\(\int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} (-48-104 x+24 x^2)+e^{5+x} (-192 x-112 x^2+16 x^3)+(-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} (8 x-24 x^2)+e^{5+x} (16 x^2-16 x^3)) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} (-2 x^3+6 x^4)+e^{5+x} (-4 x^4+4 x^5)} \, dx\) [856]

Optimal result

Integrand size = 222, antiderivative size = 27 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=6+\frac {4 \left (-\frac {12}{x-\left (e^{5+x}+x\right )^2}+\log (x)\right )}{x} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {4 \left (\frac {12}{e^{2 (5+x)}+2 e^{5+x} x+(-1+x) x}+\log (x)\right )}{x} \] Input:

Integrate[(4*E^(20 + 4*x) + 96*x + 16*E^(15 + 3*x)*x - 140*x^2 - 8*x^3 + 4 
*x^4 + E^(10 + 2*x)*(-48 - 104*x + 24*x^2) + E^(5 + x)*(-192*x - 112*x^2 + 
 16*x^3) + (-4*E^(20 + 4*x) - 16*E^(15 + 3*x)*x - 4*x^2 + 8*x^3 - 4*x^4 + 
E^(10 + 2*x)*(8*x - 24*x^2) + E^(5 + x)*(16*x^2 - 16*x^3))*Log[x])/(E^(20 
+ 4*x)*x^2 + 4*E^(15 + 3*x)*x^3 + x^4 - 2*x^5 + x^6 + E^(10 + 2*x)*(-2*x^3 
 + 6*x^4) + E^(5 + x)*(-4*x^4 + 4*x^5)),x]
 

Output:

(4*(12/(E^(2*(5 + x)) + 2*E^(5 + x)*x + (-1 + x)*x) + Log[x]))/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.

\(\displaystyle \int \frac {4 x^4-8 x^3-140 x^2+e^{2 x+10} \left (24 x^2-104 x-48\right )+e^{x+5} \left (16 x^3-112 x^2-192 x\right )+\left (-4 x^4+8 x^3-4 x^2+e^{2 x+10} \left (8 x-24 x^2\right )+e^{x+5} \left (16 x^2-16 x^3\right )-16 e^{3 x+15} x-4 e^{4 x+20}\right ) \log (x)+16 e^{3 x+15} x+96 x+4 e^{4 x+20}}{x^6-2 x^5+x^4+4 e^{3 x+15} x^3+e^{4 x+20} x^2+e^{x+5} \left (4 x^5-4 x^4\right )+e^{2 x+10} \left (6 x^4-2 x^3\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {16 e^{x+5} x \left (x^2-7 x-12\right )+8 e^{2 (x+5)} \left (3 x^2-13 x-6\right )+4 x \left (x^3-2 x^2-35 x+24\right )+4 e^{4 (x+5)}+16 e^{3 (x+5)} x-4 \left (2 e^{x+5} x+(x-1) x+e^{2 (x+5)}\right )^2 \log (x)}{x^2 \left (2 e^{x+5} x+(x-1) x+e^{2 (x+5)}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {48 (2 x+1)}{x^2 \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )}+\frac {48 \left (2 x^2+2 e^{x+5} x-4 x-2 e^{x+5}+1\right )}{x \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}-\frac {4 (\log (x)-1)}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -192 \int \frac {1}{\left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}dx+96 \int \frac {e^{x+5}}{\left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}dx+48 \int \frac {1}{x \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}dx-96 \int \frac {e^{x+5}}{x \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}dx+96 \int \frac {x}{\left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )^2}dx-48 \int \frac {1}{x^2 \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )}dx-96 \int \frac {1}{x \left (x^2+2 e^{x+5} x-x+e^{2 x+10}\right )}dx+\frac {4 \log (x)}{x}\)

Input:

Int[(4*E^(20 + 4*x) + 96*x + 16*E^(15 + 3*x)*x - 140*x^2 - 8*x^3 + 4*x^4 + 
 E^(10 + 2*x)*(-48 - 104*x + 24*x^2) + E^(5 + x)*(-192*x - 112*x^2 + 16*x^ 
3) + (-4*E^(20 + 4*x) - 16*E^(15 + 3*x)*x - 4*x^2 + 8*x^3 - 4*x^4 + E^(10 
+ 2*x)*(8*x - 24*x^2) + E^(5 + x)*(16*x^2 - 16*x^3))*Log[x])/(E^(20 + 4*x) 
*x^2 + 4*E^(15 + 3*x)*x^3 + x^4 - 2*x^5 + x^6 + E^(10 + 2*x)*(-2*x^3 + 6*x 
^4) + E^(5 + x)*(-4*x^4 + 4*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33

\[\frac {4 \ln \left (x \right )}{x}+\frac {48}{x \left ({\mathrm e}^{2 x +10}+2 x \,{\mathrm e}^{5+x}+x^{2}-x \right )}\]

Input:

int(((-4*exp(5+x)^4-16*x*exp(5+x)^3+(-24*x^2+8*x)*exp(5+x)^2+(-16*x^3+16*x 
^2)*exp(5+x)-4*x^4+8*x^3-4*x^2)*ln(x)+4*exp(5+x)^4+16*x*exp(5+x)^3+(24*x^2 
-104*x-48)*exp(5+x)^2+(16*x^3-112*x^2-192*x)*exp(5+x)+4*x^4-8*x^3-140*x^2+ 
96*x)/(x^2*exp(5+x)^4+4*x^3*exp(5+x)^3+(6*x^4-2*x^3)*exp(5+x)^2+(4*x^5-4*x 
^4)*exp(5+x)+x^6-2*x^5+x^4),x)
 

Output:

4*ln(x)/x+48/x/(exp(2*x+10)+2*x*exp(5+x)+x^2-x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {4 \, {\left ({\left (x^{2} + 2 \, x e^{\left (x + 5\right )} - x + e^{\left (2 \, x + 10\right )}\right )} \log \left (x\right ) + 12\right )}}{x^{3} + 2 \, x^{2} e^{\left (x + 5\right )} - x^{2} + x e^{\left (2 \, x + 10\right )}} \] Input:

integrate(((-4*exp(5+x)^4-16*x*exp(5+x)^3+(-24*x^2+8*x)*exp(5+x)^2+(-16*x^ 
3+16*x^2)*exp(5+x)-4*x^4+8*x^3-4*x^2)*log(x)+4*exp(5+x)^4+16*x*exp(5+x)^3+ 
(24*x^2-104*x-48)*exp(5+x)^2+(16*x^3-112*x^2-192*x)*exp(5+x)+4*x^4-8*x^3-1 
40*x^2+96*x)/(x^2*exp(5+x)^4+4*x^3*exp(5+x)^3+(6*x^4-2*x^3)*exp(5+x)^2+(4* 
x^5-4*x^4)*exp(5+x)+x^6-2*x^5+x^4),x, algorithm="fricas")
 

Output:

4*((x^2 + 2*x*e^(x + 5) - x + e^(2*x + 10))*log(x) + 12)/(x^3 + 2*x^2*e^(x 
 + 5) - x^2 + x*e^(2*x + 10))
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {48}{x^{3} + 2 x^{2} e^{x + 5} - x^{2} + x e^{2 x + 10}} + \frac {4 \log {\left (x \right )}}{x} \] Input:

integrate(((-4*exp(5+x)**4-16*x*exp(5+x)**3+(-24*x**2+8*x)*exp(5+x)**2+(-1 
6*x**3+16*x**2)*exp(5+x)-4*x**4+8*x**3-4*x**2)*ln(x)+4*exp(5+x)**4+16*x*ex 
p(5+x)**3+(24*x**2-104*x-48)*exp(5+x)**2+(16*x**3-112*x**2-192*x)*exp(5+x) 
+4*x**4-8*x**3-140*x**2+96*x)/(x**2*exp(5+x)**4+4*x**3*exp(5+x)**3+(6*x**4 
-2*x**3)*exp(5+x)**2+(4*x**5-4*x**4)*exp(5+x)+x**6-2*x**5+x**4),x)
 

Output:

48/(x**3 + 2*x**2*exp(x + 5) - x**2 + x*exp(2*x + 10)) + 4*log(x)/x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {4 \, {\left (2 \, x e^{\left (x + 5\right )} \log \left (x\right ) + {\left (x^{2} - x\right )} \log \left (x\right ) + e^{\left (2 \, x + 10\right )} \log \left (x\right ) + 12\right )}}{x^{3} + 2 \, x^{2} e^{\left (x + 5\right )} - x^{2} + x e^{\left (2 \, x + 10\right )}} \] Input:

integrate(((-4*exp(5+x)^4-16*x*exp(5+x)^3+(-24*x^2+8*x)*exp(5+x)^2+(-16*x^ 
3+16*x^2)*exp(5+x)-4*x^4+8*x^3-4*x^2)*log(x)+4*exp(5+x)^4+16*x*exp(5+x)^3+ 
(24*x^2-104*x-48)*exp(5+x)^2+(16*x^3-112*x^2-192*x)*exp(5+x)+4*x^4-8*x^3-1 
40*x^2+96*x)/(x^2*exp(5+x)^4+4*x^3*exp(5+x)^3+(6*x^4-2*x^3)*exp(5+x)^2+(4* 
x^5-4*x^4)*exp(5+x)+x^6-2*x^5+x^4),x, algorithm="maxima")
 

Output:

4*(2*x*e^(x + 5)*log(x) + (x^2 - x)*log(x) + e^(2*x + 10)*log(x) + 12)/(x^ 
3 + 2*x^2*e^(x + 5) - x^2 + x*e^(2*x + 10))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 284, normalized size of antiderivative = 10.52 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {4 \, {\left (2 \, x^{5} \log \left (x\right ) + 6 \, x^{4} e^{\left (x + 5\right )} \log \left (x\right ) + 2 \, x^{5} + 2 \, x^{4} e^{\left (x + 5\right )} - 12 \, x^{4} \log \left (x\right ) + 4 \, x^{3} e^{\left (2 \, x + 10\right )} \log \left (x\right ) - 18 \, x^{3} e^{\left (x + 5\right )} \log \left (x\right ) - 4 \, x^{4} - 6 \, x^{3} e^{\left (x + 5\right )} + 19 \, x^{3} \log \left (x\right ) - 12 \, x^{2} e^{\left (2 \, x + 10\right )} \log \left (x\right ) + 10 \, x^{2} e^{\left (x + 5\right )} \log \left (x\right ) - 23 \, x^{3} - 66 \, x^{2} e^{\left (x + 5\right )} - 11 \, x^{2} \log \left (x\right ) + 8 \, x e^{\left (2 \, x + 10\right )} \log \left (x\right ) - 94 \, x^{2} + 118 \, x e^{\left (x + 5\right )} + 2 \, x \log \left (x\right ) - e^{\left (2 \, x + 10\right )} \log \left (x\right ) + 131 \, x - 48 \, e^{\left (x + 5\right )} - 36\right )}}{4 \, x^{6} + 8 \, x^{5} e^{\left (x + 5\right )} - 16 \, x^{5} + 4 \, x^{4} e^{\left (2 \, x + 10\right )} - 24 \, x^{4} e^{\left (x + 5\right )} + 20 \, x^{4} - 12 \, x^{3} e^{\left (2 \, x + 10\right )} + 16 \, x^{3} e^{\left (x + 5\right )} - 9 \, x^{3} + 8 \, x^{2} e^{\left (2 \, x + 10\right )} - 2 \, x^{2} e^{\left (x + 5\right )} + x^{2} - x e^{\left (2 \, x + 10\right )}} \] Input:

integrate(((-4*exp(5+x)^4-16*x*exp(5+x)^3+(-24*x^2+8*x)*exp(5+x)^2+(-16*x^ 
3+16*x^2)*exp(5+x)-4*x^4+8*x^3-4*x^2)*log(x)+4*exp(5+x)^4+16*x*exp(5+x)^3+ 
(24*x^2-104*x-48)*exp(5+x)^2+(16*x^3-112*x^2-192*x)*exp(5+x)+4*x^4-8*x^3-1 
40*x^2+96*x)/(x^2*exp(5+x)^4+4*x^3*exp(5+x)^3+(6*x^4-2*x^3)*exp(5+x)^2+(4* 
x^5-4*x^4)*exp(5+x)+x^6-2*x^5+x^4),x, algorithm="giac")
 

Output:

4*(2*x^5*log(x) + 6*x^4*e^(x + 5)*log(x) + 2*x^5 + 2*x^4*e^(x + 5) - 12*x^ 
4*log(x) + 4*x^3*e^(2*x + 10)*log(x) - 18*x^3*e^(x + 5)*log(x) - 4*x^4 - 6 
*x^3*e^(x + 5) + 19*x^3*log(x) - 12*x^2*e^(2*x + 10)*log(x) + 10*x^2*e^(x 
+ 5)*log(x) - 23*x^3 - 66*x^2*e^(x + 5) - 11*x^2*log(x) + 8*x*e^(2*x + 10) 
*log(x) - 94*x^2 + 118*x*e^(x + 5) + 2*x*log(x) - e^(2*x + 10)*log(x) + 13 
1*x - 48*e^(x + 5) - 36)/(4*x^6 + 8*x^5*e^(x + 5) - 16*x^5 + 4*x^4*e^(2*x 
+ 10) - 24*x^4*e^(x + 5) + 20*x^4 - 12*x^3*e^(2*x + 10) + 16*x^3*e^(x + 5) 
 - 9*x^3 + 8*x^2*e^(2*x + 10) - 2*x^2*e^(x + 5) + x^2 - x*e^(2*x + 10))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\int -\frac {{\mathrm {e}}^{x+5}\,\left (-16\,x^3+112\,x^2+192\,x\right )-4\,{\mathrm {e}}^{4\,x+20}-96\,x+{\mathrm {e}}^{2\,x+10}\,\left (-24\,x^2+104\,x+48\right )-16\,x\,{\mathrm {e}}^{3\,x+15}+\ln \left (x\right )\,\left (4\,{\mathrm {e}}^{4\,x+20}-{\mathrm {e}}^{2\,x+10}\,\left (8\,x-24\,x^2\right )-{\mathrm {e}}^{x+5}\,\left (16\,x^2-16\,x^3\right )+16\,x\,{\mathrm {e}}^{3\,x+15}+4\,x^2-8\,x^3+4\,x^4\right )+140\,x^2+8\,x^3-4\,x^4}{4\,x^3\,{\mathrm {e}}^{3\,x+15}-{\mathrm {e}}^{2\,x+10}\,\left (2\,x^3-6\,x^4\right )-{\mathrm {e}}^{x+5}\,\left (4\,x^4-4\,x^5\right )+x^2\,{\mathrm {e}}^{4\,x+20}+x^4-2\,x^5+x^6} \,d x \] Input:

int(-(exp(x + 5)*(192*x + 112*x^2 - 16*x^3) - 4*exp(4*x + 20) - 96*x + exp 
(2*x + 10)*(104*x - 24*x^2 + 48) - 16*x*exp(3*x + 15) + log(x)*(4*exp(4*x 
+ 20) - exp(2*x + 10)*(8*x - 24*x^2) - exp(x + 5)*(16*x^2 - 16*x^3) + 16*x 
*exp(3*x + 15) + 4*x^2 - 8*x^3 + 4*x^4) + 140*x^2 + 8*x^3 - 4*x^4)/(4*x^3* 
exp(3*x + 15) - exp(2*x + 10)*(2*x^3 - 6*x^4) - exp(x + 5)*(4*x^4 - 4*x^5) 
 + x^2*exp(4*x + 20) + x^4 - 2*x^5 + x^6),x)
 

Output:

int(-(exp(x + 5)*(192*x + 112*x^2 - 16*x^3) - 4*exp(4*x + 20) - 96*x + exp 
(2*x + 10)*(104*x - 24*x^2 + 48) - 16*x*exp(3*x + 15) + log(x)*(4*exp(4*x 
+ 20) - exp(2*x + 10)*(8*x - 24*x^2) - exp(x + 5)*(16*x^2 - 16*x^3) + 16*x 
*exp(3*x + 15) + 4*x^2 - 8*x^3 + 4*x^4) + 140*x^2 + 8*x^3 - 4*x^4)/(4*x^3* 
exp(3*x + 15) - exp(2*x + 10)*(2*x^3 - 6*x^4) - exp(x + 5)*(4*x^4 - 4*x^5) 
 + x^2*exp(4*x + 20) + x^4 - 2*x^5 + x^6), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.48 \[ \int \frac {4 e^{20+4 x}+96 x+16 e^{15+3 x} x-140 x^2-8 x^3+4 x^4+e^{10+2 x} \left (-48-104 x+24 x^2\right )+e^{5+x} \left (-192 x-112 x^2+16 x^3\right )+\left (-4 e^{20+4 x}-16 e^{15+3 x} x-4 x^2+8 x^3-4 x^4+e^{10+2 x} \left (8 x-24 x^2\right )+e^{5+x} \left (16 x^2-16 x^3\right )\right ) \log (x)}{e^{20+4 x} x^2+4 e^{15+3 x} x^3+x^4-2 x^5+x^6+e^{10+2 x} \left (-2 x^3+6 x^4\right )+e^{5+x} \left (-4 x^4+4 x^5\right )} \, dx=\frac {4 e^{2 x} \mathrm {log}\left (x \right ) e^{10}+8 e^{x} \mathrm {log}\left (x \right ) e^{5} x +4 \,\mathrm {log}\left (x \right ) x^{2}-4 \,\mathrm {log}\left (x \right ) x +48}{x \left (e^{2 x} e^{10}+2 e^{x} e^{5} x +x^{2}-x \right )} \] Input:

int(((-4*exp(5+x)^4-16*x*exp(5+x)^3+(-24*x^2+8*x)*exp(5+x)^2+(-16*x^3+16*x 
^2)*exp(5+x)-4*x^4+8*x^3-4*x^2)*log(x)+4*exp(5+x)^4+16*x*exp(5+x)^3+(24*x^ 
2-104*x-48)*exp(5+x)^2+(16*x^3-112*x^2-192*x)*exp(5+x)+4*x^4-8*x^3-140*x^2 
+96*x)/(x^2*exp(5+x)^4+4*x^3*exp(5+x)^3+(6*x^4-2*x^3)*exp(5+x)^2+(4*x^5-4* 
x^4)*exp(5+x)+x^6-2*x^5+x^4),x)
 

Output:

(4*(e**(2*x)*log(x)*e**10 + 2*e**x*log(x)*e**5*x + log(x)*x**2 - log(x)*x 
+ 12))/(x*(e**(2*x)*e**10 + 2*e**x*e**5*x + x**2 - x))