\(\int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log (-5 x^2+x^3)}{-45 x^4+9 x^5+(-480 x^2+96 x^3) \log (2)+(-1280+256 x) \log ^2(2)+(1440 x^2-288 x^3+(7680-1536 x) \log (2)) \log (-5 x^2+x^3)+(-11520+2304 x) \log ^2(-5 x^2+x^3)} \, dx\) [1808]

Optimal result

Integrand size = 125, antiderivative size = 33 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=-1+\frac {x}{4 \left (\frac {x^2}{16}+\frac {\log (2)}{3}-\log \left ((-5+x) x^2\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 5.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=-\frac {12 x}{-3 x^2-16 \log (2)+48 \log \left ((-5+x) x^2\right )} \] Input:

Integrate[(-5760 + 1728*x + 180*x^2 - 36*x^3 + (-960 + 192*x)*Log[2] + (28 
80 - 576*x)*Log[-5*x^2 + x^3])/(-45*x^4 + 9*x^5 + (-480*x^2 + 96*x^3)*Log[ 
2] + (-1280 + 256*x)*Log[2]^2 + (1440*x^2 - 288*x^3 + (7680 - 1536*x)*Log[ 
2])*Log[-5*x^2 + x^3] + (-11520 + 2304*x)*Log[-5*x^2 + x^3]^2),x]
 

Output:

(-12*x)/(-3*x^2 - 16*Log[2] + 48*Log[(-5 + x)*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.

\(\displaystyle \int \frac {-36 x^3+180 x^2+(2880-576 x) \log \left (x^3-5 x^2\right )+1728 x+(192 x-960) \log (2)-5760}{9 x^5-45 x^4+(2304 x-11520) \log ^2\left (x^3-5 x^2\right )+\left (-288 x^3+1440 x^2+(7680-1536 x) \log (2)\right ) \log \left (x^3-5 x^2\right )+\left (96 x^3-480 x^2\right ) \log (2)+(256 x-1280) \log ^2(2)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {12 \left (3 x^3-15 x^2+48 (x-5) \log \left ((x-5) x^2\right )-16 x (9+\log (2))+80 (6+\log (2))\right )}{(5-x) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 12 \int -\frac {-3 x^3+15 x^2+16 (9+\log (2)) x+48 (5-x) \log \left (-\left ((5-x) x^2\right )\right )-80 (6+\log (2))}{(5-x) \left (3 x^2-48 \log \left (-\left ((5-x) x^2\right )\right )+16 \log (2)\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -12 \int \frac {-3 x^3+15 x^2+16 (9+\log (2)) x+48 (5-x) \log \left (-\left ((5-x) x^2\right )\right )-80 (6+\log (2))}{(5-x) \left (3 x^2-48 \log \left (-\left ((5-x) x^2\right )\right )+16 \log (2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -12 \int \left (\frac {6 \left (x^3-5 x^2-24 x+80\right )}{(x-5) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}+\frac {1}{-3 x^2+48 \log \left ((x-5) x^2\right )-16 \log (2)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -12 \left (-144 \int \frac {1}{\left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx-240 \int \frac {1}{(x-5) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx+6 \int \frac {x^2}{\left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx+\int \frac {1}{-3 x^2+48 \log \left ((x-5) x^2\right )-16 \log (2)}dx\right )\)

Input:

Int[(-5760 + 1728*x + 180*x^2 - 36*x^3 + (-960 + 192*x)*Log[2] + (2880 - 5 
76*x)*Log[-5*x^2 + x^3])/(-45*x^4 + 9*x^5 + (-480*x^2 + 96*x^3)*Log[2] + ( 
-1280 + 256*x)*Log[2]^2 + (1440*x^2 - 288*x^3 + (7680 - 1536*x)*Log[2])*Lo 
g[-5*x^2 + x^3] + (-11520 + 2304*x)*Log[-5*x^2 + x^3]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{2} \left (-5+x \right )\right )}\)	\(26\)
norman	\(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{3}-5 x^{2}\right )}\)	\(28\)
risch	\(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{3}-5 x^{2}\right )}\)	\(28\)

Input:

int(((-576*x+2880)*ln(x^3-5*x^2)+(192*x-960)*ln(2)-36*x^3+180*x^2+1728*x-5 
760)/((2304*x-11520)*ln(x^3-5*x^2)^2+((-1536*x+7680)*ln(2)-288*x^3+1440*x^ 
2)*ln(x^3-5*x^2)+(256*x-1280)*ln(2)^2+(96*x^3-480*x^2)*ln(2)+9*x^5-45*x^4) 
,x,method=_RETURNVERBOSE)
 

Output:

12*x/(3*x^2+16*ln(2)-48*ln(x^2*(-5+x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x^{3} - 5 \, x^{2}\right )} \] Input:

integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 
1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x 
^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 
9*x^5-45*x^4),x, algorithm="fricas")
 

Output:

12*x/(3*x^2 + 16*log(2) - 48*log(x^3 - 5*x^2))
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=- \frac {12 x}{- 3 x^{2} + 48 \log {\left (x^{3} - 5 x^{2} \right )} - 16 \log {\left (2 \right )}} \] Input:

integrate(((-576*x+2880)*ln(x**3-5*x**2)+(192*x-960)*ln(2)-36*x**3+180*x** 
2+1728*x-5760)/((2304*x-11520)*ln(x**3-5*x**2)**2+((-1536*x+7680)*ln(2)-28 
8*x**3+1440*x**2)*ln(x**3-5*x**2)+(256*x-1280)*ln(2)**2+(96*x**3-480*x**2) 
*ln(2)+9*x**5-45*x**4),x)
 

Output:

-12*x/(-3*x**2 + 48*log(x**3 - 5*x**2) - 16*log(2))
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x - 5\right ) - 96 \, \log \left (x\right )} \] Input:

integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 
1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x 
^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 
9*x^5-45*x^4),x, algorithm="maxima")
 

Output:

12*x/(3*x^2 + 16*log(2) - 48*log(x - 5) - 96*log(x))
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x^{3} - 5 \, x^{2}\right )} \] Input:

integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 
1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x 
^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 
9*x^5-45*x^4),x, algorithm="giac")
 

Output:

12*x/(3*x^2 + 16*log(2) - 48*log(x^3 - 5*x^2))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\int -\frac {1728\,x+\ln \left (2\right )\,\left (192\,x-960\right )-\ln \left (x^3-5\,x^2\right )\,\left (576\,x-2880\right )+180\,x^2-36\,x^3-5760}{\ln \left (x^3-5\,x^2\right )\,\left (\ln \left (2\right )\,\left (1536\,x-7680\right )-1440\,x^2+288\,x^3\right )-{\ln \left (2\right )}^2\,\left (256\,x-1280\right )+\ln \left (2\right )\,\left (480\,x^2-96\,x^3\right )-{\ln \left (x^3-5\,x^2\right )}^2\,\left (2304\,x-11520\right )+45\,x^4-9\,x^5} \,d x \] Input:

int(-(1728*x + log(2)*(192*x - 960) - log(x^3 - 5*x^2)*(576*x - 2880) + 18 
0*x^2 - 36*x^3 - 5760)/(log(x^3 - 5*x^2)*(log(2)*(1536*x - 7680) - 1440*x^ 
2 + 288*x^3) - log(2)^2*(256*x - 1280) + log(2)*(480*x^2 - 96*x^3) - log(x 
^3 - 5*x^2)^2*(2304*x - 11520) + 45*x^4 - 9*x^5),x)
 

Output:

int(-(1728*x + log(2)*(192*x - 960) - log(x^3 - 5*x^2)*(576*x - 2880) + 18 
0*x^2 - 36*x^3 - 5760)/(log(x^3 - 5*x^2)*(log(2)*(1536*x - 7680) - 1440*x^ 
2 + 288*x^3) - log(2)^2*(256*x - 1280) + log(2)*(480*x^2 - 96*x^3) - log(x 
^3 - 5*x^2)^2*(2304*x - 11520) + 45*x^4 - 9*x^5), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=-\frac {12 x}{48 \,\mathrm {log}\left (x^{3}-5 x^{2}\right )-16 \,\mathrm {log}\left (2\right )-3 x^{2}} \] Input:

int(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+1728*x 
-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x^3+144 
0*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+9*x^5- 
45*x^4),x)
 

Output:

( - 12*x)/(48*log(x**3 - 5*x**2) - 16*log(2) - 3*x**2)