\(\int \frac {25-20 x-20 x^2+4 x^4+(-25+10 x+20 x^2-4 x^3-4 x^4) \log (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2})}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+(-750 x+300 x^2+600 x^3-120 x^4-120 x^5) \log (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2})+(125-50 x-100 x^2+20 x^3+20 x^4) \log ^2(\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2})} \, dx\) [923]

Optimal result

Integrand size = 188, antiderivative size = 32 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \left (3 x-\log \left (2 x+\frac {2 x^2}{-\frac {5}{2}+x^2}\right )\right )} \] Output:

x/(15*x-5*ln(2*x^2/(x^2-5/2)+2*x))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=-\frac {x}{5 \left (-3 x+\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )} \] Input:

Integrate[(25 - 20*x - 20*x^2 + 4*x^4 + (-25 + 10*x + 20*x^2 - 4*x^3 - 4*x 
^4)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)])/(1125*x^2 - 450*x^3 - 900*x 
^4 + 180*x^5 + 180*x^6 + (-750*x + 300*x^2 + 600*x^3 - 120*x^4 - 120*x^5)* 
Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)] + (125 - 50*x - 100*x^2 + 20*x^3 
 + 20*x^4)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)]^2),x]
 

Output:

-1/5*x/(-3*x + Log[(2*x*(-5 + 2*x + 2*x^2))/(-5 + 2*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.

\(\displaystyle \int \frac {4 x^4-20 x^2+\left (-4 x^4-4 x^3+20 x^2+10 x-25\right ) \log \left (\frac {4 x^3+4 x^2-10 x}{2 x^2-5}\right )-20 x+25}{180 x^6+180 x^5-900 x^4-450 x^3+1125 x^2+\left (20 x^4+20 x^3-100 x^2-50 x+125\right ) \log ^2\left (\frac {4 x^3+4 x^2-10 x}{2 x^2-5}\right )+\left (-120 x^5-120 x^4+600 x^3+300 x^2-750 x\right ) \log \left (\frac {4 x^3+4 x^2-10 x}{2 x^2-5}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {4 x^4-20 x^2+\left (-4 x^4-4 x^3+20 x^2+10 x-25\right ) \log \left (\frac {4 x^3+4 x^2-10 x}{2 x^2-5}\right )-20 x+25}{5 \left (4 x^4+4 x^3-20 x^2-10 x+25\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {4 x^4-20 x^2-20 x-\left (4 x^4+4 x^3-20 x^2-10 x+25\right ) \log \left (\frac {2 \left (-2 x^3-2 x^2+5 x\right )}{5-2 x^2}\right )+25}{\left (4 x^4+4 x^3-20 x^2-10 x+25\right ) \left (3 x-\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )\right )^2}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \frac {1}{5} \int \left (\frac {x \left (4 x^4-20 x^2-20 x-\left (4 x^4+4 x^3-20 x^2-10 x+25\right ) \log \left (\frac {2 \left (-2 x^3-2 x^2+5 x\right )}{5-2 x^2}\right )+25\right )}{5 \left (2 x^2-5\right ) \left (3 x-\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )\right )^2}+\frac {(-x-1) \left (4 x^4-20 x^2-20 x-\left (4 x^4+4 x^3-20 x^2-10 x+25\right ) \log \left (\frac {2 \left (-2 x^3-2 x^2+5 x\right )}{5-2 x^2}\right )+25\right )}{5 \left (2 x^2+2 x-5\right ) \left (3 x-\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} \left (\int \frac {1}{\left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx-\frac {20 \int \frac {1}{\left (-4 x+2 \sqrt {11}-2\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx}{\sqrt {11}}-3 \int \frac {x}{\left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx-\frac {2}{11} \left (11-\sqrt {11}\right ) \int \frac {1}{\left (4 x-2 \sqrt {11}+2\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx-\frac {2}{11} \left (11+\sqrt {11}\right ) \int \frac {1}{\left (4 x+2 \sqrt {11}+2\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx-\frac {20 \int \frac {1}{\left (4 x+2 \sqrt {11}+2\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx}{\sqrt {11}}+\sqrt {5} \int \frac {1}{\left (\sqrt {5}-\sqrt {2} x\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx+\sqrt {5} \int \frac {1}{\left (\sqrt {2} x+\sqrt {5}\right ) \left (3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )\right )^2}dx+\int \frac {1}{3 x-\log \left (\frac {2 x \left (2 x^2+2 x-5\right )}{2 x^2-5}\right )}dx\right )\)

Input:

Int[(25 - 20*x - 20*x^2 + 4*x^4 + (-25 + 10*x + 20*x^2 - 4*x^3 - 4*x^4)*Lo 
g[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)])/(1125*x^2 - 450*x^3 - 900*x^4 + 1 
80*x^5 + 180*x^6 + (-750*x + 300*x^2 + 600*x^3 - 120*x^4 - 120*x^5)*Log[(- 
10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)] + (125 - 50*x - 100*x^2 + 20*x^3 + 20* 
x^4)*Log[(-10*x + 4*x^2 + 4*x^3)/(-5 + 2*x^2)]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16

method	result	size
risch	\(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\)	\(37\)
parallelrisch	\(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\)	\(37\)
norman	\(\frac {\ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}{45 x -15 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\)	\(61\)

Input:

int(((-4*x^4-4*x^3+20*x^2+10*x-25)*ln((4*x^3+4*x^2-10*x)/(2*x^2-5))+4*x^4- 
20*x^2-20*x+25)/((20*x^4+20*x^3-100*x^2-50*x+125)*ln((4*x^3+4*x^2-10*x)/(2 
*x^2-5))^2+(-120*x^5-120*x^4+600*x^3+300*x^2-750*x)*ln((4*x^3+4*x^2-10*x)/ 
(2*x^2-5))+180*x^6+180*x^5-900*x^4-450*x^3+1125*x^2),x,method=_RETURNVERBO 
SE)
 

Output:

1/5*x/(3*x-ln((4*x^3+4*x^2-10*x)/(2*x^2-5)))
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \] Input:

integrate(((-4*x^4-4*x^3+20*x^2+10*x-25)*log((4*x^3+4*x^2-10*x)/(2*x^2-5)) 
+4*x^4-20*x^2-20*x+25)/((20*x^4+20*x^3-100*x^2-50*x+125)*log((4*x^3+4*x^2- 
10*x)/(2*x^2-5))^2+(-120*x^5-120*x^4+600*x^3+300*x^2-750*x)*log((4*x^3+4*x 
^2-10*x)/(2*x^2-5))+180*x^6+180*x^5-900*x^4-450*x^3+1125*x^2),x, algorithm 
="fricas")
 

Output:

1/5*x/(3*x - log(2*(2*x^3 + 2*x^2 - 5*x)/(2*x^2 - 5)))
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=- \frac {x}{- 15 x + 5 \log {\left (\frac {4 x^{3} + 4 x^{2} - 10 x}{2 x^{2} - 5} \right )}} \] Input:

integrate(((-4*x**4-4*x**3+20*x**2+10*x-25)*ln((4*x**3+4*x**2-10*x)/(2*x** 
2-5))+4*x**4-20*x**2-20*x+25)/((20*x**4+20*x**3-100*x**2-50*x+125)*ln((4*x 
**3+4*x**2-10*x)/(2*x**2-5))**2+(-120*x**5-120*x**4+600*x**3+300*x**2-750* 
x)*ln((4*x**3+4*x**2-10*x)/(2*x**2-5))+180*x**6+180*x**5-900*x**4-450*x**3 
+1125*x**2),x)
 

Output:

-x/(-15*x + 5*log((4*x**3 + 4*x**2 - 10*x)/(2*x**2 - 5)))
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (2\right ) - \log \left (2 \, x^{2} + 2 \, x - 5\right ) + \log \left (2 \, x^{2} - 5\right ) - \log \left (x\right )\right )}} \] Input:

integrate(((-4*x^4-4*x^3+20*x^2+10*x-25)*log((4*x^3+4*x^2-10*x)/(2*x^2-5)) 
+4*x^4-20*x^2-20*x+25)/((20*x^4+20*x^3-100*x^2-50*x+125)*log((4*x^3+4*x^2- 
10*x)/(2*x^2-5))^2+(-120*x^5-120*x^4+600*x^3+300*x^2-750*x)*log((4*x^3+4*x 
^2-10*x)/(2*x^2-5))+180*x^6+180*x^5-900*x^4-450*x^3+1125*x^2),x, algorithm 
="maxima")
 

Output:

1/5*x/(3*x - log(2) - log(2*x^2 + 2*x - 5) + log(2*x^2 - 5) - log(x))
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \] Input:

integrate(((-4*x^4-4*x^3+20*x^2+10*x-25)*log((4*x^3+4*x^2-10*x)/(2*x^2-5)) 
+4*x^4-20*x^2-20*x+25)/((20*x^4+20*x^3-100*x^2-50*x+125)*log((4*x^3+4*x^2- 
10*x)/(2*x^2-5))^2+(-120*x^5-120*x^4+600*x^3+300*x^2-750*x)*log((4*x^3+4*x 
^2-10*x)/(2*x^2-5))+180*x^6+180*x^5-900*x^4-450*x^3+1125*x^2),x, algorithm 
="giac")
 

Output:

1/5*x/(3*x - log(2*(2*x^3 + 2*x^2 - 5*x)/(2*x^2 - 5)))
 

Mupad [B] (verification not implemented)

Time = 0.97 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5\,\left (3\,x-\ln \left (\frac {4\,x^3+4\,x^2-10\,x}{2\,x^2-5}\right )\right )} \] Input:

int(-(20*x + log((4*x^2 - 10*x + 4*x^3)/(2*x^2 - 5))*(4*x^3 - 20*x^2 - 10* 
x + 4*x^4 + 25) + 20*x^2 - 4*x^4 - 25)/(log((4*x^2 - 10*x + 4*x^3)/(2*x^2 
- 5))^2*(20*x^3 - 100*x^2 - 50*x + 20*x^4 + 125) + 1125*x^2 - 450*x^3 - 90 
0*x^4 + 180*x^5 + 180*x^6 - log((4*x^2 - 10*x + 4*x^3)/(2*x^2 - 5))*(750*x 
 - 300*x^2 - 600*x^3 + 120*x^4 + 120*x^5)),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=-\frac {\mathrm {log}\left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}{15 \,\mathrm {log}\left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )-45 x} \] Input:

int(((-4*x^4-4*x^3+20*x^2+10*x-25)*log((4*x^3+4*x^2-10*x)/(2*x^2-5))+4*x^4 
-20*x^2-20*x+25)/((20*x^4+20*x^3-100*x^2-50*x+125)*log((4*x^3+4*x^2-10*x)/ 
(2*x^2-5))^2+(-120*x^5-120*x^4+600*x^3+300*x^2-750*x)*log((4*x^3+4*x^2-10* 
x)/(2*x^2-5))+180*x^6+180*x^5-900*x^4-450*x^3+1125*x^2),x)
 

Output:

( - log((4*x**3 + 4*x**2 - 10*x)/(2*x**2 - 5)))/(15*(log((4*x**3 + 4*x**2 
- 10*x)/(2*x**2 - 5)) - 3*x))