\(\int \frac {-250 x^2-25 x^3-125 x^5+(-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6) \log (\frac {37+4 x}{8+x})}{(29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8) \log ^2(\frac {37+4 x}{8+x})} \, dx\) [2054]

Optimal result

Integrand size = 115, antiderivative size = 30 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {x}{\left (-\frac {1}{5}-\frac {2}{x}-x^2\right ) \log \left (4+\frac {5}{8+x}\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 x^2}{\left (10+x+5 x^3\right ) \log \left (\frac {37+4 x}{8+x}\right )} \] Input:

Integrate[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 
7380*x^4 + 1725*x^5 + 100*x^6)*Log[(37 + 4*x)/(8 + x)])/((29600 + 12820*x 
+ 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100*x 
^8)*Log[(37 + 4*x)/(8 + x)]^2),x]
 

Output:

(-5*x^2)/((10 + x + 5*x^3)*Log[(37 + 4*x)/(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.

\(\displaystyle \int \frac {-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )}{\left (100 x^8+1725 x^7+7440 x^6+1090 x^5+9864 x^4+29749 x^3+2076 x^2+12820 x+29600\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )}{32716820 (x+8) \log ^2\left (\frac {4 x+37}{x+8}\right )}-\frac {16384 \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{320594245445 (4 x+37) \log ^2\left (\frac {4 x+37}{x+8}\right )}+\frac {\left (-37317230145 x^2+265035817480 x-1817856266829\right ) \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{419552968850395396 \left (5 x^3+x+10\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )}+\frac {\left (89505 x^2-511400 x+2216181\right ) \left (-125 x^5-25 x^3-250 x^2+\left (100 x^6+1725 x^5+7380 x^4-745 x^3-8380 x^2-29600 x\right ) \log \left (\frac {4 x+37}{x+8}\right )\right )}{647729086 \left (5 x^3+x+10\right )^2 \log ^2\left (\frac {4 x+37}{x+8}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {16384 \int \frac {x (x+8) \left (5 x^3-x-20\right )}{\log \left (\frac {4 x+37}{x+8}\right )}dx}{64118849089}+\frac {\int \frac {x (4 x+37) \left (5 x^3-x-20\right )}{\log \left (\frac {4 x+37}{x+8}\right )}dx}{6543364}+\frac {25 \int \frac {x^2 \left (37317230145 x^2-265035817480 x+1817856266829\right )}{\log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{419552968850395396}-\frac {25 \int \frac {x^2 \left (89505 x^2-511400 x+2216181\right )}{\left (5 x^3+x+10\right ) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{647729086}-\frac {5 \int \frac {x^2 \left (5 x^3+x+10\right )}{(x+8) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{6543364}+\frac {81920 \int \frac {x^2 \left (5 x^3+x+10\right )}{(4 x+37) \log ^2\left (\frac {4 x+37}{x+8}\right )}dx}{64118849089}+\frac {5 \int \frac {x (x+8) (4 x+37) \left (89505 x^2-511400 x+2216181\right ) \left (5 x^3-x-20\right )}{\left (5 x^3+x+10\right )^2 \log \left (\frac {4 x+37}{x+8}\right )}dx}{647729086}-\frac {5 \int \frac {x (x+8) (4 x+37) \left (37317230145 x^2-265035817480 x+1817856266829\right ) \left (5 x^3-x-20\right )}{\left (5 x^3+x+10\right ) \log \left (\frac {4 x+37}{x+8}\right )}dx}{419552968850395396}\)

Input:

Int[(-250*x^2 - 25*x^3 - 125*x^5 + (-29600*x - 8380*x^2 - 745*x^3 + 7380*x 
^4 + 1725*x^5 + 100*x^6)*Log[(37 + 4*x)/(8 + x)])/((29600 + 12820*x + 2076 
*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725*x^7 + 100*x^8)*Lo 
g[(37 + 4*x)/(8 + x)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 8.80 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\)	\(30\)
parallelrisch	\(-\frac {5 x^{2}}{\left (5 x^{3}+x +10\right ) \ln \left (\frac {4 x +37}{x +8}\right )}\)	\(30\)
derivativedivides	\(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\)	\(89\)
default	\(\frac {160}{1279 \ln \left (4+\frac {5}{x +8}\right )}+\frac {\frac {257200 \left (4+\frac {5}{x +8}\right )^{2}}{1279}-\frac {4015200}{1279}-\frac {11888625}{1279 \left (x +8\right )}}{\left (2558 \left (4+\frac {5}{x +8}\right )^{3}-35501 \left (4+\frac {5}{x +8}\right )^{2}+403679+\frac {821120}{x +8}\right ) \ln \left (4+\frac {5}{x +8}\right )}\)	\(89\)

Input:

int(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*ln((4*x+37)/(x+8 
))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+29 
749*x^3+2076*x^2+12820*x+29600)/ln((4*x+37)/(x+8))^2,x,method=_RETURNVERBO 
SE)
 

Output:

-5*x^2/(5*x^3+x+10)/ln((4*x+37)/(x+8))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )} \] Input:

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 
7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 
*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= 
"fricas")
 

Output:

-5*x^2/((5*x^3 + x + 10)*log((4*x + 37)/(x + 8)))
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=- \frac {5 x^{2}}{\left (5 x^{3} + x + 10\right ) \log {\left (\frac {4 x + 37}{x + 8} \right )}} \] Input:

integrate(((100*x**6+1725*x**5+7380*x**4-745*x**3-8380*x**2-29600*x)*ln((4 
*x+37)/(x+8))-125*x**5-25*x**3-250*x**2)/(100*x**8+1725*x**7+7440*x**6+109 
0*x**5+9864*x**4+29749*x**3+2076*x**2+12820*x+29600)/ln((4*x+37)/(x+8))**2 
,x)
 

Output:

-5*x**2/((5*x**3 + x + 10)*log((4*x + 37)/(x + 8)))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 \, x^{2}}{{\left (5 \, x^{3} + x + 10\right )} \log \left (4 \, x + 37\right ) - {\left (5 \, x^{3} + x + 10\right )} \log \left (x + 8\right )} \] Input:

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 
7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 
*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= 
"maxima")
 

Output:

-5*x^2/((5*x^3 + x + 10)*log(4*x + 37) - (5*x^3 + x + 10)*log(x + 8))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (29) = 58\).

Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.57 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {5 \, {\left (\frac {64 \, {\left (4 \, x + 37\right )}^{3}}{{\left (x + 8\right )}^{3}} - \frac {848 \, {\left (4 \, x + 37\right )}^{2}}{{\left (x + 8\right )}^{2}} + \frac {3737 \, {\left (4 \, x + 37\right )}}{x + 8} - 5476\right )}}{\frac {2558 \, {\left (4 \, x + 37\right )}^{3} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{3}} - \frac {35501 \, {\left (4 \, x + 37\right )}^{2} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{{\left (x + 8\right )}^{2}} + \frac {164224 \, {\left (4 \, x + 37\right )} \log \left (\frac {4 \, x + 37}{x + 8}\right )}{x + 8} - 253217 \, \log \left (\frac {4 \, x + 37}{x + 8}\right )} \] Input:

integrate(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+3 
7)/(x+8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864 
*x^4+29749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x, algorithm= 
"giac")
 

Output:

5*(64*(4*x + 37)^3/(x + 8)^3 - 848*(4*x + 37)^2/(x + 8)^2 + 3737*(4*x + 37 
)/(x + 8) - 5476)/(2558*(4*x + 37)^3*log((4*x + 37)/(x + 8))/(x + 8)^3 - 3 
5501*(4*x + 37)^2*log((4*x + 37)/(x + 8))/(x + 8)^2 + 164224*(4*x + 37)*lo 
g((4*x + 37)/(x + 8))/(x + 8) - 253217*log((4*x + 37)/(x + 8)))
 

Mupad [B] (verification not implemented)

Time = 3.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=\frac {4\,x^3+\frac {4\,x}{5}+8}{5\,x^3+x+10}-\frac {5\,x^2}{\ln \left (\frac {4\,x+37}{x+8}\right )\,\left (5\,x^3+x+10\right )} \] Input:

int(-(250*x^2 + 25*x^3 + 125*x^5 + log((4*x + 37)/(x + 8))*(29600*x + 8380 
*x^2 + 745*x^3 - 7380*x^4 - 1725*x^5 - 100*x^6))/(log((4*x + 37)/(x + 8))^ 
2*(12820*x + 2076*x^2 + 29749*x^3 + 9864*x^4 + 1090*x^5 + 7440*x^6 + 1725* 
x^7 + 100*x^8 + 29600)),x)
 

Output:

((4*x)/5 + 4*x^3 + 8)/(x + 5*x^3 + 10) - (5*x^2)/(log((4*x + 37)/(x + 8))* 
(x + 5*x^3 + 10))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-250 x^2-25 x^3-125 x^5+\left (-29600 x-8380 x^2-745 x^3+7380 x^4+1725 x^5+100 x^6\right ) \log \left (\frac {37+4 x}{8+x}\right )}{\left (29600+12820 x+2076 x^2+29749 x^3+9864 x^4+1090 x^5+7440 x^6+1725 x^7+100 x^8\right ) \log ^2\left (\frac {37+4 x}{8+x}\right )} \, dx=-\frac {5 x^{2}}{\mathrm {log}\left (\frac {4 x +37}{x +8}\right ) \left (5 x^{3}+x +10\right )} \] Input:

int(((100*x^6+1725*x^5+7380*x^4-745*x^3-8380*x^2-29600*x)*log((4*x+37)/(x+ 
8))-125*x^5-25*x^3-250*x^2)/(100*x^8+1725*x^7+7440*x^6+1090*x^5+9864*x^4+2 
9749*x^3+2076*x^2+12820*x+29600)/log((4*x+37)/(x+8))^2,x)
 

Output:

( - 5*x**2)/(log((4*x + 37)/(x + 8))*(5*x**3 + x + 10))