\(\int \frac {-80+184 x+41 x^2+(-18+38 x+4 x^2) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+(4000+2440 x+488 x^2+32 x^3) \log (x)+(600+242 x+24 x^2) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx\) [1699]

Optimal result

Integrand size = 103, antiderivative size = 25 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {\left (1-\frac {1}{x}\right ) x^2}{x+(2 (5+x)+\log (x))^2} \] Output:

(1-1/x)*x^2/(x+(ln(x)+2*x+10)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {(-1+x) x}{100+41 x+4 x^2+4 (5+x) \log (x)+\log ^2(x)} \] Input:

Integrate[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x) 
*Log[x]^2)/(10000 + 8200*x + 2481*x^2 + 328*x^3 + 16*x^4 + (4000 + 2440*x 
+ 488*x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x)* 
Log[x]^3 + Log[x]^4),x]
 

Output:

((-1 + x)*x)/(100 + 41*x + 4*x^2 + 4*(5 + x)*Log[x] + Log[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.

\(\displaystyle \int \frac {41 x^2+\left (4 x^2+38 x-18\right ) \log (x)+184 x+(2 x-1) \log ^2(x)-80}{16 x^4+328 x^3+2481 x^2+\left (24 x^2+242 x+600\right ) \log ^2(x)+\left (32 x^3+488 x^2+2440 x+4000\right ) \log (x)+8200 x+\log ^4(x)+(8 x+40) \log ^3(x)+10000} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {41 x^2+2 \left (2 x^2+19 x-9\right ) \log (x)+184 x+(2 x-1) \log ^2(x)-80}{\left (4 x^2+41 x+\log ^2(x)+4 (x+5) \log (x)+100\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x-1}{4 x^2+41 x+\log ^2(x)+4 x \log (x)+20 \log (x)+100}-\frac {(x-1) \left (8 x^2+45 x+4 x \log (x)+2 \log (x)+20\right )}{\left (4 x^2+41 x+\log ^2(x)+4 x \log (x)+20 \log (x)+100\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 20 \int \frac {1}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+25 \int \frac {x}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-37 \int \frac {x^2}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+2 \int \frac {\log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx+2 \int \frac {x \log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-4 \int \frac {x^2 \log (x)}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx-\int \frac {1}{4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100}dx+2 \int \frac {x}{4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100}dx-8 \int \frac {x^3}{\left (4 x^2+4 \log (x) x+41 x+\log ^2(x)+20 \log (x)+100\right )^2}dx\)

Input:

Int[(-80 + 184*x + 41*x^2 + (-18 + 38*x + 4*x^2)*Log[x] + (-1 + 2*x)*Log[x 
]^2)/(10000 + 8200*x + 2481*x^2 + 328*x^3 + 16*x^4 + (4000 + 2440*x + 488* 
x^2 + 32*x^3)*Log[x] + (600 + 242*x + 24*x^2)*Log[x]^2 + (40 + 8*x)*Log[x] 
^3 + Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24

method	result	size
risch	\(\frac {x \left (-1+x \right )}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\)	\(31\)
default	\(\frac {x^{2}-x}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\)	\(34\)
parallelrisch	\(\frac {x^{2}-x}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\)	\(34\)
norman	\(\frac {-5 \ln \left (x \right )-\frac {45 x}{4}-\frac {\ln \left (x \right )^{2}}{4}-x \ln \left (x \right )-25}{\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}+20 \ln \left (x \right )+41 x +100}\)	\(47\)

Input:

int(((-1+2*x)*ln(x)^2+(4*x^2+38*x-18)*ln(x)+41*x^2+184*x-80)/(ln(x)^4+(8*x 
+40)*ln(x)^3+(24*x^2+242*x+600)*ln(x)^2+(32*x^3+488*x^2+2440*x+4000)*ln(x) 
+16*x^4+328*x^3+2481*x^2+8200*x+10000),x,method=_RETURNVERBOSE)
 

Output:

x*(-1+x)/(ln(x)^2+4*x*ln(x)+4*x^2+20*ln(x)+41*x+100)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 100} \] Input:

integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( 
x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 
4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="fricas")
 

Output:

(x^2 - x)/(4*x^2 + 4*(x + 5)*log(x) + log(x)^2 + 41*x + 100)
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 x^{2} + 41 x + \left (4 x + 20\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 100} \] Input:

integrate(((-1+2*x)*ln(x)**2+(4*x**2+38*x-18)*ln(x)+41*x**2+184*x-80)/(ln( 
x)**4+(8*x+40)*ln(x)**3+(24*x**2+242*x+600)*ln(x)**2+(32*x**3+488*x**2+244 
0*x+4000)*ln(x)+16*x**4+328*x**3+2481*x**2+8200*x+10000),x)
 

Output:

(x**2 - x)/(4*x**2 + 41*x + (4*x + 20)*log(x) + log(x)**2 + 100)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, {\left (x + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 100} \] Input:

integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( 
x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 
4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="maxima")
 

Output:

(x^2 - x)/(4*x^2 + 4*(x + 5)*log(x) + log(x)^2 + 41*x + 100)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\frac {x^{2} - x}{4 \, x^{2} + 4 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 41 \, x + 20 \, \log \left (x\right ) + 100} \] Input:

integrate(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log( 
x)^4+(8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+ 
4000)*log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x, algorithm="giac")
 

Output:

(x^2 - x)/(4*x^2 + 4*x*log(x) + log(x)^2 + 41*x + 20*log(x) + 100)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx=\int \frac {184\,x+\ln \left (x\right )\,\left (4\,x^2+38\,x-18\right )+41\,x^2+{\ln \left (x\right )}^2\,\left (2\,x-1\right )-80}{8200\,x+{\ln \left (x\right )}^2\,\left (24\,x^2+242\,x+600\right )+{\ln \left (x\right )}^4+2481\,x^2+328\,x^3+16\,x^4+{\ln \left (x\right )}^3\,\left (8\,x+40\right )+\ln \left (x\right )\,\left (32\,x^3+488\,x^2+2440\,x+4000\right )+10000} \,d x \] Input:

int((184*x + log(x)*(38*x + 4*x^2 - 18) + 41*x^2 + log(x)^2*(2*x - 1) - 80 
)/(8200*x + log(x)^2*(242*x + 24*x^2 + 600) + log(x)^4 + 2481*x^2 + 328*x^ 
3 + 16*x^4 + log(x)^3*(8*x + 40) + log(x)*(2440*x + 488*x^2 + 32*x^3 + 400 
0) + 10000),x)
 

Output:

int((184*x + log(x)*(38*x + 4*x^2 - 18) + 41*x^2 + log(x)^2*(2*x - 1) - 80 
)/(8200*x + log(x)^2*(242*x + 24*x^2 + 600) + log(x)^4 + 2481*x^2 + 328*x^ 
3 + 16*x^4 + log(x)^3*(8*x + 40) + log(x)*(2440*x + 488*x^2 + 32*x^3 + 400 
0) + 10000), x)
 

Reduce [F]

\[ \int \frac {-80+184 x+41 x^2+\left (-18+38 x+4 x^2\right ) \log (x)+(-1+2 x) \log ^2(x)}{10000+8200 x+2481 x^2+328 x^3+16 x^4+\left (4000+2440 x+488 x^2+32 x^3\right ) \log (x)+\left (600+242 x+24 x^2\right ) \log ^2(x)+(40+8 x) \log ^3(x)+\log ^4(x)} \, dx =\text {Too large to display} \] Input:

int(((-1+2*x)*log(x)^2+(4*x^2+38*x-18)*log(x)+41*x^2+184*x-80)/(log(x)^4+( 
8*x+40)*log(x)^3+(24*x^2+242*x+600)*log(x)^2+(32*x^3+488*x^2+2440*x+4000)* 
log(x)+16*x^4+328*x^3+2481*x^2+8200*x+10000),x)
 

Output:

 - int(log(x)**2/(log(x)**4 + 8*log(x)**3*x + 40*log(x)**3 + 24*log(x)**2* 
x**2 + 242*log(x)**2*x + 600*log(x)**2 + 32*log(x)*x**3 + 488*log(x)*x**2 
+ 2440*log(x)*x + 4000*log(x) + 16*x**4 + 328*x**3 + 2481*x**2 + 8200*x + 
10000),x) + 41*int(x**2/(log(x)**4 + 8*log(x)**3*x + 40*log(x)**3 + 24*log 
(x)**2*x**2 + 242*log(x)**2*x + 600*log(x)**2 + 32*log(x)*x**3 + 488*log(x 
)*x**2 + 2440*log(x)*x + 4000*log(x) + 16*x**4 + 328*x**3 + 2481*x**2 + 82 
00*x + 10000),x) - 18*int(log(x)/(log(x)**4 + 8*log(x)**3*x + 40*log(x)**3 
 + 24*log(x)**2*x**2 + 242*log(x)**2*x + 600*log(x)**2 + 32*log(x)*x**3 + 
488*log(x)*x**2 + 2440*log(x)*x + 4000*log(x) + 16*x**4 + 328*x**3 + 2481* 
x**2 + 8200*x + 10000),x) + 2*int((log(x)**2*x)/(log(x)**4 + 8*log(x)**3*x 
 + 40*log(x)**3 + 24*log(x)**2*x**2 + 242*log(x)**2*x + 600*log(x)**2 + 32 
*log(x)*x**3 + 488*log(x)*x**2 + 2440*log(x)*x + 4000*log(x) + 16*x**4 + 3 
28*x**3 + 2481*x**2 + 8200*x + 10000),x) + 4*int((log(x)*x**2)/(log(x)**4 
+ 8*log(x)**3*x + 40*log(x)**3 + 24*log(x)**2*x**2 + 242*log(x)**2*x + 600 
*log(x)**2 + 32*log(x)*x**3 + 488*log(x)*x**2 + 2440*log(x)*x + 4000*log(x 
) + 16*x**4 + 328*x**3 + 2481*x**2 + 8200*x + 10000),x) + 38*int((log(x)*x 
)/(log(x)**4 + 8*log(x)**3*x + 40*log(x)**3 + 24*log(x)**2*x**2 + 242*log( 
x)**2*x + 600*log(x)**2 + 32*log(x)*x**3 + 488*log(x)*x**2 + 2440*log(x)*x 
 + 4000*log(x) + 16*x**4 + 328*x**3 + 2481*x**2 + 8200*x + 10000),x) + 184 
*int(x/(log(x)**4 + 8*log(x)**3*x + 40*log(x)**3 + 24*log(x)**2*x**2 + ...