Integrand size = 124, antiderivative size = 30 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {1}{2+x^2 \left (1+\frac {(-20+x)^2}{4-x}+x+\log \left (5 x^2\right )\right )} \]
Time = 0.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {-4+x}{-8+2 x-404 x^2+37 x^3+(-4+x) x^2 \log \left (5 x^2\right )} \]
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]
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 |
\(\displaystyle -2 \left (16 \int \frac {1}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx-32 \int \frac {1}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+14 \int \frac {x}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+120 \int \frac {x^2}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+\int \frac {x^3}{\left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )^2}dx+\int \frac {1}{\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8}dx-4 \int \frac {1}{x \left (\log \left (5 x^2\right ) x^3+37 x^3-4 \log \left (5 x^2\right ) x^2-404 x^2+2 x-8\right )}dx\right )\) |
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]
3.16.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.55 (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 | \(\frac {2 x -8}{2 x^{3} \ln \left (5 x^{2}\right )-8 \ln \left (5 x^{2}\right ) x^{2}+74 x^{3}-808 x^{2}+4 x -16}\) | \(46\) |
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 )
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 \, x^{3} - 404 \, x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} \log \left (5 \, x^{2}\right ) + 2 \, x - 8} \]
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=\
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{37 x^{3} - 404 x^{2} + 2 x + \left (x^{3} - 4 x^{2}\right ) \log {\left (5 x^{2} \right )} - 8} \]
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)
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} {\left (\log \left (5\right ) + 37\right )} - 4 \, x^{2} {\left (\log \left (5\right ) + 101\right )} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} \log \left (x\right ) + 2 \, x - 8} \]
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=\
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\frac {x - 4}{x^{3} \log \left (5 \, x^{2}\right ) + 37 \, x^{3} - 4 \, x^{2} \log \left (5 \, x^{2}\right ) - 404 \, x^{2} + 2 \, x - 8} \]
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=\
Timed out. \[ \int \frac {-3264 x+864 x^2-76 x^3+\left (-32 x+16 x^2-2 x^3\right ) \log \left (5 x^2\right )}{64-32 x+6468 x^2-2208 x^3+163364 x^4-29896 x^5+1369 x^6+\left (64 x^2-32 x^3+3236 x^4-1104 x^5+74 x^6\right ) \log \left (5 x^2\right )+\left (16 x^4-8 x^5+x^6\right ) \log ^2\left (5 x^2\right )} \, dx=\int -\frac {3264\,x+\ln \left (5\,x^2\right )\,\left (2\,x^3-16\,x^2+32\,x\right )-864\,x^2+76\,x^3}{{\ln \left (5\,x^2\right )}^2\,\left (x^6-8\,x^5+16\,x^4\right )-32\,x+\ln \left (5\,x^2\right )\,\left (74\,x^6-1104\,x^5+3236\,x^4-32\,x^3+64\,x^2\right )+6468\,x^2-2208\,x^3+163364\,x^4-29896\,x^5+1369\,x^6+64} \,d x \]
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)