\(\int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} (-640 x^2-80 x^3)+e^{4 x} (-32 x^2-4 x^3)+e^{2 x} (-4800 x^2-600 x^3+e (32+68 x+8 x^2))+e^x (-16000 x^2-2000 x^3+e (320+360 x+40 x^2))+(e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} (64 x^2+8 x^3)+e^x (640 x^2+80 x^3)) \log (8+x)+(-32 x^2-4 x^3) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e (400 x^2+50 x^3)+e^{4 x} (8 x^4+x^5)+e^{3 x} (160 x^4+20 x^5)+e^{2 x} (1200 x^4+150 x^5+e (16 x^2+2 x^3))+e^x (4000 x^4+500 x^5+e (160 x^2+20 x^3))+(-400 x^4-50 x^5+e (-16 x^2-2 x^3)+e^x (-160 x^4-20 x^5)+e^{2 x} (-16 x^4-2 x^5)) \log (8+x)+(8 x^4+x^5) \log ^2(8+x)} \, dx\) [2141]

Optimal result

Integrand size = 382, antiderivative size = 27 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=\frac {4 x}{x^2+\frac {e}{\left (5+e^x\right )^2-\log (8+x)}} \] Output:

4*x/(exp(1)/((exp(x)+5)^2-ln(x+8))+x^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=-4 \left (-\frac {1}{x}-\frac {e}{x \left (-e-25 x^2-10 e^x x^2-e^{2 x} x^2+x^2 \log (8+x)\right )}\right ) \] Input:

Integrate[(-20000*x^2 - 2500*x^3 + E*(800 + 96*x) + E^(3*x)*(-640*x^2 - 80 
*x^3) + E^(4*x)*(-32*x^2 - 4*x^3) + E^(2*x)*(-4800*x^2 - 600*x^3 + E*(32 + 
 68*x + 8*x^2)) + E^x*(-16000*x^2 - 2000*x^3 + E*(320 + 360*x + 40*x^2)) + 
 (E*(-32 - 4*x) + 1600*x^2 + 200*x^3 + E^(2*x)*(64*x^2 + 8*x^3) + E^x*(640 
*x^2 + 80*x^3))*Log[8 + x] + (-32*x^2 - 4*x^3)*Log[8 + x]^2)/(5000*x^4 + 6 
25*x^5 + E^2*(8 + x) + E*(400*x^2 + 50*x^3) + E^(4*x)*(8*x^4 + x^5) + E^(3 
*x)*(160*x^4 + 20*x^5) + E^(2*x)*(1200*x^4 + 150*x^5 + E*(16*x^2 + 2*x^3)) 
 + E^x*(4000*x^4 + 500*x^5 + E*(160*x^2 + 20*x^3)) + (-400*x^4 - 50*x^5 + 
E*(-16*x^2 - 2*x^3) + E^x*(-160*x^4 - 20*x^5) + E^(2*x)*(-16*x^4 - 2*x^5)) 
*Log[8 + x] + (8*x^4 + x^5)*Log[8 + x]^2),x]
 

Output:

-4*(-x^(-1) - E/(x*(-E - 25*x^2 - 10*E^x*x^2 - E^(2*x)*x^2 + x^2*Log[8 + 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.

\(\displaystyle \int \frac {-2500 x^3-20000 x^2+e^{3 x} \left (-80 x^3-640 x^2\right )+e^{4 x} \left (-4 x^3-32 x^2\right )+e^{2 x} \left (-600 x^3-4800 x^2+e \left (8 x^2+68 x+32\right )\right )+e^x \left (-2000 x^3-16000 x^2+e \left (40 x^2+360 x+320\right )\right )+\left (-4 x^3-32 x^2\right ) \log ^2(x+8)+\left (200 x^3+1600 x^2+e^{2 x} \left (8 x^3+64 x^2\right )+e^x \left (80 x^3+640 x^2\right )+e (-4 x-32)\right ) \log (x+8)+e (96 x+800)}{625 x^5+5000 x^4+e^{4 x} \left (x^5+8 x^4\right )+e^{3 x} \left (20 x^5+160 x^4\right )+\left (x^5+8 x^4\right ) \log ^2(x+8)+e \left (50 x^3+400 x^2\right )+e^{2 x} \left (150 x^5+1200 x^4+e \left (2 x^3+16 x^2\right )\right )+e^x \left (500 x^5+4000 x^4+e \left (20 x^3+160 x^2\right )\right )+\left (-50 x^5-400 x^4+e^x \left (-20 x^5-160 x^4\right )+e^{2 x} \left (-2 x^5-16 x^4\right )+e \left (-2 x^3-16 x^2\right )\right ) \log (x+8)+e^2 (x+8)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (-500 e^x (x+8) x^2-150 e^{2 x} (x+8) x^2-20 e^{3 x} (x+8) x^2-e^{4 x} (x+8) x^2-625 (x+8) x^2+10 e^{x+1} \left (x^2+9 x+8\right )+e^{2 x+1} \left (2 x^2+17 x+8\right )-\left ((x+8) x^2 \log ^2(x+8)\right )-(x+8) \left (-20 e^x x^2-2 e^{2 x} x^2-50 x^2+e\right ) \log (x+8)+8 e (3 x+25)\right )}{(x+8) \left (10 e^x x^2+e^{2 x} x^2+25 x^2-x^2 \log (x+8)+e\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \int -\frac {(x+8) \log ^2(x+8) x^2+500 e^x (x+8) x^2+150 e^{2 x} (x+8) x^2+20 e^{3 x} (x+8) x^2+e^{4 x} (x+8) x^2+625 (x+8) x^2-8 e (3 x+25)-10 e^{x+1} \left (x^2+9 x+8\right )-e^{2 x+1} \left (2 x^2+17 x+8\right )+(x+8) \left (-20 e^x x^2-2 e^{2 x} x^2-50 x^2+e\right ) \log (x+8)}{(x+8) \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -4 \int \frac {(x+8) \log ^2(x+8) x^2+500 e^x (x+8) x^2+150 e^{2 x} (x+8) x^2+20 e^{3 x} (x+8) x^2+e^{4 x} (x+8) x^2+625 (x+8) x^2-8 e (3 x+25)-10 e^{x+1} \left (x^2+9 x+8\right )-e^{2 x+1} \left (2 x^2+17 x+8\right )+(x+8) \left (-20 e^x x^2-2 e^{2 x} x^2-50 x^2+e\right ) \log (x+8)}{(x+8) \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -4 \int \left (-\frac {e (2 x+3)}{x^2 \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )}+\frac {e \left (10 e^x x^4-2 \log (x+8) x^4+50 x^4+80 e^x x^3-16 \log (x+8) x^3+401 x^3+2 e x^2+18 e x+16 e\right )}{x^2 (x+8) \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )^2}+\frac {1}{x^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle -4 \int \left (-\frac {e (2 x+3)}{x^2 \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )}+\frac {e \left (10 e^x x^4-2 \log (x+8) x^4+50 x^4+80 e^x x^3-16 \log (x+8) x^3+401 x^3+2 e x^2+18 e x+16 e\right )}{x^2 (x+8) \left (10 e^x x^2+e^{2 x} x^2-\log (x+8) x^2+25 x^2+e\right )^2}+\frac {1}{x^2}\right )dx\)

Input:

Int[(-20000*x^2 - 2500*x^3 + E*(800 + 96*x) + E^(3*x)*(-640*x^2 - 80*x^3) 
+ E^(4*x)*(-32*x^2 - 4*x^3) + E^(2*x)*(-4800*x^2 - 600*x^3 + E*(32 + 68*x 
+ 8*x^2)) + E^x*(-16000*x^2 - 2000*x^3 + E*(320 + 360*x + 40*x^2)) + (E*(- 
32 - 4*x) + 1600*x^2 + 200*x^3 + E^(2*x)*(64*x^2 + 8*x^3) + E^x*(640*x^2 + 
 80*x^3))*Log[8 + x] + (-32*x^2 - 4*x^3)*Log[8 + x]^2)/(5000*x^4 + 625*x^5 
 + E^2*(8 + x) + E*(400*x^2 + 50*x^3) + E^(4*x)*(8*x^4 + x^5) + E^(3*x)*(1 
60*x^4 + 20*x^5) + E^(2*x)*(1200*x^4 + 150*x^5 + E*(16*x^2 + 2*x^3)) + E^x 
*(4000*x^4 + 500*x^5 + E*(160*x^2 + 20*x^3)) + (-400*x^4 - 50*x^5 + E*(-16 
*x^2 - 2*x^3) + E^x*(-160*x^4 - 20*x^5) + E^(2*x)*(-16*x^4 - 2*x^5))*Log[8 
 + x] + (8*x^4 + x^5)*Log[8 + x]^2),x]
 

Output:

$Aborted
 

Maple [F(-1)]

Timed out.

hanged

Input:

int(((-4*x^3-32*x^2)*ln(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640*x^2)*e 
xp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*ln(x+8)+(-4*x^3-32*x^2)*exp(x)^4+ 
(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^2)*exp(x 
)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+800)*exp(1 
)-2500*x^3-20000*x^2)/((x^5+8*x^4)*ln(x+8)^2+((-2*x^5-16*x^4)*exp(x)^2+(-2 
0*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*ln(x+8)+(x^5+ 
8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1)+150*x^5+1 
200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp(x)+(x+8)* 
exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x)
 

Output:

int(((-4*x^3-32*x^2)*ln(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640*x^2)*e 
xp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*ln(x+8)+(-4*x^3-32*x^2)*exp(x)^4+ 
(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^2)*exp(x 
)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+800)*exp(1 
)-2500*x^3-20000*x^2)/((x^5+8*x^4)*ln(x+8)^2+((-2*x^5-16*x^4)*exp(x)^2+(-2 
0*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*ln(x+8)+(x^5+ 
8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1)+150*x^5+1 
200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp(x)+(x+8)* 
exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=\frac {4 \, {\left (x e^{\left (2 \, x\right )} + 10 \, x e^{x} - x \log \left (x + 8\right ) + 25 \, x\right )}}{x^{2} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{x} - x^{2} \log \left (x + 8\right ) + 25 \, x^{2} + e} \] Input:

integrate(((-4*x^3-32*x^2)*log(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640 
*x^2)*exp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*log(x+8)+(-4*x^3-32*x^2)*e 
xp(x)^4+(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^ 
2)*exp(x)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+80 
0)*exp(1)-2500*x^3-20000*x^2)/((x^5+8*x^4)*log(x+8)^2+((-2*x^5-16*x^4)*exp 
(x)^2+(-20*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*log( 
x+8)+(x^5+8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1) 
+150*x^5+1200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp 
(x)+(x+8)*exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x, algorithm= 
"fricas")
 

Output:

4*(x*e^(2*x) + 10*x*e^x - x*log(x + 8) + 25*x)/(x^2*e^(2*x) + 10*x^2*e^x - 
 x^2*log(x + 8) + 25*x^2 + e)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).

Time = 0.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=- \frac {4 e}{x^{3} e^{2 x} + 10 x^{3} e^{x} - x^{3} \log {\left (x + 8 \right )} + 25 x^{3} + e x} + \frac {4}{x} \] Input:

integrate(((-4*x**3-32*x**2)*ln(x+8)**2+((8*x**3+64*x**2)*exp(x)**2+(80*x* 
*3+640*x**2)*exp(x)+(-4*x-32)*exp(1)+200*x**3+1600*x**2)*ln(x+8)+(-4*x**3- 
32*x**2)*exp(x)**4+(-80*x**3-640*x**2)*exp(x)**3+((8*x**2+68*x+32)*exp(1)- 
600*x**3-4800*x**2)*exp(x)**2+((40*x**2+360*x+320)*exp(1)-2000*x**3-16000* 
x**2)*exp(x)+(96*x+800)*exp(1)-2500*x**3-20000*x**2)/((x**5+8*x**4)*ln(x+8 
)**2+((-2*x**5-16*x**4)*exp(x)**2+(-20*x**5-160*x**4)*exp(x)+(-2*x**3-16*x 
**2)*exp(1)-50*x**5-400*x**4)*ln(x+8)+(x**5+8*x**4)*exp(x)**4+(20*x**5+160 
*x**4)*exp(x)**3+((2*x**3+16*x**2)*exp(1)+150*x**5+1200*x**4)*exp(x)**2+(( 
20*x**3+160*x**2)*exp(1)+500*x**5+4000*x**4)*exp(x)+(x+8)*exp(1)**2+(50*x* 
*3+400*x**2)*exp(1)+625*x**5+5000*x**4),x)
 

Output:

-4*E/(x**3*exp(2*x) + 10*x**3*exp(x) - x**3*log(x + 8) + 25*x**3 + E*x) + 
4/x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (27) = 54\).

Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=\frac {4 \, {\left (x e^{\left (2 \, x\right )} + 10 \, x e^{x} - x \log \left (x + 8\right ) + 25 \, x\right )}}{x^{2} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{x} - x^{2} \log \left (x + 8\right ) + 25 \, x^{2} + e} \] Input:

integrate(((-4*x^3-32*x^2)*log(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640 
*x^2)*exp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*log(x+8)+(-4*x^3-32*x^2)*e 
xp(x)^4+(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^ 
2)*exp(x)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+80 
0)*exp(1)-2500*x^3-20000*x^2)/((x^5+8*x^4)*log(x+8)^2+((-2*x^5-16*x^4)*exp 
(x)^2+(-20*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*log( 
x+8)+(x^5+8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1) 
+150*x^5+1200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp 
(x)+(x+8)*exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x, algorithm= 
"maxima")
 

Output:

4*(x*e^(2*x) + 10*x*e^x - x*log(x + 8) + 25*x)/(x^2*e^(2*x) + 10*x^2*e^x - 
 x^2*log(x + 8) + 25*x^2 + e)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4300 vs. \(2 (27) = 54\).

Time = 6.47 (sec) , antiderivative size = 4300, normalized size of antiderivative = 159.26 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=\text {Too large to display} \] Input:

integrate(((-4*x^3-32*x^2)*log(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640 
*x^2)*exp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*log(x+8)+(-4*x^3-32*x^2)*e 
xp(x)^4+(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^ 
2)*exp(x)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+80 
0)*exp(1)-2500*x^3-20000*x^2)/((x^5+8*x^4)*log(x+8)^2+((-2*x^5-16*x^4)*exp 
(x)^2+(-20*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*log( 
x+8)+(x^5+8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1) 
+150*x^5+1200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp 
(x)+(x+8)*exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x, algorithm= 
"giac")
 

Output:

4*(4*(x + 8)^9*e^16*log(x + 8)^3 - 200*(x + 8)^9*e^16*log(x + 8)^2 - 4*(x 
+ 8)^9*e^(2*x + 16)*log(x + 8)^2 - 40*(x + 8)^9*e^(x + 16)*log(x + 8)^2 - 
226*(x + 8)^8*e^16*log(x + 8)^3 + 2500*(x + 8)^9*e^16*log(x + 8) + 100*(x 
+ 8)^9*e^(2*x + 16)*log(x + 8) + 1000*(x + 8)^9*e^(x + 16)*log(x + 8) + 11 
246*(x + 8)^8*e^16*log(x + 8)^2 + 224*(x + 8)^8*e^(2*x + 16)*log(x + 8)^2 
+ 2230*(x + 8)^8*e^(x + 16)*log(x + 8)^2 + 5472*(x + 8)^7*e^16*log(x + 8)^ 
3 - 138650*(x + 8)^8*e^16*log(x + 8) - 5596*(x + 8)^8*e^(2*x + 16)*log(x + 
 8) - 55460*(x + 8)^8*e^(x + 16)*log(x + 8) - 8*(x + 8)^7*e^17*log(x + 8)^ 
2 - 270975*(x + 8)^7*e^16*log(x + 8)^2 - 5376*(x + 8)^7*e^(2*x + 16)*log(x 
 + 8)^2 - 53280*(x + 8)^7*e^(x + 16)*log(x + 8)^2 - 73600*(x + 8)^6*e^16*l 
og(x + 8)^3 - 31250*(x + 8)^8*e^16 - 6250*(x + 8)^8*e^(x + 16) + 300*(x + 
8)^7*e^17*log(x + 8) + 3294351*(x + 8)^7*e^16*log(x + 8) + 8*(x + 8)^7*e^( 
2*x + 17)*log(x + 8) + 134176*(x + 8)^7*e^(2*x + 16)*log(x + 8) + 80*(x + 
8)^7*e^(x + 17)*log(x + 8) + 1317760*(x + 8)^7*e^(x + 16)*log(x + 8) + 312 
*(x + 8)^6*e^17*log(x + 8)^2 + 3626576*(x + 8)^6*e^16*log(x + 8)^2 + 71680 
*(x + 8)^6*e^(2*x + 16)*log(x + 8)^2 + 707200*(x + 8)^6*e^(x + 16)*log(x + 
 8)^2 + 593920*(x + 8)^5*e^16*log(x + 8)^3 - 2500*(x + 8)^7*e^17 + 1500600 
*(x + 8)^7*e^16 - 100*(x + 8)^7*e^(2*x + 17) - (x + 8)^7*e^(2*x + 16) - 10 
00*(x + 8)^7*e^(x + 17) + 299990*(x + 8)^7*e^(x + 16) - 11898*(x + 8)^6*e^ 
17*log(x + 8) - 43463256*(x + 8)^6*e^16*log(x + 8) - 312*(x + 8)^6*e^(2...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=-\int \frac {{\mathrm {e}}^x\,\left (16000\,x^2-\mathrm {e}\,\left (40\,x^2+360\,x+320\right )+2000\,x^3\right )+{\mathrm {e}}^{4\,x}\,\left (4\,x^3+32\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (80\,x^3+640\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (4800\,x^2-\mathrm {e}\,\left (8\,x^2+68\,x+32\right )+600\,x^3\right )+{\ln \left (x+8\right )}^2\,\left (4\,x^3+32\,x^2\right )+20000\,x^2+2500\,x^3-\ln \left (x+8\right )\,\left ({\mathrm {e}}^x\,\left (80\,x^3+640\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (8\,x^3+64\,x^2\right )+1600\,x^2+200\,x^3-\mathrm {e}\,\left (4\,x+32\right )\right )-\mathrm {e}\,\left (96\,x+800\right )}{{\mathrm {e}}^{4\,x}\,\left (x^5+8\,x^4\right )+{\mathrm {e}}^{3\,x}\,\left (20\,x^5+160\,x^4\right )+{\ln \left (x+8\right )}^2\,\left (x^5+8\,x^4\right )-\ln \left (x+8\right )\,\left ({\mathrm {e}}^x\,\left (20\,x^5+160\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (2\,x^5+16\,x^4\right )+\mathrm {e}\,\left (2\,x^3+16\,x^2\right )+400\,x^4+50\,x^5\right )+\mathrm {e}\,\left (50\,x^3+400\,x^2\right )+{\mathrm {e}}^x\,\left (\mathrm {e}\,\left (20\,x^3+160\,x^2\right )+4000\,x^4+500\,x^5\right )+{\mathrm {e}}^2\,\left (x+8\right )+5000\,x^4+625\,x^5+{\mathrm {e}}^{2\,x}\,\left (\mathrm {e}\,\left (2\,x^3+16\,x^2\right )+1200\,x^4+150\,x^5\right )} \,d x \] Input:

int(-(exp(x)*(16000*x^2 - exp(1)*(360*x + 40*x^2 + 320) + 2000*x^3) + exp( 
4*x)*(32*x^2 + 4*x^3) + exp(3*x)*(640*x^2 + 80*x^3) + exp(2*x)*(4800*x^2 - 
 exp(1)*(68*x + 8*x^2 + 32) + 600*x^3) + log(x + 8)^2*(32*x^2 + 4*x^3) + 2 
0000*x^2 + 2500*x^3 - log(x + 8)*(exp(x)*(640*x^2 + 80*x^3) + exp(2*x)*(64 
*x^2 + 8*x^3) + 1600*x^2 + 200*x^3 - exp(1)*(4*x + 32)) - exp(1)*(96*x + 8 
00))/(exp(4*x)*(8*x^4 + x^5) + exp(3*x)*(160*x^4 + 20*x^5) + log(x + 8)^2* 
(8*x^4 + x^5) - log(x + 8)*(exp(x)*(160*x^4 + 20*x^5) + exp(2*x)*(16*x^4 + 
 2*x^5) + exp(1)*(16*x^2 + 2*x^3) + 400*x^4 + 50*x^5) + exp(1)*(400*x^2 + 
50*x^3) + exp(x)*(exp(1)*(160*x^2 + 20*x^3) + 4000*x^4 + 500*x^5) + exp(2) 
*(x + 8) + 5000*x^4 + 625*x^5 + exp(2*x)*(exp(1)*(16*x^2 + 2*x^3) + 1200*x 
^4 + 150*x^5)),x)
 

Output:

-int((exp(x)*(16000*x^2 - exp(1)*(360*x + 40*x^2 + 320) + 2000*x^3) + exp( 
4*x)*(32*x^2 + 4*x^3) + exp(3*x)*(640*x^2 + 80*x^3) + exp(2*x)*(4800*x^2 - 
 exp(1)*(68*x + 8*x^2 + 32) + 600*x^3) + log(x + 8)^2*(32*x^2 + 4*x^3) + 2 
0000*x^2 + 2500*x^3 - log(x + 8)*(exp(x)*(640*x^2 + 80*x^3) + exp(2*x)*(64 
*x^2 + 8*x^3) + 1600*x^2 + 200*x^3 - exp(1)*(4*x + 32)) - exp(1)*(96*x + 8 
00))/(exp(4*x)*(8*x^4 + x^5) + exp(3*x)*(160*x^4 + 20*x^5) + log(x + 8)^2* 
(8*x^4 + x^5) - log(x + 8)*(exp(x)*(160*x^4 + 20*x^5) + exp(2*x)*(16*x^4 + 
 2*x^5) + exp(1)*(16*x^2 + 2*x^3) + 400*x^4 + 50*x^5) + exp(1)*(400*x^2 + 
50*x^3) + exp(x)*(exp(1)*(160*x^2 + 20*x^3) + 4000*x^4 + 500*x^5) + exp(2) 
*(x + 8) + 5000*x^4 + 625*x^5 + exp(2*x)*(exp(1)*(16*x^2 + 2*x^3) + 1200*x 
^4 + 150*x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {-20000 x^2-2500 x^3+e (800+96 x)+e^{3 x} \left (-640 x^2-80 x^3\right )+e^{4 x} \left (-32 x^2-4 x^3\right )+e^{2 x} \left (-4800 x^2-600 x^3+e \left (32+68 x+8 x^2\right )\right )+e^x \left (-16000 x^2-2000 x^3+e \left (320+360 x+40 x^2\right )\right )+\left (e (-32-4 x)+1600 x^2+200 x^3+e^{2 x} \left (64 x^2+8 x^3\right )+e^x \left (640 x^2+80 x^3\right )\right ) \log (8+x)+\left (-32 x^2-4 x^3\right ) \log ^2(8+x)}{5000 x^4+625 x^5+e^2 (8+x)+e \left (400 x^2+50 x^3\right )+e^{4 x} \left (8 x^4+x^5\right )+e^{3 x} \left (160 x^4+20 x^5\right )+e^{2 x} \left (1200 x^4+150 x^5+e \left (16 x^2+2 x^3\right )\right )+e^x \left (4000 x^4+500 x^5+e \left (160 x^2+20 x^3\right )\right )+\left (-400 x^4-50 x^5+e \left (-16 x^2-2 x^3\right )+e^x \left (-160 x^4-20 x^5\right )+e^{2 x} \left (-16 x^4-2 x^5\right )\right ) \log (8+x)+\left (8 x^4+x^5\right ) \log ^2(8+x)} \, dx=\frac {4 x \left (e^{2 x}+10 e^{x}-\mathrm {log}\left (x +8\right )+25\right )}{e^{2 x} x^{2}+10 e^{x} x^{2}-\mathrm {log}\left (x +8\right ) x^{2}+e +25 x^{2}} \] Input:

int(((-4*x^3-32*x^2)*log(x+8)^2+((8*x^3+64*x^2)*exp(x)^2+(80*x^3+640*x^2)* 
exp(x)+(-4*x-32)*exp(1)+200*x^3+1600*x^2)*log(x+8)+(-4*x^3-32*x^2)*exp(x)^ 
4+(-80*x^3-640*x^2)*exp(x)^3+((8*x^2+68*x+32)*exp(1)-600*x^3-4800*x^2)*exp 
(x)^2+((40*x^2+360*x+320)*exp(1)-2000*x^3-16000*x^2)*exp(x)+(96*x+800)*exp 
(1)-2500*x^3-20000*x^2)/((x^5+8*x^4)*log(x+8)^2+((-2*x^5-16*x^4)*exp(x)^2+ 
(-20*x^5-160*x^4)*exp(x)+(-2*x^3-16*x^2)*exp(1)-50*x^5-400*x^4)*log(x+8)+( 
x^5+8*x^4)*exp(x)^4+(20*x^5+160*x^4)*exp(x)^3+((2*x^3+16*x^2)*exp(1)+150*x 
^5+1200*x^4)*exp(x)^2+((20*x^3+160*x^2)*exp(1)+500*x^5+4000*x^4)*exp(x)+(x 
+8)*exp(1)^2+(50*x^3+400*x^2)*exp(1)+625*x^5+5000*x^4),x)
 

Output:

(4*x*(e**(2*x) + 10*e**x - log(x + 8) + 25))/(e**(2*x)*x**2 + 10*e**x*x**2 
 - log(x + 8)*x**2 + e + 25*x**2)