\(\int \frac {e^{\frac {e^{5+4 x} (5 x+x^2)+e^5 (x+6 x^2+6 x^3+x^4)}{e^{4 x}+x+x^2}} (e^{5+8 x} (5+2 x)+e^{5+4 x} (1+8 x+10 x^2+4 x^3)+e^5 (5 x^2+12 x^3+9 x^4+2 x^5))}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} (2 x+2 x^2)} \, dx\) [2970]

Optimal result

Integrand size = 147, antiderivative size = 26 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{e^5 x \left (5+x+\frac {1+x}{e^{4 x}+x+x^2}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{5 e^5 x+e^5 x^2+\frac {e^5 x+e^5 x^2}{e^{4 x}+x+x^2}} \] Input:

Integrate[(E^((E^(5 + 4*x)*(5*x + x^2) + E^5*(x + 6*x^2 + 6*x^3 + x^4))/(E 
^(4*x) + x + x^2))*(E^(5 + 8*x)*(5 + 2*x) + E^(5 + 4*x)*(1 + 8*x + 10*x^2 
+ 4*x^3) + E^5*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5)))/(E^(8*x) + x^2 + 2*x^3 + 
 x^4 + E^(4*x)*(2*x + 2*x^2)),x]
 

Output:

E^(5*E^5*x + E^5*x^2 + (E^5*x + E^5*x^2)/(E^(4*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.

\(\displaystyle \int \frac {\left (e^{4 x+5} \left (4 x^3+10 x^2+8 x+1\right )+e^5 \left (2 x^5+9 x^4+12 x^3+5 x^2\right )+e^{8 x+5} (2 x+5)\right ) \exp \left (\frac {e^{4 x+5} \left (x^2+5 x\right )+e^5 \left (x^4+6 x^3+6 x^2+x\right )}{x^2+x+e^{4 x}}\right )}{x^4+2 x^3+x^2+e^{4 x} \left (2 x^2+2 x\right )+e^{8 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (e^{4 x+5} \left (4 x^3+10 x^2+8 x+1\right )+e^5 \left (2 x^5+9 x^4+12 x^3+5 x^2\right )+e^{8 x+5} (2 x+5)\right ) \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}\right )}{\left (x^2+x+e^{4 x}\right )^2}dx\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(E^((E^(5 + 4*x)*(5*x + x^2) + E^5*(x + 6*x^2 + 6*x^3 + x^4))/(E^(4*x) 
 + x + x^2))*(E^(5 + 8*x)*(5 + 2*x) + E^(5 + 4*x)*(1 + 8*x + 10*x^2 + 4*x^ 
3) + E^5*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5)))/(E^(8*x) + x^2 + 2*x^3 + x^4 + 
 E^(4*x)*(2*x + 2*x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62

method	result	size
parallelrisch	\({\mathrm e}^{\frac {{\mathrm e}^{5} x \left (x^{3}+x \,{\mathrm e}^{4 x}+6 x^{2}+5 \,{\mathrm e}^{4 x}+6 x +1\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\)	\(42\)
risch	\({\mathrm e}^{\frac {x \left (x^{3} {\mathrm e}^{5}+6 x^{2} {\mathrm e}^{5}+6 x \,{\mathrm e}^{5}+x \,{\mathrm e}^{4 x +5}+{\mathrm e}^{5}+5 \,{\mathrm e}^{4 x +5}\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\)	\(52\)

Input:

int(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+(2*x^5 
+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x^3+6*x 
^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4+2*x^3 
+x^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {{\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + x\right )} e^{10} + {\left (x^{2} + 5 \, x\right )} e^{\left (4 \, x + 10\right )}}{{\left (x^{2} + x\right )} e^{5} + e^{\left (4 \, x + 5\right )}}\right )} \] Input:

integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ 
(2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x 
^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 
+2*x^3+x^2),x, algorithm="fricas")
 

Output:

e^(((x^4 + 6*x^3 + 6*x^2 + x)*e^10 + (x^2 + 5*x)*e^(4*x + 10))/((x^2 + x)* 
e^5 + e^(4*x + 5)))
 

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\frac {\left (x^{2} + 5 x\right ) e^{5} e^{4 x} + \left (x^{4} + 6 x^{3} + 6 x^{2} + x\right ) e^{5}}{x^{2} + x + e^{4 x}}} \] Input:

integrate(((5+2*x)*exp(5)*exp(4*x)**2+(4*x**3+10*x**2+8*x+1)*exp(5)*exp(4* 
x)+(2*x**5+9*x**4+12*x**3+5*x**2)*exp(5))*exp(((x**2+5*x)*exp(5)*exp(4*x)+ 
(x**4+6*x**3+6*x**2+x)*exp(5))/(exp(4*x)+x**2+x))/(exp(4*x)**2+(2*x**2+2*x 
)*exp(4*x)+x**4+2*x**3+x**2),x)
 

Output:

exp(((x**2 + 5*x)*exp(5)*exp(4*x) + (x**4 + 6*x**3 + 6*x**2 + x)*exp(5))/( 
x**2 + x + exp(4*x)))
 

Maxima [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (x^{2} e^{5} + 5 \, x e^{5} - \frac {e^{\left (4 \, x + 5\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} + e^{5}\right )} \] Input:

integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ 
(2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x 
^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 
+2*x^3+x^2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {x^{4} e^{5} + 6 \, x^{3} e^{5} + 6 \, x^{2} e^{5} + x^{2} e^{\left (4 \, x + 5\right )} + 5 \, x^{2} + x e^{5} + 5 \, x e^{\left (4 \, x + 5\right )} + 5 \, x + 5 \, e^{\left (4 \, x\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} - 5\right )} \] Input:

integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ 
(2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x 
^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 
+2*x^3+x^2),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.81 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.46 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx={\mathrm {e}}^{\frac {5\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^2\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^3\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}} \] Input:

int((exp((exp(5)*(x + 6*x^2 + 6*x^3 + x^4) + exp(4*x)*exp(5)*(5*x + x^2))/ 
(x + exp(4*x) + x^2))*(exp(5)*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5) + exp(8*x)* 
exp(5)*(2*x + 5) + exp(4*x)*exp(5)*(8*x + 10*x^2 + 4*x^3 + 1)))/(exp(8*x) 
+ exp(4*x)*(2*x + 2*x^2) + x^2 + 2*x^3 + x^4),x)
 

Output:

exp((5*x*exp(4*x)*exp(5))/(x + exp(4*x) + x^2))*exp((x^4*exp(5))/(x + exp( 
4*x) + x^2))*exp((6*x^2*exp(5))/(x + exp(4*x) + x^2))*exp((6*x^3*exp(5))/( 
x + exp(4*x) + x^2))*exp((x^2*exp(4*x)*exp(5))/(x + exp(4*x) + x^2))*exp(( 
x*exp(5))/(x + exp(4*x) + x^2))
 

Reduce [F]

\[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx =\text {Too large to display} \] Input:

int(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+(2*x^5 
+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x^3+6*x 
^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4+2*x^3 
+x^2),x)
 

Output:

e**5*(5*int(e**((e**(4*x)*e**5*x**2 + 5*e**(4*x)*e**5*x + 8*e**(4*x)*x + e 
**5*x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5*x + 8*x**3 + 8*x**2)/(e**(4*x) 
 + x**2 + x))/(e**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)*x + x**4 + 2*x**3 + 
 x**2),x) + int(e**((e**(4*x)*e**5*x**2 + 5*e**(4*x)*e**5*x + 4*e**(4*x)*x 
 + e**5*x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5*x + 4*x**3 + 4*x**2)/(e**( 
4*x) + x**2 + x))/(e**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)*x + x**4 + 2*x* 
*3 + x**2),x) + 2*int((e**((e**(4*x)*e**5*x**2 + 5*e**(4*x)*e**5*x + e**5* 
x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5*x)/(e**(4*x) + x**2 + x))*x**5)/(e 
**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)*x + x**4 + 2*x**3 + x**2),x) + 9*in 
t((e**((e**(4*x)*e**5*x**2 + 5*e**(4*x)*e**5*x + e**5*x**4 + 6*e**5*x**3 + 
 6*e**5*x**2 + e**5*x)/(e**(4*x) + x**2 + x))*x**4)/(e**(8*x) + 2*e**(4*x) 
*x**2 + 2*e**(4*x)*x + x**4 + 2*x**3 + x**2),x) + 12*int((e**((e**(4*x)*e* 
*5*x**2 + 5*e**(4*x)*e**5*x + e**5*x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5 
*x)/(e**(4*x) + x**2 + x))*x**3)/(e**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)* 
x + x**4 + 2*x**3 + x**2),x) + 5*int((e**((e**(4*x)*e**5*x**2 + 5*e**(4*x) 
*e**5*x + e**5*x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5*x)/(e**(4*x) + x**2 
 + x))*x**2)/(e**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)*x + x**4 + 2*x**3 + 
x**2),x) + 2*int((e**((e**(4*x)*e**5*x**2 + 5*e**(4*x)*e**5*x + 8*e**(4*x) 
*x + e**5*x**4 + 6*e**5*x**3 + 6*e**5*x**2 + e**5*x + 8*x**3 + 8*x**2)/(e* 
*(4*x) + x**2 + x))*x)/(e**(8*x) + 2*e**(4*x)*x**2 + 2*e**(4*x)*x + x**...