\(\int \frac {(8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7) \log ^2(\frac {1}{x})+(8-50 x-2 x^2) \log (4-25 x-x^2)+\log (\frac {1}{x}) (-50 x-4 x^2+(-8+50 x+2 x^2) \log (4-25 x-x^2))}{(-4 x^2+25 x^3+x^4) \log ^2(\frac {1}{x})} \, dx\) [249]

Optimal result

Integrand size = 120, antiderivative size = 27 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=x^4-2 \left (x+\frac {\log (4-x (25+x))}{x \log \left (\frac {1}{x}\right )}\right ) \] Output:

x^4-2*x-2*ln(4-x*(x+25))/x/ln(1/x)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=2 \left (-x+\frac {x^4}{2}-\frac {\log \left (4-25 x-x^2\right )}{x \log \left (\frac {1}{x}\right )}\right ) \] Input:

Integrate[((8*x^2 - 50*x^3 - 2*x^4 - 16*x^5 + 100*x^6 + 4*x^7)*Log[x^(-1)] 
^2 + (8 - 50*x - 2*x^2)*Log[4 - 25*x - x^2] + Log[x^(-1)]*(-50*x - 4*x^2 + 
 (-8 + 50*x + 2*x^2)*Log[4 - 25*x - x^2]))/((-4*x^2 + 25*x^3 + x^4)*Log[x^ 
(-1)]^2),x]
 

Output:

2*(-x + x^4/2 - Log[4 - 25*x - x^2]/(x*Log[x^(-1)]))
 

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.

\(\displaystyle \int \frac {\left (-4 x^2+\left (2 x^2+50 x-8\right ) \log \left (-x^2-25 x+4\right )-50 x\right ) \log \left (\frac {1}{x}\right )+\left (-2 x^2-50 x+8\right ) \log \left (-x^2-25 x+4\right )+\left (4 x^7+100 x^6-16 x^5-2 x^4-50 x^3+8 x^2\right ) \log ^2\left (\frac {1}{x}\right )}{\left (x^4+25 x^3-4 x^2\right ) \log ^2\left (\frac {1}{x}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (-4 x^2+\left (2 x^2+50 x-8\right ) \log \left (-x^2-25 x+4\right )-50 x\right ) \log \left (\frac {1}{x}\right )+\left (-2 x^2-50 x+8\right ) \log \left (-x^2-25 x+4\right )+\left (4 x^7+100 x^6-16 x^5-2 x^4-50 x^3+8 x^2\right ) \log ^2\left (\frac {1}{x}\right )}{x^2 \left (x^2+25 x-4\right ) \log ^2\left (\frac {1}{x}\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {2 \left (\log \left (\frac {1}{x}\right )-1\right ) \log \left (-x^2-25 x+4\right )}{x^2 \log ^2\left (\frac {1}{x}\right )}+\frac {2 \left (2 x^6 \log \left (\frac {1}{x}\right )+50 x^5 \log \left (\frac {1}{x}\right )-8 x^4 \log \left (\frac {1}{x}\right )-x^3 \log \left (\frac {1}{x}\right )-25 x^2 \log \left (\frac {1}{x}\right )-2 x+4 x \log \left (\frac {1}{x}\right )-25\right )}{x \left (x^2+25 x-4\right ) \log \left (\frac {1}{x}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {100 \int \frac {\operatorname {LogIntegral}\left (\frac {1}{x}\right )}{-2 x+\sqrt {641}-25}dx}{\sqrt {641}}+\frac {4}{641} \left (641-25 \sqrt {641}\right ) \int \frac {\operatorname {LogIntegral}\left (\frac {1}{x}\right )}{2 x-\sqrt {641}+25}dx+\frac {4}{641} \left (641+25 \sqrt {641}\right ) \int \frac {\operatorname {LogIntegral}\left (\frac {1}{x}\right )}{2 x+\sqrt {641}+25}dx-\frac {100 \int \frac {\operatorname {LogIntegral}\left (\frac {1}{x}\right )}{2 x+\sqrt {641}+25}dx}{\sqrt {641}}-2 \int \frac {\log \left (-x^2-25 x+4\right )}{x^2 \log ^2\left (\frac {1}{x}\right )}dx+2 \int \frac {-2 x-25}{x \left (x^2+25 x-4\right ) \log \left (\frac {1}{x}\right )}dx-2 \operatorname {LogIntegral}\left (\frac {1}{x}\right ) \log \left (-x^2-25 x+4\right )+x^4-2 x-\frac {25}{641} \left (641-25 \sqrt {641}\right ) \log \left (2 x-\sqrt {641}+25\right )+\frac {1}{641} \left (16025-633 \sqrt {641}\right ) \log \left (2 x-\sqrt {641}+25\right )-\frac {8}{641} \left (403189-15925 \sqrt {641}\right ) \log \left (2 x-\sqrt {641}+25\right )-\frac {50}{641} \left (10143825-400657 \sqrt {641}\right ) \log \left (2 x-\sqrt {641}+25\right )+\frac {2}{641} \left (255208381-10080125 \sqrt {641}\right ) \log \left (2 x-\sqrt {641}+25\right )+\frac {8 \log \left (2 x-\sqrt {641}+25\right )}{\sqrt {641}}+\frac {2}{641} \left (255208381+10080125 \sqrt {641}\right ) \log \left (2 x+\sqrt {641}+25\right )-\frac {50}{641} \left (10143825+400657 \sqrt {641}\right ) \log \left (2 x+\sqrt {641}+25\right )-\frac {8}{641} \left (403189+15925 \sqrt {641}\right ) \log \left (2 x+\sqrt {641}+25\right )+\frac {1}{641} \left (16025+633 \sqrt {641}\right ) \log \left (2 x+\sqrt {641}+25\right )-\frac {25}{641} \left (641+25 \sqrt {641}\right ) \log \left (2 x+\sqrt {641}+25\right )-\frac {8 \log \left (2 x+\sqrt {641}+25\right )}{\sqrt {641}}\)

Input:

Int[((8*x^2 - 50*x^3 - 2*x^4 - 16*x^5 + 100*x^6 + 4*x^7)*Log[x^(-1)]^2 + ( 
8 - 50*x - 2*x^2)*Log[4 - 25*x - x^2] + Log[x^(-1)]*(-50*x - 4*x^2 + (-8 + 
 50*x + 2*x^2)*Log[4 - 25*x - x^2]))/((-4*x^2 + 25*x^3 + x^4)*Log[x^(-1)]^ 
2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 21.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {2 \ln \left (-x^{2}-25 x +4\right )}{x \ln \left (x \right )}+x^{4}-2 x\)	\(28\)
parallelrisch	\(\frac {50 \ln \left (\frac {1}{x}\right ) x^{5}-100 x^{2} \ln \left (\frac {1}{x}\right )+1234 x \ln \left (\frac {1}{x}\right )-100 \ln \left (-x^{2}-25 x +4\right )}{50 x \ln \left (\frac {1}{x}\right )}\)	\(51\)

Input:

int(((4*x^7+100*x^6-16*x^5-2*x^4-50*x^3+8*x^2)*ln(1/x)^2+((2*x^2+50*x-8)*l 
n(-x^2-25*x+4)-4*x^2-50*x)*ln(1/x)+(-2*x^2-50*x+8)*ln(-x^2-25*x+4))/(x^4+2 
5*x^3-4*x^2)/ln(1/x)^2,x,method=_RETURNVERBOSE)
 

Output:

2/x/ln(x)*ln(-x^2-25*x+4)+x^4-2*x
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=\frac {{\left (x^{5} - 2 \, x^{2}\right )} \log \left (\frac {1}{x}\right ) - 2 \, \log \left (-x^{2} - 25 \, x + 4\right )}{x \log \left (\frac {1}{x}\right )} \] Input:

integrate(((4*x^7+100*x^6-16*x^5-2*x^4-50*x^3+8*x^2)*log(1/x)^2+((2*x^2+50 
*x-8)*log(-x^2-25*x+4)-4*x^2-50*x)*log(1/x)+(-2*x^2-50*x+8)*log(-x^2-25*x+ 
4))/(x^4+25*x^3-4*x^2)/log(1/x)^2,x, algorithm="fricas")
 

Output:

((x^5 - 2*x^2)*log(1/x) - 2*log(-x^2 - 25*x + 4))/(x*log(1/x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((4*x**7+100*x**6-16*x**5-2*x**4-50*x**3+8*x**2)*ln(1/x)**2+((2* 
x**2+50*x-8)*ln(-x**2-25*x+4)-4*x**2-50*x)*ln(1/x)+(-2*x**2-50*x+8)*ln(-x* 
*2-25*x+4))/(x**4+25*x**3-4*x**2)/ln(1/x)**2,x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=\frac {{\left (x^{5} - 2 \, x^{2}\right )} \log \left (x\right ) + 2 \, \log \left (-x^{2} - 25 \, x + 4\right )}{x \log \left (x\right )} \] Input:

integrate(((4*x^7+100*x^6-16*x^5-2*x^4-50*x^3+8*x^2)*log(1/x)^2+((2*x^2+50 
*x-8)*log(-x^2-25*x+4)-4*x^2-50*x)*log(1/x)+(-2*x^2-50*x+8)*log(-x^2-25*x+ 
4))/(x^4+25*x^3-4*x^2)/log(1/x)^2,x, algorithm="maxima")
 

Output:

((x^5 - 2*x^2)*log(x) + 2*log(-x^2 - 25*x + 4))/(x*log(x))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=x^{4} - 2 \, x + \frac {2 \, \log \left (-x^{2} - 25 \, x + 4\right )}{x \log \left (x\right )} \] Input:

integrate(((4*x^7+100*x^6-16*x^5-2*x^4-50*x^3+8*x^2)*log(1/x)^2+((2*x^2+50 
*x-8)*log(-x^2-25*x+4)-4*x^2-50*x)*log(1/x)+(-2*x^2-50*x+8)*log(-x^2-25*x+ 
4))/(x^4+25*x^3-4*x^2)/log(1/x)^2,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.91 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=x^4-2\,x-\frac {2\,\ln \left (-x^2-25\,x+4\right )}{x\,\ln \left (\frac {1}{x}\right )} \] Input:

int(-(log(4 - x^2 - 25*x)*(50*x + 2*x^2 - 8) - log(1/x)^2*(8*x^2 - 50*x^3 
- 2*x^4 - 16*x^5 + 100*x^6 + 4*x^7) + log(1/x)*(50*x - log(4 - x^2 - 25*x) 
*(50*x + 2*x^2 - 8) + 4*x^2))/(log(1/x)^2*(25*x^3 - 4*x^2 + x^4)),x)
 

Output:

x^4 - 2*x - (2*log(4 - x^2 - 25*x))/(x*log(1/x))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {\left (8 x^2-50 x^3-2 x^4-16 x^5+100 x^6+4 x^7\right ) \log ^2\left (\frac {1}{x}\right )+\left (8-50 x-2 x^2\right ) \log \left (4-25 x-x^2\right )+\log \left (\frac {1}{x}\right ) \left (-50 x-4 x^2+\left (-8+50 x+2 x^2\right ) \log \left (4-25 x-x^2\right )\right )}{\left (-4 x^2+25 x^3+x^4\right ) \log ^2\left (\frac {1}{x}\right )} \, dx=\frac {2 \,\mathrm {log}\left (-x^{2}-25 x +4\right )+\mathrm {log}\left (x \right ) x^{5}-2 \,\mathrm {log}\left (x \right ) x^{2}}{\mathrm {log}\left (x \right ) x} \] Input:

int(((4*x^7+100*x^6-16*x^5-2*x^4-50*x^3+8*x^2)*log(1/x)^2+((2*x^2+50*x-8)* 
log(-x^2-25*x+4)-4*x^2-50*x)*log(1/x)+(-2*x^2-50*x+8)*log(-x^2-25*x+4))/(x 
^4+25*x^3-4*x^2)/log(1/x)^2,x)
 

Output:

(2*log( - x**2 - 25*x + 4) + log(x)*x**5 - 2*log(x)*x**2)/(log(x)*x)