Integrand size = 97, antiderivative size = 29 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=\frac {x+5 \left (-3+\frac {4+x}{x \log \left ((-3+x)^2\right )}\right )}{1+2 x} \]
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=\frac {-31+\frac {10 (4+x)}{x \log \left ((-3+x)^2\right )}}{2 (1+2 x)} \]
Integrate[(-40*x - 90*x^2 - 20*x^3 + (60 + 220*x - 50*x^2 - 10*x^3)*Log[9 - 6*x + x^2] + (-93*x^2 + 31*x^3)*Log[9 - 6*x + x^2]^2)/((-3*x^2 - 11*x^3 - 8*x^4 + 4*x^5)*Log[9 - 6*x + 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.
\(\displaystyle \int \frac {-20 x^3-90 x^2+\left (31 x^3-93 x^2\right ) \log ^2\left (x^2-6 x+9\right )+\left (-10 x^3-50 x^2+220 x+60\right ) \log \left (x^2-6 x+9\right )-40 x}{\left (4 x^5-8 x^4-11 x^3-3 x^2\right ) \log ^2\left (x^2-6 x+9\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-20 x^3-90 x^2+\left (31 x^3-93 x^2\right ) \log ^2\left (x^2-6 x+9\right )+\left (-10 x^3-50 x^2+220 x+60\right ) \log \left (x^2-6 x+9\right )-40 x}{x^2 \left (4 x^3-8 x^2-11 x-3\right ) \log ^2\left (x^2-6 x+9\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {2 \left (-20 x^3-90 x^2+\left (31 x^3-93 x^2\right ) \log ^2\left (x^2-6 x+9\right )+\left (-10 x^3-50 x^2+220 x+60\right ) \log \left (x^2-6 x+9\right )-40 x\right )}{49 x^2 (2 x+1) \log ^2\left (x^2-6 x+9\right )}+\frac {-20 x^3-90 x^2+\left (31 x^3-93 x^2\right ) \log ^2\left (x^2-6 x+9\right )+\left (-10 x^3-50 x^2+220 x+60\right ) \log \left (x^2-6 x+9\right )-40 x}{49 (x-3) x^2 \log ^2\left (x^2-6 x+9\right )}-\frac {2 \left (-20 x^3-90 x^2+\left (31 x^3-93 x^2\right ) \log ^2\left (x^2-6 x+9\right )+\left (-10 x^3-50 x^2+220 x+60\right ) \log \left (x^2-6 x+9\right )-40 x\right )}{7 x^2 (2 x+1)^2 \log ^2\left (x^2-6 x+9\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -20 \int \frac {1}{x^2 \log \left ((x-3)^2\right )}dx+\frac {40}{3} \int \frac {1}{x \log ^2\left ((x-3)^2\right )}dx-20 \int \frac {1}{(2 x+1) \log ^2\left ((x-3)^2\right )}dx+70 \int \frac {1}{(2 x+1)^2 \log \left ((x-3)^2\right )}dx-\frac {31}{2 (2 x+1)}+\frac {5}{3 \log \left ((x-3)^2\right )}\) |
Int[(-40*x - 90*x^2 - 20*x^3 + (60 + 220*x - 50*x^2 - 10*x^3)*Log[9 - 6*x + x^2] + (-93*x^2 + 31*x^3)*Log[9 - 6*x + x^2]^2)/((-3*x^2 - 11*x^3 - 8*x^ 4 + 4*x^5)*Log[9 - 6*x + x^2]^2),x]
3.12.1.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 37.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {31}{2 \left (1+2 x \right )}+\frac {20+5 x}{x \left (1+2 x \right ) \ln \left (x^{2}-6 x +9\right )}\) | \(37\) |
norman | \(\frac {20-\frac {31 \ln \left (x^{2}-6 x +9\right ) x}{2}+5 x}{x \left (1+2 x \right ) \ln \left (x^{2}-6 x +9\right )}\) | \(40\) |
parallelrisch | \(\frac {40-31 \ln \left (x^{2}-6 x +9\right ) x +10 x}{2 \ln \left (x^{2}-6 x +9\right ) x \left (1+2 x \right )}\) | \(41\) |
int(((31*x^3-93*x^2)*ln(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*ln(x^2-6*x+ 9)-20*x^3-90*x^2-40*x)/(4*x^5-8*x^4-11*x^3-3*x^2)/ln(x^2-6*x+9)^2,x,method =_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=-\frac {31 \, x \log \left (x^{2} - 6 \, x + 9\right ) - 10 \, x - 40}{2 \, {\left (2 \, x^{2} + x\right )} \log \left (x^{2} - 6 \, x + 9\right )} \]
integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log( x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2 ,x, algorithm=\
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=\frac {5 x + 20}{\left (2 x^{2} + x\right ) \log {\left (x^{2} - 6 x + 9 \right )}} - \frac {31}{4 x + 2} \]
integrate(((31*x**3-93*x**2)*ln(x**2-6*x+9)**2+(-10*x**3-50*x**2+220*x+60) *ln(x**2-6*x+9)-20*x**3-90*x**2-40*x)/(4*x**5-8*x**4-11*x**3-3*x**2)/ln(x* *2-6*x+9)**2,x)
Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=-\frac {31 \, x \log \left (x - 3\right ) - 5 \, x - 20}{2 \, {\left (2 \, x^{2} + x\right )} \log \left (x - 3\right )} \]
integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log( x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2 ,x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=\frac {5 \, {\left (x + 4\right )}}{2 \, x^{2} \log \left (x^{2} - 6 \, x + 9\right ) + x \log \left (x^{2} - 6 \, x + 9\right )} - \frac {31}{2 \, {\left (2 \, x + 1\right )}} \]
integrate(((31*x^3-93*x^2)*log(x^2-6*x+9)^2+(-10*x^3-50*x^2+220*x+60)*log( x^2-6*x+9)-20*x^3-90*x^2-40*x)/(4*x^5-8*x^4-11*x^3-3*x^2)/log(x^2-6*x+9)^2 ,x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-40 x-90 x^2-20 x^3+\left (60+220 x-50 x^2-10 x^3\right ) \log \left (9-6 x+x^2\right )+\left (-93 x^2+31 x^3\right ) \log ^2\left (9-6 x+x^2\right )}{\left (-3 x^2-11 x^3-8 x^4+4 x^5\right ) \log ^2\left (9-6 x+x^2\right )} \, dx=\frac {31\,x}{2\,x+1}+\frac {5\,x+20}{x\,\ln \left (x^2-6\,x+9\right )\,\left (2\,x+1\right )} \]