Integrand size = 24, antiderivative size = 168 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d}+\frac {2 n^2 \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {2 n^2 \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )}{d} \]
ln(-b*x/a)*ln(c*(b*x+a)^n)^2/d-ln(c*(b*x+a)^n)^2*ln(b*(e*x+d)/(-a*e+b*d))/ d-2*n*ln(c*(b*x+a)^n)*polylog(2,-e*(b*x+a)/(-a*e+b*d))/d+2*n*ln(c*(b*x+a)^ n)*polylog(2,1+b*x/a)/d+2*n^2*polylog(3,-e*(b*x+a)/(-a*e+b*d))/d-2*n^2*pol ylog(3,1+b*x/a)/d
Time = 0.09 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.74 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log (x) \left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2-\left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2 \log (d+e x)-2 n \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (\log (x) \left (\log (a+b x)-\log \left (1+\frac {b x}{a}\right )\right )-\log (a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )\right )+n^2 \left (\log \left (-\frac {b x}{a}\right ) \log ^2(a+b x)-\log ^2(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \log (a+b x) \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )+2 \log (a+b x) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+2 \operatorname {PolyLog}\left (3,\frac {e (a+b x)}{-b d+a e}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )\right )}{d} \]
(Log[x]*(-(n*Log[a + b*x]) + Log[c*(a + b*x)^n])^2 - (-(n*Log[a + b*x]) + Log[c*(a + b*x)^n])^2*Log[d + e*x] - 2*n*(n*Log[a + b*x] - Log[c*(a + b*x) ^n])*(Log[x]*(Log[a + b*x] - Log[1 + (b*x)/a]) - Log[a + b*x]*Log[(b*(d + e*x))/(b*d - a*e)] - PolyLog[2, -((b*x)/a)] - PolyLog[2, (e*(a + b*x))/(-( b*d) + a*e)]) + n^2*(Log[-((b*x)/a)]*Log[a + b*x]^2 - Log[a + b*x]^2*Log[( b*(d + e*x))/(b*d - a*e)] - 2*Log[a + b*x]*PolyLog[2, (e*(a + b*x))/(-(b*d ) + a*e)] + 2*Log[a + b*x]*PolyLog[2, 1 + (b*x)/a] + 2*PolyLog[3, (e*(a + b*x))/(-(b*d) + a*e)] - 2*PolyLog[3, 1 + (b*x)/a]))/d
Time = 0.46 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2026, 2863, 2009}
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.
| \(\displaystyle \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\log ^2\left (c (a+b x)^n\right )}{x (d+e x)}dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\log ^2\left (c (a+b x)^n\right )}{d x}-\frac {e \log ^2\left (c (a+b x)^n\right )}{d (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {2 n \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{d}+\frac {2 n^2 \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {b x}{a}+1\right )}{d}\) |
(Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^2)/d - (Log[c*(a + b*x)^n]^2*Log[(b*(d + e*x))/(b*d - a*e)])/d - (2*n*Log[c*(a + b*x)^n]*PolyLog[2, -((e*(a + b* x))/(b*d - a*e))])/d + (2*n*Log[c*(a + b*x)^n]*PolyLog[2, 1 + (b*x)/a])/d + (2*n^2*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (2*n^2*PolyLog[3, 1 + (b*x)/a])/d
3.4.44.3.1 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[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.28 (sec) , antiderivative size = 761, normalized size of antiderivative = 4.53
| method | result | size |
| risch | \(\frac {{\left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right )}^{2} \ln \left (b x \right )}{d}-\frac {{\left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right )}^{2} \ln \left (e \left (b x +a \right )-a e +b d \right )}{d}+\frac {n^{2} \ln \left (b x +a \right )^{2} \ln \left (1-\frac {b x +a}{a}\right )}{d}+\frac {2 n^{2} \ln \left (b x +a \right ) \operatorname {Li}_{2}\left (\frac {b x +a}{a}\right )}{d}-\frac {2 n^{2} \operatorname {Li}_{3}\left (\frac {b x +a}{a}\right )}{d}-\frac {n^{2} \ln \left (b x +a \right )^{2} \ln \left (1+\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}-\frac {2 n^{2} \ln \left (b x +a \right ) \operatorname {Li}_{2}\left (-\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}+\frac {2 n^{2} \operatorname {Li}_{3}\left (-\frac {e \left (b x +a \right )}{-a e +b d}\right )}{d}+2 b n \left (\ln \left (\left (b x +a \right )^{n}\right )-n \ln \left (b x +a \right )\right ) \left (\frac {\operatorname {dilog}\left (-\frac {x b}{a}\right )+\ln \left (b x +a \right ) \ln \left (-\frac {x b}{a}\right )}{b d}-\frac {e \left (\frac {\operatorname {dilog}\left (\frac {e \left (b x +a \right )-a e +b d}{-a e +b d}\right )}{e}+\frac {\ln \left (b x +a \right ) \ln \left (\frac {e \left (b x +a \right )-a e +b d}{-a e +b d}\right )}{e}\right )}{b d}\right )+\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right ) \ln \left (\left (b x +a \right )^{n}\right )}{d}+\frac {\ln \left (\left (b x +a \right )^{n}\right ) \ln \left (x \right )}{d}-b n \left (\frac {\operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d b}+\frac {\ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d b}-\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d b}\right )\right )+\frac {{\left (-i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{3}+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right )+i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,\operatorname {csgn}\left (i c \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2} \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )}{4}\) | \(761\) |
(ln((b*x+a)^n)-n*ln(b*x+a))^2/d*ln(b*x)-(ln((b*x+a)^n)-n*ln(b*x+a))^2/d*ln (e*(b*x+a)-a*e+b*d)+n^2/d*ln(b*x+a)^2*ln(1-(b*x+a)/a)+2*n^2/d*ln(b*x+a)*po lylog(2,(b*x+a)/a)-2*n^2/d*polylog(3,(b*x+a)/a)-n^2/d*ln(b*x+a)^2*ln(1+e*( b*x+a)/(-a*e+b*d))-2*n^2/d*ln(b*x+a)*polylog(2,-e*(b*x+a)/(-a*e+b*d))+2*n^ 2*polylog(3,-e*(b*x+a)/(-a*e+b*d))/d+2*b*n*(ln((b*x+a)^n)-n*ln(b*x+a))*(1/ b/d*(dilog(-x/a*b)+ln(b*x+a)*ln(-x/a*b))-e/b/d*(dilog((e*(b*x+a)-a*e+b*d)/ (-a*e+b*d))/e+ln(b*x+a)*ln((e*(b*x+a)-a*e+b*d)/(-a*e+b*d))/e))+(-I*Pi*csgn (I*c*(b*x+a)^n)^3+I*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)+I*Pi*csgn(I *c*(b*x+a)^n)^2*csgn(I*c)-I*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*(b*x+a)^n)*csgn( I*c)+2*ln(c))*(-1/d*ln(e*x+d)*ln((b*x+a)^n)+ln((b*x+a)^n)/d*ln(x)-b*n*(1/d *dilog((b*x+a)/a)/b+1/d*ln(x)*ln((b*x+a)/a)/b-1/d*dilog(((e*x+d)*b+a*e-b*d )/(a*e-b*d))/b-1/d*ln(e*x+d)*ln(((e*x+d)*b+a*e-b*d)/(a*e-b*d))/b))+1/4*(-I *Pi*csgn(I*c*(b*x+a)^n)^3+I*Pi*csgn(I*c*(b*x+a)^n)^2*csgn(I*(b*x+a)^n)+I*P i*csgn(I*c*(b*x+a)^n)^2*csgn(I*c)-I*Pi*csgn(I*c*(b*x+a)^n)*csgn(I*(b*x+a)^ n)*csgn(I*c)+2*ln(c))^2*(-1/d*ln(e*x+d)+1/d*ln(x))
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{e x^{2} + d x} \,d x } \]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{x \left (d + e x\right )}\, dx \]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{e x^{2} + d x} \,d x } \]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{e x^{2} + d x} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{e\,x^2+d\,x} \,d x \]