\(\int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} (-1+x+30 x^2)}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} (e^{3 x} (3-2 x)+e^{2 x} (4-5 x+2 x^2))}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx\) [2923]

Optimal result

Integrand size = 217, antiderivative size = 35 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=5+e^{1+\frac {e^{2 x} \left (-x+x^2\right )}{5 x \left (e^x+x\right )^4}}-x \] Output:

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

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=\frac {1}{5} \left (5 e^{\frac {1}{5} \left (5+\frac {(-1+x) x^2}{\left (e^x+x\right )^4}-\frac {2 (-1+x) x}{\left (e^x+x\right )^3}+\frac {-1+x}{\left (e^x+x\right )^2}\right )}-5 x\right ) \] Input:

Integrate[(-5*E^(5*x) - 25*E^(4*x)*x - 50*E^(3*x)*x^2 - 50*E^(2*x)*x^3 - 2 
5*E^x*x^4 - 5*x^5 + E^((5*E^(4*x) + 20*E^(3*x)*x + 20*E^x*x^3 + 5*x^4 + E^ 
(2*x)*(-1 + x + 30*x^2))/(5*E^(4*x) + 20*E^(3*x)*x + 30*E^(2*x)*x^2 + 20*E 
^x*x^3 + 5*x^4))*(E^(3*x)*(3 - 2*x) + E^(2*x)*(4 - 5*x + 2*x^2)))/(5*E^(5* 
x) + 25*E^(4*x)*x + 50*E^(3*x)*x^2 + 50*E^(2*x)*x^3 + 25*E^x*x^4 + 5*x^5), 
x]
 

Output:

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

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.

\(\displaystyle \int \frac {\left (e^{2 x} \left (2 x^2-5 x+4\right )+e^{3 x} (3-2 x)\right ) \exp \left (\frac {5 x^4+20 e^x x^3+e^{2 x} \left (30 x^2+x-1\right )+20 e^{3 x} x+5 e^{4 x}}{5 x^4+20 e^x x^3+30 e^{2 x} x^2+20 e^{3 x} x+5 e^{4 x}}\right )-5 x^5-25 e^x x^4-50 e^{2 x} x^3-50 e^{3 x} x^2-25 e^{4 x} x-5 e^{5 x}}{5 x^5+25 e^x x^4+50 e^{2 x} x^3+50 e^{3 x} x^2+25 e^{4 x} x+5 e^{5 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (e^{2 x} \left (2 x^2-5 x+4\right )+e^{3 x} (3-2 x)\right ) \exp \left (\frac {5 x^4+20 e^x x^3+e^{2 x} \left (30 x^2+x-1\right )+20 e^{3 x} x+5 e^{4 x}}{5 x^4+20 e^x x^3+30 e^{2 x} x^2+20 e^{3 x} x+5 e^{4 x}}\right )-5 x^5-25 e^x x^4-50 e^{2 x} x^3-50 e^{3 x} x^2-25 e^{4 x} x-5 e^{5 x}}{5 \left (x+e^x\right )^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int -\frac {5 x^5+25 e^x x^4+50 e^{2 x} x^3+50 e^{3 x} x^2+25 e^{4 x} x+5 e^{5 x}-\exp \left (\frac {5 x^4+20 e^x x^3+20 e^{3 x} x+5 e^{4 x}-e^{2 x} \left (-30 x^2-x+1\right )}{5 \left (x^4+4 e^x x^3+6 e^{2 x} x^2+4 e^{3 x} x+e^{4 x}\right )}\right ) \left (e^{3 x} (3-2 x)+e^{2 x} \left (2 x^2-5 x+4\right )\right )}{\left (x+e^x\right )^5}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{5} \int \frac {5 x^5+25 e^x x^4+50 e^{2 x} x^3+50 e^{3 x} x^2+25 e^{4 x} x+5 e^{5 x}-\exp \left (\frac {5 x^4+20 e^x x^3+20 e^{3 x} x+5 e^{4 x}-e^{2 x} \left (-30 x^2-x+1\right )}{5 \left (x^4+4 e^x x^3+6 e^{2 x} x^2+4 e^{3 x} x+e^{4 x}\right )}\right ) \left (e^{3 x} (3-2 x)+e^{2 x} \left (2 x^2-5 x+4\right )\right )}{\left (x+e^x\right )^5}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{5} \int \left (\frac {5 x^5}{\left (x+e^x\right )^5}+\frac {25 e^x x^4}{\left (x+e^x\right )^5}+\frac {50 e^{2 x} x^3}{\left (x+e^x\right )^5}+\frac {50 e^{3 x} x^2}{\left (x+e^x\right )^5}+\frac {25 e^{4 x} x}{\left (x+e^x\right )^5}+\frac {\exp \left (\frac {10 x^5+40 e^x x^4+5 x^4+20 e^x x^3+60 e^{2 x} x^3+30 e^{2 x} x^2+40 e^{3 x} x^2+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x-e^{2 x}+5 e^{4 x}}{5 \left (x+e^x\right )^4}\right ) \left (-2 x^2+2 e^x x+5 x-3 e^x-4\right )}{\left (x+e^x\right )^5}+\frac {5 e^{5 x}}{\left (x+e^x\right )^5}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {1}{5} \int \left (\frac {5 x^5}{\left (x+e^x\right )^5}+\frac {25 e^x x^4}{\left (x+e^x\right )^5}+\frac {50 e^{2 x} x^3}{\left (x+e^x\right )^5}+\frac {50 e^{3 x} x^2}{\left (x+e^x\right )^5}+\frac {25 e^{4 x} x}{\left (x+e^x\right )^5}+\frac {\exp \left (\frac {10 x^5+40 e^x x^4+5 x^4+20 e^x x^3+60 e^{2 x} x^3+30 e^{2 x} x^2+40 e^{3 x} x^2+e^{2 x} x+20 e^{3 x} x+10 e^{4 x} x-e^{2 x}+5 e^{4 x}}{5 \left (x+e^x\right )^4}\right ) \left (-2 x^2+2 e^x x+5 x-3 e^x-4\right )}{\left (x+e^x\right )^5}+\frac {5 e^{5 x}}{\left (x+e^x\right )^5}\right )dx\)

Input:

Int[(-5*E^(5*x) - 25*E^(4*x)*x - 50*E^(3*x)*x^2 - 50*E^(2*x)*x^3 - 25*E^x* 
x^4 - 5*x^5 + E^((5*E^(4*x) + 20*E^(3*x)*x + 20*E^x*x^3 + 5*x^4 + E^(2*x)* 
(-1 + x + 30*x^2))/(5*E^(4*x) + 20*E^(3*x)*x + 30*E^(2*x)*x^2 + 20*E^x*x^3 
 + 5*x^4))*(E^(3*x)*(3 - 2*x) + E^(2*x)*(4 - 5*x + 2*x^2)))/(5*E^(5*x) + 2 
5*E^(4*x)*x + 50*E^(3*x)*x^2 + 50*E^(2*x)*x^3 + 25*E^x*x^4 + 5*x^5),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(30)=60\).

Time = 1.60 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.29

method	result	size
parallelrisch	\(-x +{\mathrm e}^{\frac {5 \,{\mathrm e}^{4 x}+20 x \,{\mathrm e}^{3 x}+\left (30 x^{2}+x -1\right ) {\mathrm e}^{2 x}+20 \,{\mathrm e}^{x} x^{3}+5 x^{4}}{5 \,{\mathrm e}^{4 x}+20 x \,{\mathrm e}^{3 x}+30 \,{\mathrm e}^{2 x} x^{2}+20 \,{\mathrm e}^{x} x^{3}+5 x^{4}}}\)	\(80\)
risch	\(-x +{\mathrm e}^{\frac {20 \,{\mathrm e}^{x} x^{3}+5 x^{4}+30 \,{\mathrm e}^{2 x} x^{2}+20 x \,{\mathrm e}^{3 x}+x \,{\mathrm e}^{2 x}+5 \,{\mathrm e}^{4 x}-{\mathrm e}^{2 x}}{5 \,{\mathrm e}^{4 x}+20 x \,{\mathrm e}^{3 x}+30 \,{\mathrm e}^{2 x} x^{2}+20 \,{\mathrm e}^{x} x^{3}+5 x^{4}}}\)	\(88\)

Input:

int((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x) 
^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30 
*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2*exp(x) 
^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*e 
xp(x)^3+50*exp(x)^2*x^3+25*exp(x)*x^4+5*x^5),x,method=_RETURNVERBOSE)
 

Output:

-x+exp(1/5/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x^4)*(5*exp( 
x)^4+20*x*exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.26 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=-x + e^{\left (\frac {5 \, x^{4} + 20 \, x^{3} e^{x} + 20 \, x e^{\left (3 \, x\right )} + {\left (30 \, x^{2} + x - 1\right )} e^{\left (2 \, x\right )} + 5 \, e^{\left (4 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} \] Input:

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x* 
exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x 
)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2* 
exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50 
*x^2*exp(x)^3+50*exp(x)^2*x^3+25*exp(x)*x^4+5*x^5),x, algorithm="fricas")
 

Output:

-x + e^(1/5*(5*x^4 + 20*x^3*e^x + 20*x*e^(3*x) + (30*x^2 + x - 1)*e^(2*x) 
+ 5*e^(4*x))/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).

Time = 0.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.37 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=- x + e^{\frac {5 x^{4} + 20 x^{3} e^{x} + 20 x e^{3 x} + \left (30 x^{2} + x - 1\right ) e^{2 x} + 5 e^{4 x}}{5 x^{4} + 20 x^{3} e^{x} + 30 x^{2} e^{2 x} + 20 x e^{3 x} + 5 e^{4 x}}} \] Input:

integrate((((3-2*x)*exp(x)**3+(2*x**2-5*x+4)*exp(x)**2)*exp((5*exp(x)**4+2 
0*x*exp(x)**3+(30*x**2+x-1)*exp(x)**2+20*exp(x)*x**3+5*x**4)/(5*exp(x)**4+ 
20*x*exp(x)**3+30*exp(x)**2*x**2+20*exp(x)*x**3+5*x**4))-5*exp(x)**5-25*x* 
exp(x)**4-50*x**2*exp(x)**3-50*exp(x)**2*x**3-25*exp(x)*x**4-5*x**5)/(5*ex 
p(x)**5+25*x*exp(x)**4+50*x**2*exp(x)**3+50*exp(x)**2*x**3+25*exp(x)*x**4+ 
5*x**5),x)
 

Output:

-x + exp((5*x**4 + 20*x**3*exp(x) + 20*x*exp(3*x) + (30*x**2 + x - 1)*exp( 
2*x) + 5*exp(4*x))/(5*x**4 + 20*x**3*exp(x) + 30*x**2*exp(2*x) + 20*x*exp( 
3*x) + 5*exp(4*x)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (30) = 60\).

Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.54 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=-{\left (x e^{\left (\frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} - e^{\left (-\frac {e^{\left (3 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}} + \frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{3} + 3 \, x^{2} e^{x} + 3 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )}} + 1\right )}\right )} e^{\left (-\frac {e^{\left (2 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} \] Input:

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x* 
exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x 
)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2* 
exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50 
*x^2*exp(x)^3+50*exp(x)^2*x^3+25*exp(x)*x^4+5*x^5),x, algorithm="maxima")
 

Output:

-(x*e^(1/5*e^(2*x)/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x 
))) - e^(-1/5*e^(3*x)/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^( 
4*x)) + 1/5*e^(2*x)/(x^3 + 3*x^2*e^x + 3*x*e^(2*x) + e^(3*x)) + 1))*e^(-1/ 
5*e^(2*x)/(x^4 + 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)))
 

Giac [F]

\[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=\int { -\frac {5 \, x^{5} + 25 \, x^{4} e^{x} + 50 \, x^{3} e^{\left (2 \, x\right )} + 50 \, x^{2} e^{\left (3 \, x\right )} + 25 \, x e^{\left (4 \, x\right )} + {\left ({\left (2 \, x - 3\right )} e^{\left (3 \, x\right )} - {\left (2 \, x^{2} - 5 \, x + 4\right )} e^{\left (2 \, x\right )}\right )} e^{\left (\frac {5 \, x^{4} + 20 \, x^{3} e^{x} + 20 \, x e^{\left (3 \, x\right )} + {\left (30 \, x^{2} + x - 1\right )} e^{\left (2 \, x\right )} + 5 \, e^{\left (4 \, x\right )}}{5 \, {\left (x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}\right )}}\right )} + 5 \, e^{\left (5 \, x\right )}}{5 \, {\left (x^{5} + 5 \, x^{4} e^{x} + 10 \, x^{3} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{\left (3 \, x\right )} + 5 \, x e^{\left (4 \, x\right )} + e^{\left (5 \, x\right )}\right )}} \,d x } \] Input:

integrate((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x* 
exp(x)^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x 
)^3+30*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2* 
exp(x)^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50 
*x^2*exp(x)^3+50*exp(x)^2*x^3+25*exp(x)*x^4+5*x^5),x, algorithm="giac")
 

Output:

integrate(-1/5*(5*x^5 + 25*x^4*e^x + 50*x^3*e^(2*x) + 50*x^2*e^(3*x) + 25* 
x*e^(4*x) + ((2*x - 3)*e^(3*x) - (2*x^2 - 5*x + 4)*e^(2*x))*e^(1/5*(5*x^4 
+ 20*x^3*e^x + 20*x*e^(3*x) + (30*x^2 + x - 1)*e^(2*x) + 5*e^(4*x))/(x^4 + 
 4*x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x))) + 5*e^(5*x))/(x^5 + 5 
*x^4*e^x + 10*x^3*e^(2*x) + 10*x^2*e^(3*x) + 5*x*e^(4*x) + e^(5*x)), x)
 

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 317, normalized size of antiderivative = 9.06 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx={\mathrm {e}}^{\frac {30\,x^2\,{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {5\,{\mathrm {e}}^{4\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{2\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {20\,x\,{\mathrm {e}}^{3\,x}}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {20\,x^3\,{\mathrm {e}}^x}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}\,{\mathrm {e}}^{\frac {5\,x^4}{5\,{\mathrm {e}}^{4\,x}+20\,x\,{\mathrm {e}}^{3\,x}+20\,x^3\,{\mathrm {e}}^x+30\,x^2\,{\mathrm {e}}^{2\,x}+5\,x^4}}-x \] Input:

int(-(5*exp(5*x) + 25*x*exp(4*x) + 25*x^4*exp(x) + 50*x^2*exp(3*x) + 50*x^ 
3*exp(2*x) - exp((5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + exp(2*x)*(x 
 + 30*x^2 - 1) + 5*x^4)/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x 
^2*exp(2*x) + 5*x^4))*(exp(2*x)*(2*x^2 - 5*x + 4) - exp(3*x)*(2*x - 3)) + 
5*x^5)/(5*exp(5*x) + 25*x*exp(4*x) + 25*x^4*exp(x) + 50*x^2*exp(3*x) + 50* 
x^3*exp(2*x) + 5*x^5),x)
 

Output:

exp((30*x^2*exp(2*x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2 
*exp(2*x) + 5*x^4))*exp(-exp(2*x)/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp 
(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((5*exp(4*x))/(5*exp(4*x) + 20*x*exp(3* 
x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((x*exp(2*x))/(5*exp(4*x 
) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5*x^4))*exp((20*x*ex 
p(3*x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30*x^2*exp(2*x) + 5* 
x^4))*exp((20*x^3*exp(x))/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*exp(x) + 30 
*x^2*exp(2*x) + 5*x^4))*exp((5*x^4)/(5*exp(4*x) + 20*x*exp(3*x) + 20*x^3*e 
xp(x) + 30*x^2*exp(2*x) + 5*x^4)) - x
 

Reduce [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 157, normalized size of antiderivative = 4.49 \[ \int \frac {-5 e^{5 x}-25 e^{4 x} x-50 e^{3 x} x^2-50 e^{2 x} x^3-25 e^x x^4-5 x^5+e^{\frac {5 e^{4 x}+20 e^{3 x} x+20 e^x x^3+5 x^4+e^{2 x} \left (-1+x+30 x^2\right )}{5 e^{4 x}+20 e^{3 x} x+30 e^{2 x} x^2+20 e^x x^3+5 x^4}} \left (e^{3 x} (3-2 x)+e^{2 x} \left (4-5 x+2 x^2\right )\right )}{5 e^{5 x}+25 e^{4 x} x+50 e^{3 x} x^2+50 e^{2 x} x^3+25 e^x x^4+5 x^5} \, dx=\frac {-e^{\frac {e^{2 x}}{5 e^{4 x}+20 e^{3 x} x +30 e^{2 x} x^{2}+20 e^{x} x^{3}+5 x^{4}}} x +e^{\frac {e^{2 x} x}{5 e^{4 x}+20 e^{3 x} x +30 e^{2 x} x^{2}+20 e^{x} x^{3}+5 x^{4}}} e}{e^{\frac {e^{2 x}}{5 e^{4 x}+20 e^{3 x} x +30 e^{2 x} x^{2}+20 e^{x} x^{3}+5 x^{4}}}} \] Input:

int((((3-2*x)*exp(x)^3+(2*x^2-5*x+4)*exp(x)^2)*exp((5*exp(x)^4+20*x*exp(x) 
^3+(30*x^2+x-1)*exp(x)^2+20*exp(x)*x^3+5*x^4)/(5*exp(x)^4+20*x*exp(x)^3+30 
*exp(x)^2*x^2+20*exp(x)*x^3+5*x^4))-5*exp(x)^5-25*x*exp(x)^4-50*x^2*exp(x) 
^3-50*exp(x)^2*x^3-25*exp(x)*x^4-5*x^5)/(5*exp(x)^5+25*x*exp(x)^4+50*x^2*e 
xp(x)^3+50*exp(x)^2*x^3+25*exp(x)*x^4+5*x^5),x)
 

Output:

( - e**(e**(2*x)/(5*e**(4*x) + 20*e**(3*x)*x + 30*e**(2*x)*x**2 + 20*e**x* 
x**3 + 5*x**4))*x + e**((e**(2*x)*x)/(5*e**(4*x) + 20*e**(3*x)*x + 30*e**( 
2*x)*x**2 + 20*e**x*x**3 + 5*x**4))*e)/e**(e**(2*x)/(5*e**(4*x) + 20*e**(3 
*x)*x + 30*e**(2*x)*x**2 + 20*e**x*x**3 + 5*x**4))