 

Output:

(4*E^(2 + (8*x^2)/(9*E^(2*x))))/(E^(2*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.

\(\displaystyle \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (e^{2 x} \left (e^2 \left (64 x-64 x^2\right )-144 e^{2 x+2}\right )+e^2 \left (64 x^2-64 x^3\right )-72 e^{2 x+2}\right )}{9 e^{2 x} x^3+27 e^{4 x} x^2+27 e^{6 x} x+9 e^{8 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {8 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (-8 x^3-8 e^{2 x} x^2+8 x^2+8 e^{2 x} x-9 e^{2 x}-18 e^{4 x}\right )}{9 \left (x+e^{2 x}\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {8}{9} \int -\frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^3+8 e^{2 x} x^2-8 x^2-8 e^{2 x} x+9 e^{2 x}+18 e^{4 x}\right )}{\left (x+e^{2 x}\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {8}{9} \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^3+8 e^{2 x} x^2-8 x^2-8 e^{2 x} x+9 e^{2 x}+18 e^{4 x}\right )}{\left (x+e^{2 x}\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {8}{9} \int \left (\frac {9 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x (2 x-1)}{\left (x+e^{2 x}\right )^3}+\frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} \left (8 x^2-44 x+9\right )}{\left (x+e^{2 x}\right )^2}+\frac {18 e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{x+e^{2 x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {8}{9} \left (-9 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x}{\left (x+e^{2 x}\right )^3}dx+18 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x^2}{\left (x+e^{2 x}\right )^3}dx+9 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{\left (x+e^{2 x}\right )^2}dx-44 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x}{\left (x+e^{2 x}\right )^2}dx+8 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2} x^2}{\left (x+e^{2 x}\right )^2}dx+18 \int \frac {e^{\frac {8}{9} e^{-2 x} x^2-2 x+2}}{x+e^{2 x}}dx\right )\)

Input:

Int[(E^((8*x^2)/(9*E^(2*x)))*(-72*E^(2 + 2*x) + E^2*(64*x^2 - 64*x^3) + E^ 
(2*x)*(-144*E^(2 + 2*x) + E^2*(64*x - 64*x^2))))/(9*E^(8*x) + 27*E^(6*x)*x 
 + 27*E^(4*x)*x^2 + 9*E^(2*x)*x^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.54 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
risch	\(\frac {4 \,{\mathrm e}^{2+\frac {8 \,{\mathrm e}^{-2 x} x^{2}}{9}}}{\left ({\mathrm e}^{2 x}+x \right )^{2}}\)	\(23\)
parallelrisch	\(\frac {4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {8 \,{\mathrm e}^{-2 x} x^{2}}{9}}}{x^{2}+2 x \,{\mathrm e}^{2 x}+{\mathrm e}^{4 x}}\)	\(38\)

Input:

int(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2 
*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*e 
xp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x)^2*x^3 
),x,method=_RETURNVERBOSE)
 

Output:

4/(exp(2*x)+x)^2*exp(2+8/9*exp(-2*x)*x^2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 6\right )}}{x^{2} e^{4} + 2 \, x e^{\left (2 \, x + 4\right )} + e^{\left (4 \, x + 4\right )}} \] Input:

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex 
p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( 
x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) 
^2*x^3),x, algorithm="fricas")
 

Output:

4*e^(8/9*x^2*e^(-2*x) + 6)/(x^2*e^4 + 2*x*e^(2*x + 4) + e^(4*x + 4))
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 e^{2} e^{- 4 x} e^{\frac {8 x^{2} e^{- 2 x}}{9}}}{x^{2} e^{- 4 x} + 2 x e^{- 2 x} + 1} \] Input:

integrate(((-144*exp(1)**2*exp(x)**2+(-64*x**2+64*x)*exp(1)**2)*exp(2*x)-7 
2*exp(1)**2*exp(x)**2+(-64*x**3+64*x**2)*exp(1)**2)*exp(4/9*x**2/exp(x)**2 
)**2/(9*exp(x)**2*exp(2*x)**3+27*x*exp(x)**2*exp(2*x)**2+27*x**2*exp(x)**2 
*exp(2*x)+9*exp(x)**2*x**3),x)
 

Output:

4*exp(2)*exp(-4*x)*exp(8*x**2*exp(-2*x)/9)/(x**2*exp(-4*x) + 2*x*exp(-2*x) 
 + 1)
 

Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 2\right )}}{x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \] Input:

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex 
p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( 
x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) 
^2*x^3),x, algorithm="maxima")
 

Output:

4*e^(8/9*x^2*e^(-2*x) + 2)/(x^2 + 2*x*e^(2*x) + e^(4*x))
 

Giac [F]

\[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\int { -\frac {8 \, {\left (8 \, {\left (x^{3} - x^{2}\right )} e^{2} + 2 \, {\left (4 \, {\left (x^{2} - x\right )} e^{2} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (2 \, x\right )} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )}\right )}}{9 \, {\left (x^{3} e^{\left (2 \, x\right )} + 3 \, x^{2} e^{\left (4 \, x\right )} + 3 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}\right )}} \,d x } \] Input:

integrate(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*ex 
p(1)^2*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp( 
x)^2*exp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x) 
^2*x^3),x, algorithm="giac")
 

Output:

integrate(-8/9*(8*(x^3 - x^2)*e^2 + 2*(4*(x^2 - x)*e^2 + 9*e^(2*x + 2))*e^ 
(2*x) + 9*e^(2*x + 2))*e^(8/9*x^2*e^(-2*x))/(x^3*e^(2*x) + 3*x^2*e^(4*x) + 
 3*x*e^(6*x) + e^(8*x)), x)
 

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {4\,{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{-2\,x}}{9}}}{{\mathrm {e}}^{4\,x}+2\,x\,{\mathrm {e}}^{2\,x}+x^2} \] Input:

int(-(exp((8*x^2*exp(-2*x))/9)*(72*exp(2*x)*exp(2) + exp(2*x)*(144*exp(2*x 
)*exp(2) - exp(2)*(64*x - 64*x^2)) - exp(2)*(64*x^2 - 64*x^3)))/(9*exp(8*x 
) + 27*x*exp(6*x) + 9*x^3*exp(2*x) + 27*x^2*exp(4*x)),x)
 

Output:

(4*exp(2)*exp((8*x^2*exp(-2*x))/9))/(exp(4*x) + 2*x*exp(2*x) + x^2)
 

Reduce [F]

\[ \int \frac {e^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx=\frac {8 e^{2} \left (-18 \left (\int \frac {e^{\frac {18 e^{2 x} x +8 x^{2}}{9 e^{2 x}}}}{e^{6 x}+3 e^{4 x} x +3 e^{2 x} x^{2}+x^{3}}d x \right )-9 \left (\int \frac {e^{\frac {8 x^{2}}{9 e^{2 x}}}}{e^{6 x}+3 e^{4 x} x +3 e^{2 x} x^{2}+x^{3}}d x \right )-8 \left (\int \frac {e^{\frac {8 x^{2}}{9 e^{2 x}}} x^{3}}{e^{8 x}+3 e^{6 x} x +3 e^{4 x} x^{2}+e^{2 x} x^{3}}d x \right )+8 \left (\int \frac {e^{\frac {8 x^{2}}{9 e^{2 x}}} x^{2}}{e^{8 x}+3 e^{6 x} x +3 e^{4 x} x^{2}+e^{2 x} x^{3}}d x \right )-8 \left (\int \frac {e^{\frac {8 x^{2}}{9 e^{2 x}}} x^{2}}{e^{6 x}+3 e^{4 x} x +3 e^{2 x} x^{2}+x^{3}}d x \right )+8 \left (\int \frac {e^{\frac {8 x^{2}}{9 e^{2 x}}} x}{e^{6 x}+3 e^{4 x} x +3 e^{2 x} x^{2}+x^{3}}d x \right )\right )}{9} \] Input:

int(((-144*exp(1)^2*exp(x)^2+(-64*x^2+64*x)*exp(1)^2)*exp(2*x)-72*exp(1)^2 
*exp(x)^2+(-64*x^3+64*x^2)*exp(1)^2)*exp(4/9*x^2/exp(x)^2)^2/(9*exp(x)^2*e 
xp(2*x)^3+27*x*exp(x)^2*exp(2*x)^2+27*x^2*exp(x)^2*exp(2*x)+9*exp(x)^2*x^3 
),x)
 

Output:

(8*e**2*( - 18*int(e**((18*e**(2*x)*x + 8*x**2)/(9*e**(2*x)))/(e**(6*x) + 
3*e**(4*x)*x + 3*e**(2*x)*x**2 + x**3),x) - 9*int(e**((8*x**2)/(9*e**(2*x) 
))/(e**(6*x) + 3*e**(4*x)*x + 3*e**(2*x)*x**2 + x**3),x) - 8*int((e**((8*x 
**2)/(9*e**(2*x)))*x**3)/(e**(8*x) + 3*e**(6*x)*x + 3*e**(4*x)*x**2 + e**( 
2*x)*x**3),x) + 8*int((e**((8*x**2)/(9*e**(2*x)))*x**2)/(e**(8*x) + 3*e**( 
6*x)*x + 3*e**(4*x)*x**2 + e**(2*x)*x**3),x) - 8*int((e**((8*x**2)/(9*e**( 
2*x)))*x**2)/(e**(6*x) + 3*e**(4*x)*x + 3*e**(2*x)*x**2 + x**3),x) + 8*int 
((e**((8*x**2)/(9*e**(2*x)))*x)/(e**(6*x) + 3*e**(4*x)*x + 3*e**(2*x)*x**2 
 + x**3),x)))/9