\(\int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3)}+\frac {1}{2} (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3)} (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} (-15 x^2+e^{\frac {e^5+x^2}{x}} (10 e^5 x-10 x^3))) \, dx\) [394]

Optimal result

Integrand size = 127, antiderivative size = 35 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=e^{\frac {5}{4} e^{\frac {1}{2} x \left (x-e^{2 e^{\frac {e^5}{x}+x}} x^2\right )}} \] Output:

exp(5/4*exp(1/2*x*(x-exp(exp(exp(5)/x+x))^2*x^2)))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=e^{\frac {5}{4} e^{-\frac {1}{2} x^2 \left (-1+e^{2 e^{\frac {e^5}{x}+x}} x\right )}} \] Input:

Integrate[(E^((5*E^((x^2 - E^(2*E^((E^5 + x^2)/x))*x^3)/2))/4 + (x^2 - E^( 
2*E^((E^5 + x^2)/x))*x^3)/2)*(10*x + E^(2*E^((E^5 + x^2)/x))*(-15*x^2 + E^ 
((E^5 + x^2)/x)*(10*E^5*x - 10*x^3))))/8,x]
 

Output:

E^(5/(4*E^((x^2*(-1 + E^(2*E^(E^5/x + 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.

\(\displaystyle \int \frac {1}{8} \left (e^{2 e^{\frac {x^2+e^5}{x}}} \left (e^{\frac {x^2+e^5}{x}} \left (10 e^5 x-10 x^3\right )-15 x^2\right )+10 x\right ) \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int 5 \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) \left (2 x-e^{2 e^{\frac {x^2+e^5}{x}}} \left (3 x^2-2 e^{\frac {x^2+e^5}{x}} \left (e^5 x-x^3\right )\right )\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} \int \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) \left (2 x-e^{2 e^{\frac {x^2+e^5}{x}}} \left (3 x^2-2 e^{\frac {x^2+e^5}{x}} \left (e^5 x-x^3\right )\right )\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {5}{8} \int \left (2 \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) x-\exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+2 e^{x+\frac {e^5}{x}}+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) x \left (2 e^{x+\frac {e^5}{x}} x^2+3 x-2 e^{x+5+\frac {e^5}{x}}\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5}{8} \left (2 \int \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) xdx+2 \int \exp \left (x+2 e^{x+\frac {e^5}{x}}+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+5+\frac {e^5}{x}\right ) xdx-3 \int \exp \left (\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+2 e^{x+\frac {e^5}{x}}+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}\right ) x^2dx-2 \int \exp \left (x+2 e^{x+\frac {e^5}{x}}+\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {x^2+e^5}{x}}} x^3\right )+\frac {e^5}{x}\right ) x^3dx\right )\)

Input:

Int[(E^((5*E^((x^2 - E^(2*E^((E^5 + x^2)/x))*x^3)/2))/4 + (x^2 - E^(2*E^(( 
E^5 + x^2)/x))*x^3)/2)*(10*x + E^(2*E^((E^5 + x^2)/x))*(-15*x^2 + E^((E^5 
+ x^2)/x)*(10*E^5*x - 10*x^3))))/8,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80

\[{\mathrm e}^{\frac {5 \,{\mathrm e}^{-\frac {x^{2} \left ({\mathrm e}^{2 \,{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{5}}{x}}} x -1\right )}{2}}}{4}}\]

Input:

int(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp((x^2+ex 
p(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*exp(5/4 
*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x)
 

Output:

exp(5/4*exp(-1/2*x^2*(exp(2*exp((x^2+exp(5))/x))*x-1)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=e^{\left (\frac {5}{4} \, e^{\left (-\frac {1}{2} \, x^{3} e^{\left (2 \, e^{\left (\frac {x^{2} + e^{5}}{x}\right )}\right )} + \frac {1}{2} \, x^{2}\right )}\right )} \] Input:

integrate(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp(( 
x^2+exp(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*e 
xp(5/4*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x, algorithm="fri 
cas")
 

Output:

e^(5/4*e^(-1/2*x^3*e^(2*e^((x^2 + e^5)/x)) + 1/2*x^2))
 

Sympy [A] (verification not implemented)

Time = 22.72 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=e^{\frac {5 e^{- \frac {x^{3} e^{2 e^{\frac {x^{2} + e^{5}}{x}}}}{2} + \frac {x^{2}}{2}}}{4}} \] Input:

integrate(1/8*(((10*x*exp(5)-10*x**3)*exp((x**2+exp(5))/x)-15*x**2)*exp(ex 
p((x**2+exp(5))/x))**2+10*x)*exp(-1/2*x**3*exp(exp((x**2+exp(5))/x))**2+1/ 
2*x**2)*exp(5/4*exp(-1/2*x**3*exp(exp((x**2+exp(5))/x))**2+1/2*x**2)),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=e^{\left (\frac {5}{4} \, e^{\left (-\frac {1}{2} \, x^{3} e^{\left (2 \, e^{\left (x + \frac {e^{5}}{x}\right )}\right )} + \frac {1}{2} \, x^{2}\right )}\right )} \] Input:

integrate(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp(( 
x^2+exp(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*e 
xp(5/4*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x, algorithm="max 
ima")
 

Output:

e^(5/4*e^(-1/2*x^3*e^(2*e^(x + e^5/x)) + 1/2*x^2))
 

Giac [F]

\[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=\int { -\frac {5}{8} \, {\left ({\left (3 \, x^{2} + 2 \, {\left (x^{3} - x e^{5}\right )} e^{\left (\frac {x^{2} + e^{5}}{x}\right )}\right )} e^{\left (2 \, e^{\left (\frac {x^{2} + e^{5}}{x}\right )}\right )} - 2 \, x\right )} e^{\left (-\frac {1}{2} \, x^{3} e^{\left (2 \, e^{\left (\frac {x^{2} + e^{5}}{x}\right )}\right )} + \frac {1}{2} \, x^{2} + \frac {5}{4} \, e^{\left (-\frac {1}{2} \, x^{3} e^{\left (2 \, e^{\left (\frac {x^{2} + e^{5}}{x}\right )}\right )} + \frac {1}{2} \, x^{2}\right )}\right )} \,d x } \] Input:

integrate(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp(( 
x^2+exp(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*e 
xp(5/4*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x, algorithm="gia 
c")
 

Output:

integrate(-5/8*((3*x^2 + 2*(x^3 - x*e^5)*e^((x^2 + e^5)/x))*e^(2*e^((x^2 + 
 e^5)/x)) - 2*x)*e^(-1/2*x^3*e^(2*e^((x^2 + e^5)/x)) + 1/2*x^2 + 5/4*e^(-1 
/2*x^3*e^(2*e^((x^2 + e^5)/x)) + 1/2*x^2)), x)
 

Mupad [B] (verification not implemented)

Time = 3.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx={\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{-\frac {x^3\,{\mathrm {e}}^{2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^5}{x}}\,{\mathrm {e}}^x}}{2}}\,{\mathrm {e}}^{\frac {x^2}{2}}}{4}} \] Input:

int((exp(x^2/2 - (x^3*exp(2*exp((exp(5) + x^2)/x)))/2)*exp((5*exp(x^2/2 - 
(x^3*exp(2*exp((exp(5) + x^2)/x)))/2))/4)*(10*x + exp(2*exp((exp(5) + x^2) 
/x))*(exp((exp(5) + x^2)/x)*(10*x*exp(5) - 10*x^3) - 15*x^2)))/8,x)
 

Output:

exp((5*exp(-(x^3*exp(2*exp(exp(5)/x)*exp(x)))/2)*exp(x^2/2))/4)
 

Reduce [F]

\[ \int \frac {1}{8} e^{\frac {5}{4} e^{\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )}+\frac {1}{2} \left (x^2-e^{2 e^{\frac {e^5+x^2}{x}}} x^3\right )} \left (10 x+e^{2 e^{\frac {e^5+x^2}{x}}} \left (-15 x^2+e^{\frac {e^5+x^2}{x}} \left (10 e^5 x-10 x^3\right )\right )\right ) \, dx=\int \frac {\left (\left (\left (10 x \,{\mathrm e}^{5}-10 x^{3}\right ) {\mathrm e}^{\frac {x^{2}+{\mathrm e}^{5}}{x}}-15 x^{2}\right ) \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{5}}{x}}}\right )^{2}+10 x \right ) {\mathrm e}^{-\frac {x^{3} \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{5}}{x}}}\right )^{2}}{2}+\frac {x^{2}}{2}} {\mathrm e}^{\frac {5 \,{\mathrm e}^{-\frac {x^{3} \left ({\mathrm e}^{{\mathrm e}^{\frac {x^{2}+{\mathrm e}^{5}}{x}}}\right )^{2}}{2}+\frac {x^{2}}{2}}}{4}}}{8}d x \] Input:

int(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp((x^2+ex 
p(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*exp(5/4 
*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x)
 

Output:

int(1/8*(((10*x*exp(5)-10*x^3)*exp((x^2+exp(5))/x)-15*x^2)*exp(exp((x^2+ex 
p(5))/x))^2+10*x)*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)*exp(5/4 
*exp(-1/2*x^3*exp(exp((x^2+exp(5))/x))^2+1/2*x^2)),x)