\(\int \frac {-800-168 x+e^x (400+164 x+16 x^2)+(-200-40 x+e^x (100+40 x+4 x^2)) \log (5+x)}{40000 x^6+8000 x^7+e^x (-40000 x^6-8000 x^7)+e^{2 x} (10000 x^6+2000 x^7)+(20000 x^6+4000 x^7+e^x (-20000 x^6-4000 x^7)+e^{2 x} (5000 x^6+1000 x^7)) \log (5+x)+(2500 x^6+500 x^7+e^x (-2500 x^6-500 x^7)+e^{2 x} (625 x^6+125 x^7)) \log ^2(5+x)} \, dx\) [1772]

Optimal result

Integrand size = 187, antiderivative size = 24 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=\frac {4}{125 \left (2-e^x\right ) x^5 (4+\log (5+x))} \] Output:

4/125/(-exp(x)+2)/(ln(5+x)+4)/x^5
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \left (-2+e^x\right ) x^5 (4+\log (5+x))} \] Input:

Integrate[(-800 - 168*x + E^x*(400 + 164*x + 16*x^2) + (-200 - 40*x + E^x* 
(100 + 40*x + 4*x^2))*Log[5 + x])/(40000*x^6 + 8000*x^7 + E^x*(-40000*x^6 
- 8000*x^7) + E^(2*x)*(10000*x^6 + 2000*x^7) + (20000*x^6 + 4000*x^7 + E^x 
*(-20000*x^6 - 4000*x^7) + E^(2*x)*(5000*x^6 + 1000*x^7))*Log[5 + x] + (25 
00*x^6 + 500*x^7 + E^x*(-2500*x^6 - 500*x^7) + E^(2*x)*(625*x^6 + 125*x^7) 
)*Log[5 + x]^2),x]
 

Output:

-4/(125*(-2 + E^x)*x^5*(4 + Log[5 + 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.

\(\displaystyle \int \frac {e^x \left (16 x^2+164 x+400\right )+\left (e^x \left (4 x^2+40 x+100\right )-40 x-200\right ) \log (x+5)-168 x-800}{8000 x^7+40000 x^6+e^x \left (-8000 x^7-40000 x^6\right )+e^{2 x} \left (2000 x^7+10000 x^6\right )+\left (500 x^7+2500 x^6+e^x \left (-500 x^7-2500 x^6\right )+e^{2 x} \left (125 x^7+625 x^6\right )\right ) \log ^2(x+5)+\left (4000 x^7+20000 x^6+e^x \left (-4000 x^7-20000 x^6\right )+e^{2 x} \left (1000 x^7+5000 x^6\right )\right ) \log (x+5)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (e^x \left (4 x^2+41 x+100\right )-42 x+(x+5) \left (e^x (x+5)-10\right ) \log (x+5)-200\right )}{125 \left (2-e^x\right )^2 x^6 (x+5) (\log (x+5)+4)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{125} \int -\frac {42 x-e^x \left (4 x^2+41 x+100\right )+(x+5) \left (10-e^x (x+5)\right ) \log (x+5)+200}{\left (2-e^x\right )^2 x^6 (x+5) (\log (x+5)+4)^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4}{125} \int \frac {42 x-e^x \left (4 x^2+41 x+100\right )+(x+5) \left (10-e^x (x+5)\right ) \log (x+5)+200}{\left (2-e^x\right )^2 x^6 (x+5) (\log (x+5)+4)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4}{125} \int \left (-\frac {\log (x+5) x^2+4 x^2+10 \log (x+5) x+41 x+25 \log (x+5)+100}{\left (-2+e^x\right ) x^6 (x+5) (\log (x+5)+4)^2}-\frac {2}{\left (-2+e^x\right )^2 x^5 (\log (x+5)+4)}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle -\frac {4}{125} \int \left (-\frac {\log (x+5) x^2+4 x^2+10 \log (x+5) x+41 x+25 \log (x+5)+100}{\left (-2+e^x\right ) x^6 (x+5) (\log (x+5)+4)^2}-\frac {2}{\left (-2+e^x\right )^2 x^5 (\log (x+5)+4)}\right )dx\)

Input:

Int[(-800 - 168*x + E^x*(400 + 164*x + 16*x^2) + (-200 - 40*x + E^x*(100 + 
 40*x + 4*x^2))*Log[5 + x])/(40000*x^6 + 8000*x^7 + E^x*(-40000*x^6 - 8000 
*x^7) + E^(2*x)*(10000*x^6 + 2000*x^7) + (20000*x^6 + 4000*x^7 + E^x*(-200 
00*x^6 - 4000*x^7) + E^(2*x)*(5000*x^6 + 1000*x^7))*Log[5 + x] + (2500*x^6 
 + 500*x^7 + E^x*(-2500*x^6 - 500*x^7) + E^(2*x)*(625*x^6 + 125*x^7))*Log[ 
5 + x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

method	result	size
risch	\(-\frac {4}{125 \left ({\mathrm e}^{x}-2\right ) x^{5} \left (\ln \left (5+x \right )+4\right )}\)	\(20\)
parallelrisch	\(-\frac {4}{125 \left ({\mathrm e}^{x}-2\right ) x^{5} \left (\ln \left (5+x \right )+4\right )}\)	\(20\)

Input:

int((((4*x^2+40*x+100)*exp(x)-40*x-200)*ln(5+x)+(16*x^2+164*x+400)*exp(x)- 
168*x-800)/(((125*x^7+625*x^6)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^7 
+2500*x^6)*ln(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20000*x^6)*e 
xp(x)+4000*x^7+20000*x^6)*ln(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000*x^7 
-40000*x^6)*exp(x)+8000*x^7+40000*x^6),x,method=_RETURNVERBOSE)
 

Output:

-4/125/(exp(x)-2)/x^5/(ln(5+x)+4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \] Input:

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)* 
exp(x)-168*x-800)/(((125*x^7+625*x^6)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+ 
500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-2000 
0*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+( 
-8000*x^7-40000*x^6)*exp(x)+8000*x^7+40000*x^6),x, algorithm="fricas")
 

Output:

-4/125/(4*x^5*e^x - 8*x^5 + (x^5*e^x - 2*x^5)*log(x + 5))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).

Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=- \frac {4}{- 250 x^{5} \log {\left (x + 5 \right )} - 1000 x^{5} + \left (125 x^{5} \log {\left (x + 5 \right )} + 500 x^{5}\right ) e^{x}} \] Input:

integrate((((4*x**2+40*x+100)*exp(x)-40*x-200)*ln(5+x)+(16*x**2+164*x+400) 
*exp(x)-168*x-800)/(((125*x**7+625*x**6)*exp(x)**2+(-500*x**7-2500*x**6)*e 
xp(x)+500*x**7+2500*x**6)*ln(5+x)**2+((1000*x**7+5000*x**6)*exp(x)**2+(-40 
00*x**7-20000*x**6)*exp(x)+4000*x**7+20000*x**6)*ln(5+x)+(2000*x**7+10000* 
x**6)*exp(x)**2+(-8000*x**7-40000*x**6)*exp(x)+8000*x**7+40000*x**6),x)
 

Output:

-4/(-250*x**5*log(x + 5) - 1000*x**5 + (125*x**5*log(x + 5) + 500*x**5)*ex 
p(x))
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (4 \, x^{5} e^{x} - 8 \, x^{5} + {\left (x^{5} e^{x} - 2 \, x^{5}\right )} \log \left (x + 5\right )\right )}} \] Input:

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)* 
exp(x)-168*x-800)/(((125*x^7+625*x^6)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+ 
500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-2000 
0*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+( 
-8000*x^7-40000*x^6)*exp(x)+8000*x^7+40000*x^6),x, algorithm="maxima")
 

Output:

-4/125/(4*x^5*e^x - 8*x^5 + (x^5*e^x - 2*x^5)*log(x + 5))
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 \, {\left (x^{5} e^{x} \log \left (x + 5\right ) + 4 \, x^{5} e^{x} - 2 \, x^{5} \log \left (x + 5\right ) - 8 \, x^{5}\right )}} \] Input:

integrate((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)* 
exp(x)-168*x-800)/(((125*x^7+625*x^6)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+ 
500*x^7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-2000 
0*x^6)*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+( 
-8000*x^7-40000*x^6)*exp(x)+8000*x^7+40000*x^6),x, algorithm="giac")
 

Output:

-4/125/(x^5*e^x*log(x + 5) + 4*x^5*e^x - 2*x^5*log(x + 5) - 8*x^5)
 

Mupad [B] (verification not implemented)

Time = 3.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125\,x^5\,\left ({\mathrm {e}}^x-2\right )\,\left (\ln \left (x+5\right )+4\right )} \] Input:

int(-(168*x + log(x + 5)*(40*x - exp(x)*(40*x + 4*x^2 + 100) + 200) - exp( 
x)*(164*x + 16*x^2 + 400) + 800)/(log(x + 5)^2*(exp(2*x)*(625*x^6 + 125*x^ 
7) - exp(x)*(2500*x^6 + 500*x^7) + 2500*x^6 + 500*x^7) - exp(x)*(40000*x^6 
 + 8000*x^7) + exp(2*x)*(10000*x^6 + 2000*x^7) + 40000*x^6 + 8000*x^7 + lo 
g(x + 5)*(exp(2*x)*(5000*x^6 + 1000*x^7) - exp(x)*(20000*x^6 + 4000*x^7) + 
 20000*x^6 + 4000*x^7)),x)
 

Output:

-4/(125*x^5*(exp(x) - 2)*(log(x + 5) + 4))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-800-168 x+e^x \left (400+164 x+16 x^2\right )+\left (-200-40 x+e^x \left (100+40 x+4 x^2\right )\right ) \log (5+x)}{40000 x^6+8000 x^7+e^x \left (-40000 x^6-8000 x^7\right )+e^{2 x} \left (10000 x^6+2000 x^7\right )+\left (20000 x^6+4000 x^7+e^x \left (-20000 x^6-4000 x^7\right )+e^{2 x} \left (5000 x^6+1000 x^7\right )\right ) \log (5+x)+\left (2500 x^6+500 x^7+e^x \left (-2500 x^6-500 x^7\right )+e^{2 x} \left (625 x^6+125 x^7\right )\right ) \log ^2(5+x)} \, dx=-\frac {4}{125 x^{5} \left (e^{x} \mathrm {log}\left (x +5\right )+4 e^{x}-2 \,\mathrm {log}\left (x +5\right )-8\right )} \] Input:

int((((4*x^2+40*x+100)*exp(x)-40*x-200)*log(5+x)+(16*x^2+164*x+400)*exp(x) 
-168*x-800)/(((125*x^7+625*x^6)*exp(x)^2+(-500*x^7-2500*x^6)*exp(x)+500*x^ 
7+2500*x^6)*log(5+x)^2+((1000*x^7+5000*x^6)*exp(x)^2+(-4000*x^7-20000*x^6) 
*exp(x)+4000*x^7+20000*x^6)*log(5+x)+(2000*x^7+10000*x^6)*exp(x)^2+(-8000* 
x^7-40000*x^6)*exp(x)+8000*x^7+40000*x^6),x)
 

Output:

( - 4)/(125*x**5*(e**x*log(x + 5) + 4*e**x - 2*log(x + 5) - 8))