\(\int e^{-78+56 x-13 x^2+x^3+e^x (-16+8 x-x^2)} (-55+82 x-29 x^2+3 x^3+e^3 (56-26 x+3 x^2)+e^x (8-14 x+7 x^2-x^3+e^3 (-8+6 x-x^2))) \, dx\) [2872]

Optimal result

Integrand size = 92, antiderivative size = 29 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx=e^{2-(4-x) (-4+x) \left (-5-e^x+x\right )} \left (-1+e^3+x\right ) \] Output:

exp(2-(-4+x)*(4-x)*(-exp(x)+x-5))*(exp(3)+x-1)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx=e^{-78-e^x (-4+x)^2+56 x-13 x^2+x^3} \left (-1+e^3+x\right ) \] Input:

Integrate[E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*(-55 + 82* 
x - 29*x^2 + 3*x^3 + E^3*(56 - 26*x + 3*x^2) + E^x*(8 - 14*x + 7*x^2 - x^3 
 + E^3*(-8 + 6*x - x^2))),x]
 

Output:

E^(-78 - E^x*(-4 + x)^2 + 56*x - 13*x^2 + x^3)*(-1 + E^3 + 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.

\(\displaystyle \int e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-78} \left (3 x^3-29 x^2+e^3 \left (3 x^2-26 x+56\right )+e^x \left (-x^3+7 x^2+e^3 \left (-x^2+6 x-8\right )-14 x+8\right )+82 x-55\right ) \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (3 e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-78} x^3-29 e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-78} x^2+82 e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-78} x-55 e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-78}-e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+57 x-78} \left (x+e^3-1\right ) \left (x^2-6 x+8\right )+e^{x^3-13 x^2+e^x \left (-x^2+8 x-16\right )+56 x-75} \left (3 x^2-26 x+56\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -55 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-78}dx+56 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-75}dx+8 \left (1-e^3\right ) \int e^{x^3-13 x^2+57 x+e^x \left (-x^2+8 x-16\right )-78}dx+82 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-78} xdx-26 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-75} xdx-2 \left (7-3 e^3\right ) \int e^{x^3-13 x^2+57 x+e^x \left (-x^2+8 x-16\right )-78} xdx-29 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-78} x^2dx+3 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-75} x^2dx+\left (7-e^3\right ) \int e^{x^3-13 x^2+57 x+e^x \left (-x^2+8 x-16\right )-78} x^2dx+3 \int e^{x^3-13 x^2+56 x+e^x \left (-x^2+8 x-16\right )-78} x^3dx-\int e^{x^3-13 x^2+57 x+e^x \left (-x^2+8 x-16\right )-78} x^3dx\)

Input:

Int[E^(-78 + 56*x - 13*x^2 + x^3 + E^x*(-16 + 8*x - x^2))*(-55 + 82*x - 29 
*x^2 + 3*x^3 + E^3*(56 - 26*x + 3*x^2) + E^x*(8 - 14*x + 7*x^2 - x^3 + E^3 
*(-8 + 6*x - x^2))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28

method	result	size
risch	\(\left ({\mathrm e}^{3}+x -1\right ) {\mathrm e}^{-{\mathrm e}^{x} x^{2}+x^{3}+8 \,{\mathrm e}^{x} x -13 x^{2}-16 \,{\mathrm e}^{x}+56 x -78}\)	\(37\)
norman	\(x \,{\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}+\left ({\mathrm e}^{3}-1\right ) {\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}\)	\(63\)
parallelrisch	\({\mathrm e}^{3} {\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}+x \,{\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}-{\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}\)	\(90\)

Input:

int((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+ 
3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x,metho 
d=_RETURNVERBOSE)
 

Output:

(exp(3)+x-1)*exp(-exp(x)*x^2+x^3+8*exp(x)*x-13*x^2-16*exp(x)+56*x-78)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx={\left (x + e^{3} - 1\right )} e^{\left (x^{3} - 13 \, x^{2} - {\left (x^{2} - 8 \, x + 16\right )} e^{x} + 56 \, x - 78\right )} \] Input:

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*e 
xp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x 
, algorithm="fricas")
 

Output:

(x + e^3 - 1)*e^(x^3 - 13*x^2 - (x^2 - 8*x + 16)*e^x + 56*x - 78)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx=\left (x - 1 + e^{3}\right ) e^{x^{3} - 13 x^{2} + 56 x + \left (- x^{2} + 8 x - 16\right ) e^{x} - 78} \] Input:

integrate((((-x**2+6*x-8)*exp(3)-x**3+7*x**2-14*x+8)*exp(x)+(3*x**2-26*x+5 
6)*exp(3)+3*x**3-29*x**2+82*x-55)*exp((-x**2+8*x-16)*exp(x)+x**3-13*x**2+5 
6*x-78),x)
 

Output:

(x - 1 + exp(3))*exp(x**3 - 13*x**2 + 56*x + (-x**2 + 8*x - 16)*exp(x) - 7 
8)
 

Maxima [A] (verification not implemented)

Time = 31.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx={\left (x + e^{3} - 1\right )} e^{\left (x^{3} - x^{2} e^{x} - 13 \, x^{2} + 8 \, x e^{x} + 56 \, x - 16 \, e^{x} - 78\right )} \] Input:

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*e 
xp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x 
, algorithm="maxima")
 

Output:

(x + e^3 - 1)*e^(x^3 - x^2*e^x - 13*x^2 + 8*x*e^x + 56*x - 16*e^x - 78)
 

Giac [F]

\[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx=\int { {\left (3 \, x^{3} - 29 \, x^{2} + {\left (3 \, x^{2} - 26 \, x + 56\right )} e^{3} - {\left (x^{3} - 7 \, x^{2} + {\left (x^{2} - 6 \, x + 8\right )} e^{3} + 14 \, x - 8\right )} e^{x} + 82 \, x - 55\right )} e^{\left (x^{3} - 13 \, x^{2} - {\left (x^{2} - 8 \, x + 16\right )} e^{x} + 56 \, x - 78\right )} \,d x } \] Input:

integrate((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*e 
xp(3)+3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x 
, algorithm="giac")
 

Output:

integrate((3*x^3 - 29*x^2 + (3*x^2 - 26*x + 56)*e^3 - (x^3 - 7*x^2 + (x^2 
- 6*x + 8)*e^3 + 14*x - 8)*e^x + 82*x - 55)*e^(x^3 - 13*x^2 - (x^2 - 8*x + 
 16)*e^x + 56*x - 78), x)
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.83 \[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx={\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-75}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x}-{\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-78}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x}+x\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{56\,x}\,{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{-78}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-13\,x^2}\,{\mathrm {e}}^{-16\,{\mathrm {e}}^x} \] Input:

int(exp(56*x - exp(x)*(x^2 - 8*x + 16) - 13*x^2 + x^3 - 78)*(82*x + exp(3) 
*(3*x^2 - 26*x + 56) - exp(x)*(14*x + exp(3)*(x^2 - 6*x + 8) - 7*x^2 + x^3 
 - 8) - 29*x^2 + 3*x^3 - 55),x)
 

Output:

exp(8*x*exp(x))*exp(56*x)*exp(x^3)*exp(-75)*exp(-x^2*exp(x))*exp(-13*x^2)* 
exp(-16*exp(x)) - exp(8*x*exp(x))*exp(56*x)*exp(x^3)*exp(-78)*exp(-x^2*exp 
(x))*exp(-13*x^2)*exp(-16*exp(x)) + x*exp(8*x*exp(x))*exp(56*x)*exp(x^3)*e 
xp(-78)*exp(-x^2*exp(x))*exp(-13*x^2)*exp(-16*exp(x))
 

Reduce [F]

\[ \int e^{-78+56 x-13 x^2+x^3+e^x \left (-16+8 x-x^2\right )} \left (-55+82 x-29 x^2+3 x^3+e^3 \left (56-26 x+3 x^2\right )+e^x \left (8-14 x+7 x^2-x^3+e^3 \left (-8+6 x-x^2\right )\right )\right ) \, dx=\int \left (\left (\left (-x^{2}+6 x -8\right ) {\mathrm e}^{3}-x^{3}+7 x^{2}-14 x +8\right ) {\mathrm e}^{x}+\left (3 x^{2}-26 x +56\right ) {\mathrm e}^{3}+3 x^{3}-29 x^{2}+82 x -55\right ) {\mathrm e}^{\left (-x^{2}+8 x -16\right ) {\mathrm e}^{x}+x^{3}-13 x^{2}+56 x -78}d x \] Input:

int((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+ 
3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x)
 

Output:

int((((-x^2+6*x-8)*exp(3)-x^3+7*x^2-14*x+8)*exp(x)+(3*x^2-26*x+56)*exp(3)+ 
3*x^3-29*x^2+82*x-55)*exp((-x^2+8*x-16)*exp(x)+x^3-13*x^2+56*x-78),x)