Integrand size = 32, antiderivative size = 637 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=f \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b g m n \log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x)-b g m \log \left (-\frac {j x}{i}\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \log (i+j x)+\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {e i-d j}{e (i+j x)}\right )-\log \left (-\frac {(e i-d j) x}{d (i+j x)}\right )\right ) \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right )-\frac {1}{2} b g m n \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right )^2-b g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right ) \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right )+a g \log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+b f n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+b g m n \left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )-b g n \left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \operatorname {PolyLog}\left (2,\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{e (i+j x)}\right )+a g m \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )-b g m \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )+b g m n \left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )-b g m n \operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )+b g m n \operatorname {PolyLog}\left (3,\frac {i (d+e x)}{d (i+j x)}\right )-b g m n \operatorname {PolyLog}\left (3,\frac {j (d+e x)}{e (i+j x)}\right )-b g m n \operatorname {PolyLog}\left (3,1+\frac {j x}{i}\right ) \]
f*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))+b*g*m*n*ln(-e*x/d)*ln(e*x+d)*ln(j*x+i)- b*g*m*ln(-j*x/i)*(n*ln(e*x+d)-ln(c*(e*x+d)^n))*ln(j*x+i)+1/2*b*g*m*n*(ln(- e*x/d)+ln((-d*j+e*i)/e/(j*x+i))-ln(-(-d*j+e*i)*x/d/(j*x+i)))*ln(d*(j*x+i)/ i/(e*x+d))^2-1/2*b*g*m*n*(ln(-e*x/d)-ln(-j*x/i))*(ln(e*x+d)+ln(d*(j*x+i)/i /(e*x+d)))^2-b*g*ln(-e*x/d)*ln(c*(e*x+d)^n)*(m*ln(j*x+i)-ln(h*(j*x+i)^m))+ a*g*ln(-j*x/i)*ln(h*(j*x+i)^m)+b*f*n*polylog(2,1+e*x/d)+b*g*m*n*(ln(j*x+i) -ln(d*(j*x+i)/i/(e*x+d)))*polylog(2,1+e*x/d)-b*g*n*(m*ln(j*x+i)-ln(h*(j*x+ i)^m))*polylog(2,1+e*x/d)+b*g*m*n*ln(d*(j*x+i)/i/(e*x+d))*polylog(2,i*(e*x +d)/d/(j*x+i))-b*g*m*n*ln(d*(j*x+i)/i/(e*x+d))*polylog(2,j*(e*x+d)/e/(j*x+ i))+a*g*m*polylog(2,1+j*x/i)-b*g*m*(n*ln(e*x+d)-ln(c*(e*x+d)^n))*polylog(2 ,1+j*x/i)+b*g*m*n*(ln(e*x+d)+ln(d*(j*x+i)/i/(e*x+d)))*polylog(2,1+j*x/i)-b *g*m*n*polylog(3,1+e*x/d)+b*g*m*n*polylog(3,i*(e*x+d)/d/(j*x+i))-b*g*m*n*p olylog(3,j*(e*x+d)/e/(j*x+i))-b*g*m*n*polylog(3,1+j*x/i)
Time = 0.19 (sec) , antiderivative size = 605, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (f-g m \log (i+j x)+g \log \left (h (i+j x)^m\right )\right )+b n \left (f-g m \log (i+j x)+g \log \left (h (i+j x)^m\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+a g m \left (\log (x) \left (\log (i+j x)-\log \left (1+\frac {j x}{i}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {j x}{i}\right )\right )+b g m \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (i+j x)-\log \left (1+\frac {j x}{i}\right )\right )-\operatorname {PolyLog}\left (2,-\frac {j x}{i}\right )\right )+b g m n \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x)+\frac {1}{2} \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right ) \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {-e i+d j}{j (d+e x)}\right )-\log \left (\frac {e i x-d j x}{d i+e i x}\right )\right )+\left (-\log \left (-\frac {e x}{d}\right )+\log \left (-\frac {j x}{i}\right )\right ) \log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \log \left (1+\frac {j x}{i}\right )+\frac {1}{2} \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \log \left (1+\frac {j x}{i}\right ) \left (-2 \log (d+e x)+\log \left (1+\frac {j x}{i}\right )\right )+\left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+\log \left (\frac {d (i+j x)}{i (d+e x)}\right ) \left (-\operatorname {PolyLog}\left (2,\frac {d (i+j x)}{i (d+e x)}\right )+\operatorname {PolyLog}\left (2,\frac {e (i+j x)}{j (d+e x)}\right )\right )+\left (\log (d+e x)+\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )-\operatorname {PolyLog}\left (3,1+\frac {e x}{d}\right )+\operatorname {PolyLog}\left (3,\frac {d (i+j x)}{i (d+e x)}\right )-\operatorname {PolyLog}\left (3,\frac {e (i+j x)}{j (d+e x)}\right )-\operatorname {PolyLog}\left (3,1+\frac {j x}{i}\right )\right ) \]
Log[x]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(f - g*m*Log[i + j*x] + g*Log[h*(i + j*x)^m]) + b*n*(f - g*m*Log[i + j*x] + g*Log[h*(i + j*x)^m ])*(Log[x]*(Log[d + e*x] - Log[1 + (e*x)/d]) - PolyLog[2, -((e*x)/d)]) + a *g*m*(Log[x]*(Log[i + j*x] - Log[1 + (j*x)/i]) - PolyLog[2, -((j*x)/i)]) + b*g*m*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])*(Log[x]*(Log[i + j*x] - Lo g[1 + (j*x)/i]) - PolyLog[2, -((j*x)/i)]) + b*g*m*n*(Log[-((e*x)/d)]*Log[d + e*x]*Log[i + j*x] + (Log[(d*(i + j*x))/(i*(d + e*x))]^2*(Log[-((e*x)/d) ] + Log[(-(e*i) + d*j)/(j*(d + e*x))] - Log[(e*i*x - d*j*x)/(d*i + e*i*x)] ))/2 + (-Log[-((e*x)/d)] + Log[-((j*x)/i)])*Log[(d*(i + j*x))/(i*(d + e*x) )]*Log[1 + (j*x)/i] + ((Log[-((e*x)/d)] - Log[-((j*x)/i)])*Log[1 + (j*x)/i ]*(-2*Log[d + e*x] + Log[1 + (j*x)/i]))/2 + (Log[i + j*x] - Log[(d*(i + j* x))/(i*(d + e*x))])*PolyLog[2, 1 + (e*x)/d] + Log[(d*(i + j*x))/(i*(d + e* x))]*(-PolyLog[2, (d*(i + j*x))/(i*(d + e*x))] + PolyLog[2, (e*(i + j*x))/ (j*(d + e*x))]) + (Log[d + e*x] + Log[(d*(i + j*x))/(i*(d + e*x))])*PolyLo g[2, 1 + (j*x)/i] - PolyLog[3, 1 + (e*x)/d] + PolyLog[3, (d*(i + j*x))/(i* (d + e*x))] - PolyLog[3, (e*(i + j*x))/(j*(d + e*x))] - PolyLog[3, 1 + (j* x)/i])
Time = 1.26 (sec) , antiderivative size = 553, normalized size of antiderivative = 0.87, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {2888, 2841, 2752, 2888, 2841, 2752, 2887, 2841, 2752, 2887, 2841, 2752, 2885}
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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 2888 |
| \(\displaystyle f \int \frac {a+b \log \left (c (d+e x)^n\right )}{x}dx+g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{x}dx\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b e n \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x}dx\right )+g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{x}dx\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (h (i+j x)^m\right )}{x}dx+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2888 |
| \(\displaystyle g \left (a \int \frac {\log \left (h (i+j x)^m\right )}{x}dx+b \int \frac {\log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )}{x}dx\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle g \left (a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )-j m \int \frac {\log \left (-\frac {j x}{i}\right )}{i+j x}dx\right )+b \int \frac {\log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )}{x}dx\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle g \left (b \int \frac {\log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )}{x}dx+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2887 |
| \(\displaystyle g \left (b \left (m \int \frac {\log \left (c (d+e x)^n\right ) \log (i+j x)}{x}dx-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \int \frac {\log \left (c (d+e x)^n\right )}{x}dx\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle g \left (b \left (m \int \frac {\log \left (c (d+e x)^n\right ) \log (i+j x)}{x}dx-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )-e n \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x}dx\right )\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle g \left (b \left (m \int \frac {\log \left (c (d+e x)^n\right ) \log (i+j x)}{x}dx-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2887 |
| \(\displaystyle g \left (b \left (m \left (n \int \frac {\log (d+e x) \log (i+j x)}{x}dx-\left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \int \frac {\log (i+j x)}{x}dx\right )-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2841 |
| \(\displaystyle g \left (b \left (m \left (n \int \frac {\log (d+e x) \log (i+j x)}{x}dx-\left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right ) \left (\log \left (-\frac {j x}{i}\right ) \log (i+j x)-j \int \frac {\log \left (-\frac {j x}{i}\right )}{i+j x}dx\right )\right )-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle g \left (b \left (m \left (n \int \frac {\log (d+e x) \log (i+j x)}{x}dx-\left (\operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )+\log \left (-\frac {j x}{i}\right ) \log (i+j x)\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right )-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\right )+a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )\right )+f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\) |
| \(\Big \downarrow \) 2885 |
| \(\displaystyle f \left (\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )+g \left (a \left (\log \left (-\frac {j x}{i}\right ) \log \left (h (i+j x)^m\right )+m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )\right )+b \left (m \left (n \left (\operatorname {PolyLog}\left (3,\frac {i (d+e x)}{d (i+j x)}\right )-\operatorname {PolyLog}\left (3,\frac {j (d+e x)}{e (i+j x)}\right )+\operatorname {PolyLog}\left (2,\frac {i (d+e x)}{d (i+j x)}\right ) \log \left (\frac {d (i+j x)}{i (d+e x)}\right )-\operatorname {PolyLog}\left (2,\frac {j (d+e x)}{e (i+j x)}\right ) \log \left (\frac {d (i+j x)}{i (d+e x)}\right )+\operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (\log (i+j x)-\log \left (\frac {d (i+j x)}{i (d+e x)}\right )\right )+\operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right ) \left (\log \left (\frac {d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )+\frac {1}{2} \left (\log \left (\frac {e i-d j}{e (i+j x)}\right )-\log \left (-\frac {x (e i-d j)}{d (i+j x)}\right )+\log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (\frac {d (i+j x)}{i (d+e x)}\right )-\frac {1}{2} \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {j x}{i}\right )\right ) \left (\log \left (\frac {d (i+j x)}{i (d+e x)}\right )+\log (d+e x)\right )^2+\log \left (-\frac {e x}{d}\right ) \log (d+e x) \log (i+j x)-\operatorname {PolyLog}\left (3,\frac {e x}{d}+1\right )-\operatorname {PolyLog}\left (3,\frac {j x}{i}+1\right )\right )-\left (\operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )+\log \left (-\frac {j x}{i}\right ) \log (i+j x)\right ) \left (n \log (d+e x)-\log \left (c (d+e x)^n\right )\right )\right )-\left (m \log (i+j x)-\log \left (h (i+j x)^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )\right )\right )\right )\) |
f*(Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]) + b*n*PolyLog[2, 1 + (e*x)/d ]) + g*(a*(Log[-((j*x)/i)]*Log[h*(i + j*x)^m] + m*PolyLog[2, 1 + (j*x)/i]) + b*(-((m*Log[i + j*x] - Log[h*(i + j*x)^m])*(Log[-((e*x)/d)]*Log[c*(d + e*x)^n] + n*PolyLog[2, 1 + (e*x)/d])) + m*(-((n*Log[d + e*x] - Log[c*(d + e*x)^n])*(Log[-((j*x)/i)]*Log[i + j*x] + PolyLog[2, 1 + (j*x)/i])) + n*(Lo g[-((e*x)/d)]*Log[d + e*x]*Log[i + j*x] + ((Log[-((e*x)/d)] + Log[(e*i - d *j)/(e*(i + j*x))] - Log[-(((e*i - d*j)*x)/(d*(i + j*x)))])*Log[(d*(i + j* x))/(i*(d + e*x))]^2)/2 - ((Log[-((e*x)/d)] - Log[-((j*x)/i)])*(Log[d + e* x] + Log[(d*(i + j*x))/(i*(d + e*x))])^2)/2 + (Log[i + j*x] - Log[(d*(i + j*x))/(i*(d + e*x))])*PolyLog[2, 1 + (e*x)/d] + Log[(d*(i + j*x))/(i*(d + e*x))]*PolyLog[2, (i*(d + e*x))/(d*(i + j*x))] - Log[(d*(i + j*x))/(i*(d + e*x))]*PolyLog[2, (j*(d + e*x))/(e*(i + j*x))] + (Log[d + e*x] + Log[(d*( i + j*x))/(i*(d + e*x))])*PolyLog[2, 1 + (j*x)/i] - PolyLog[3, 1 + (e*x)/d ] + PolyLog[3, (i*(d + e*x))/(d*(i + j*x))] - PolyLog[3, (j*(d + e*x))/(e* (i + j*x))] - PolyLog[3, 1 + (j*x)/i]))))
3.4.90.3.1 Defintions of rubi rules used
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_ )), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x )^n])/g), x] - Simp[b*e*(n/g) Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]
Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp [Log[(-b)*(x/a)]*Log[a + b*x]*Log[c + d*x], x] + (Simp[(1/2)*(Log[(-b)*(x/a )] - Log[(-(b*c - a*d))*(x/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x))] )*Log[a*((c + d*x)/(c*(a + b*x)))]^2, x] - Simp[(1/2)*(Log[(-b)*(x/a)] - Lo g[(-d)*(x/c)])*(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])^2, x] + Si mp[(Log[c + d*x] - Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + b*(x/a) ], x] + Simp[(Log[a + b*x] + Log[a*((c + d*x)/(c*(a + b*x)))])*PolyLog[2, 1 + d*(x/c)], x] + Simp[Log[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, c*((a + b*x)/(a*(c + d*x)))], x] - Simp[Log[a*((c + d*x)/(c*(a + b*x)))]*PolyLog[2, d*((a + b*x)/(b*(c + d*x)))], x] - Simp[PolyLog[3, 1 + b*(x/a)], x] - Simp [PolyLog[3, 1 + d*(x/c)], x] + Simp[PolyLog[3, c*((a + b*x)/(a*(c + d*x)))] , x] - Simp[PolyLog[3, d*((a + b*x)/(b*(c + d*x)))], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m _.)])/(x_), x_Symbol] :> Simp[m Int[Log[i + j*x]*(Log[c*(d + e*x)^n]/x), x], x] - Simp[(m*Log[i + j*x] - Log[h*(i + j*x)^m]) Int[Log[c*(d + e*x)^n ]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && N eQ[i + j*x, h*(i + j*x)^m]
Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*(Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.) + (f_)))/(x_), x_Symbol] :> Simp[f Int[(a + b *Log[c*(d + e*x)^n])/x, x], x] + Simp[g Int[Log[h*(i + j*x)^m]*((a + b*Lo g[c*(d + e*x)^n])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0]
\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]
integral((b*f*log((e*x + d)^n*c) + a*f + (b*g*log((e*x + d)^n*c) + a*g)*lo g((j*x + i)^m*h))/x, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]
a*f*log(x) + integrate(((g*log(h) + f)*b*log((e*x + d)^n) + (g*log(h) + f) *b*log(c) + a*g*log(h) + (b*g*log((e*x + d)^n) + b*g*log(c) + a*g)*log((j* x + i)^m))/x, x)
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x} \,d x \]