\(\int \frac {-3264 x+864 x^2-76 x^3+(-32 x+16 x^2-2 x^3) \log (5 x^2)}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+(64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6) \log (5 x^2)+(16 x^4-8 x^5+x^6) \log ^2(5 x^2)} \, dx\) [1535]

Optimal result

Integrand size = 124, antiderivative size = 30 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {1}{2+x^2 \left (1+\frac {(-20+x)^2}{4-x}+x+\log \left (5 x^2\right )\right )} \] Output:

1/(x^2*(ln(5*x^2)+1+(x-20)^2/(4-x)+x)+2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {-4+x}{-8+2 x-404 x^2+37 x^3+(-4+x) x^2 \log \left (5 x^2\right )} \] Input:

Integrate[(-3264*x + 864*x^2 - 76*x^3 + (-32*x + 16*x^2 - 2*x^3)*Log[5*x^2 
])/(64 - 32*x + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x^5 + 1369*x^6 + 
(64*x^2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6)*Log[5*x^2] + (16*x^4 - 8* 
x^5 + x^6)*Log[5*x^2]^2),x]
 

Output:

(-4 + x)/(-8 + 2*x - 404*x^2 + 37*x^3 + (-4 + x)*x^2*Log[5*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.

\(\displaystyle \int \frac {-76 x^3+864 x^2+\left (-2 x^3+16 x^2-32 x\right ) \log \left (5 x^2\right )-3264 x}{1369 x^6-29896 x^5+163364 x^4-2208 x^3+6468 x^2+\left (x^6-8 x^5+16 x^4\right ) \log ^2\left (5 x^2\right )+\left (74 x^6-1104 x^5+3236 x^4-32 x^3+64 x^2\right ) \log \left (5 x^2\right )-32 x+64} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x \left (-38 x^2+(x-4)^2 \left (-\log \left (5 x^2\right )\right )+432 x-1632\right )}{\left (-37 x^3+404 x^2-(x-4) x^2 \log \left (5 x^2\right )-2 x+8\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {x \left (\log \left (5 x^2\right ) (4-x)^2+38 x^2-432 x+1632\right )}{\left (-37 x^3+(4-x) \log \left (5 x^2\right ) x^2+404 x^2-2 x+8\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {x \left (\log \left (5 x^2\right ) (4-x)^2+38 x^2-432 x+1632\right )}{\left (-37 x^3+(4-x) \log \left (5 x^2\right ) x^2+404 x^2-2 x+8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {x-4}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )}+\frac {x^4+120 x^3+14 x^2+16 x-32}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (16 \int \frac {1}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx-32 \int \frac {1}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+14 \int \frac {x}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+120 \int \frac {x^2}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+\int \frac {x^3}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+\int \frac {1}{\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8}dx-4 \int \frac {1}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )}dx\right )\)

Input:

Int[(-3264*x + 864*x^2 - 76*x^3 + (-32*x + 16*x^2 - 2*x^3)*Log[5*x^2])/(64 
 - 32*x + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x^5 + 1369*x^6 + (64*x^ 
2 - 32*x^3 + 3236*x^4 - 1104*x^5 + 74*x^6)*Log[5*x^2] + (16*x^4 - 8*x^5 + 
x^6)*Log[5*x^2]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43

method	result	size
norman	\(\frac {x -4}{x^{3} \ln \left (5 x^{2}\right )-4 \ln \left (5 x^{2}\right ) x^{2}+37 x^{3}-404 x^{2}+2 x -8}\)	\(43\)
risch	\(\frac {x -4}{x^{3} \ln \left (5 x^{2}\right )-4 \ln \left (5 x^{2}\right ) x^{2}+37 x^{3}-404 x^{2}+2 x -8}\)	\(43\)
parallelrisch	\(\frac {2 x -8}{2 x^{3} \ln \left (5 x^{2}\right )-8 \ln \left (5 x^{2}\right ) x^{2}+74 x^{3}-808 x^{2}+4 x -16}\)	\(46\)

Input:

int(((-2*x^3+16*x^2-32*x)*ln(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16* 
x^4)*ln(5*x^2)^2+(74*x^6-1104*x^5+3236*x^4-32*x^3+64*x^2)*ln(5*x^2)+1369*x 
^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x,method=_RETURNVERBOSE 
)
 

Output:

(x-4)/(x^3*ln(5*x^2)-4*ln(5*x^2)*x^2+37*x^3-404*x^2+2*x-8)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 \, x^{3} - 404 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (5 \, x^{2}\right ) + 2 \, x - 8} \] Input:

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8* 
x^5+16*x^4)*log(5*x^2)^2+(74*x^6-1104*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^ 
2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm=" 
fricas")
 

Output:

(x - 4)/(37*x^3 - 404*x^2 + (x^3 - 4*x^2)*log(5*x^2) + 2*x - 8)
 

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 x^{3} - 404 x^{2} + 2 x + \left (x^{3} - 4 x^{2}\right ) \log {\left (5 x^{2} \right )} - 8} \] Input:

integrate(((-2*x**3+16*x**2-32*x)*ln(5*x**2)-76*x**3+864*x**2-3264*x)/((x* 
*6-8*x**5+16*x**4)*ln(5*x**2)**2+(74*x**6-1104*x**5+3236*x**4-32*x**3+64*x 
**2)*ln(5*x**2)+1369*x**6-29896*x**5+163364*x**4-2208*x**3+6468*x**2-32*x+ 
64),x)
 

Output:

(x - 4)/(37*x**3 - 404*x**2 + 2*x + (x**3 - 4*x**2)*log(5*x**2) - 8)
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} {\left (\log \left (5\right ) + 37\right )} - 4 \, x^{2} {\left (\log \left (5\right ) + 101\right )} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x\right ) + 2 \, x - 8} \] Input:

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8* 
x^5+16*x^4)*log(5*x^2)^2+(74*x^6-1104*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^ 
2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm=" 
maxima")
 

Output:

(x - 4)/(x^3*(log(5) + 37) - 4*x^2*(log(5) + 101) + 2*(x^3 - 4*x^2)*log(x) 
 + 2*x - 8)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} \log \left (5 \, x^{2}\right ) + 37 \, x^{3} - 4 \, x^{2} \log \left (5 \, x^{2}\right ) - 404 \, x^{2} + 2 \, x - 8} \] Input:

integrate(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8* 
x^5+16*x^4)*log(5*x^2)^2+(74*x^6-1104*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^ 
2)+1369*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x, algorithm=" 
giac")
 

Output:

(x - 4)/(x^3*log(5*x^2) + 37*x^3 - 4*x^2*log(5*x^2) - 404*x^2 + 2*x - 8)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\int -\frac {3264\,x+\ln \left (5\,x^2\right )\,\left (2\,x^3-16\,x^2+32\,x\right )-864\,x^2+76\,x^3}{{\ln \left (5\,x^2\right )}^2\,\left (x^6-8\,x^5+16\,x^4\right )-32\,x+\ln \left (5\,x^2\right )\,\left (74\,x^6-1104\,x^5+3236\,x^4-32\,x^3+64\,x^2\right )+6468\,x^2-2208\,x^3+163364\,x^4-29896\,x^5+1369\,x^6+64} \,d x \] Input:

int(-(3264*x + log(5*x^2)*(32*x - 16*x^2 + 2*x^3) - 864*x^2 + 76*x^3)/(log 
(5*x^2)^2*(16*x^4 - 8*x^5 + x^6) - 32*x + log(5*x^2)*(64*x^2 - 32*x^3 + 32 
36*x^4 - 1104*x^5 + 74*x^6) + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x^5 
 + 1369*x^6 + 64),x)
 

Output:

int(-(3264*x + log(5*x^2)*(32*x - 16*x^2 + 2*x^3) - 864*x^2 + 76*x^3)/(log 
(5*x^2)^2*(16*x^4 - 8*x^5 + x^6) - 32*x + log(5*x^2)*(64*x^2 - 32*x^3 + 32 
36*x^4 - 1104*x^5 + 74*x^6) + 6468*x^2 - 2208*x^3 + 163364*x^4 - 29896*x^5 
 + 1369*x^6 + 64), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x -4}{\mathrm {log}\left (5 x^{2}\right ) x^{3}-4 \,\mathrm {log}\left (5 x^{2}\right ) x^{2}+37 x^{3}-404 x^{2}+2 x -8} \] Input:

int(((-2*x^3+16*x^2-32*x)*log(5*x^2)-76*x^3+864*x^2-3264*x)/((x^6-8*x^5+16 
*x^4)*log(5*x^2)^2+(74*x^6-1104*x^5+3236*x^4-32*x^3+64*x^2)*log(5*x^2)+136 
9*x^6-29896*x^5+163364*x^4-2208*x^3+6468*x^2-32*x+64),x)
 

Output:

(x - 4)/(log(5*x**2)*x**3 - 4*log(5*x**2)*x**2 + 37*x**3 - 404*x**2 + 2*x 
- 8)