Integrand size = 32, antiderivative size = 270 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\frac {g j m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{i}+\frac {b e n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {b e n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}-\frac {b g j m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{i}+\frac {b g j m n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{i}-\frac {b e g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d}+\frac {b e g m n \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )}{d} \]
g*j*m*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/i-g*j*m*(a+b*ln(c*(e*x+d)^n))*ln(e* (j*x+i)/(-d*j+e*i))/i+b*e*n*ln(-j*x/i)*(f+g*ln(h*(j*x+i)^m))/d-b*e*n*ln(-j *(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i)^m))/d-(a+b*ln(c*(e*x+d)^n))*(f+g*ln (h*(j*x+i)^m))/x-b*g*j*m*n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/i+b*g*j*m*n*po lylog(2,1+e*x/d)/i-b*e*g*m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/d+b*e*g*m*n*p olylog(2,1+j*x/i)/d
Time = 0.16 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=-\frac {a d f i-b e f i n x \log (x)-a d g j m x \log \left (-\frac {j x}{i}\right )+b e f i n x \log (d+e x)-b d g j m n x \log \left (-\frac {e x}{d}\right ) \log (d+e x)+b d g j m n x \log \left (-\frac {j x}{i}\right ) \log (d+e x)+b d f i \log \left (c (d+e x)^n\right )-b d g j m x \log \left (-\frac {j x}{i}\right ) \log \left (c (d+e x)^n\right )+a d g j m x \log (i+j x)-b e g i m n x \log (d+e x) \log (i+j x)-b d g j m n x \log (d+e x) \log (i+j x)+b e g i m n x \log \left (\frac {j (d+e x)}{-e i+d j}\right ) \log (i+j x)+b d g j m x \log \left (c (d+e x)^n\right ) \log (i+j x)+b d g j m n x \log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )+a d g i \log \left (h (i+j x)^m\right )-b e g i n x \log (x) \log \left (h (i+j x)^m\right )+b e g i n x \log (d+e x) \log \left (h (i+j x)^m\right )+b d g i \log \left (c (d+e x)^n\right ) \log \left (h (i+j x)^m\right )+b e g i m n x \log (x) \log \left (1+\frac {j x}{i}\right )+b e g i m n x \operatorname {PolyLog}\left (2,-\frac {j x}{i}\right )+b d g j m n x \operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )-b d g j m n x \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )+b e g i m n x \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d i x} \]
-((a*d*f*i - b*e*f*i*n*x*Log[x] - a*d*g*j*m*x*Log[-((j*x)/i)] + b*e*f*i*n* x*Log[d + e*x] - b*d*g*j*m*n*x*Log[-((e*x)/d)]*Log[d + e*x] + b*d*g*j*m*n* x*Log[-((j*x)/i)]*Log[d + e*x] + b*d*f*i*Log[c*(d + e*x)^n] - b*d*g*j*m*x* Log[-((j*x)/i)]*Log[c*(d + e*x)^n] + a*d*g*j*m*x*Log[i + j*x] - b*e*g*i*m* n*x*Log[d + e*x]*Log[i + j*x] - b*d*g*j*m*n*x*Log[d + e*x]*Log[i + j*x] + b*e*g*i*m*n*x*Log[(j*(d + e*x))/(-(e*i) + d*j)]*Log[i + j*x] + b*d*g*j*m*x *Log[c*(d + e*x)^n]*Log[i + j*x] + b*d*g*j*m*n*x*Log[d + e*x]*Log[(e*(i + j*x))/(e*i - d*j)] + a*d*g*i*Log[h*(i + j*x)^m] - b*e*g*i*n*x*Log[x]*Log[h *(i + j*x)^m] + b*e*g*i*n*x*Log[d + e*x]*Log[h*(i + j*x)^m] + b*d*g*i*Log[ c*(d + e*x)^n]*Log[h*(i + j*x)^m] + b*e*g*i*m*n*x*Log[x]*Log[1 + (j*x)/i] + b*e*g*i*m*n*x*PolyLog[2, -((j*x)/i)] + b*d*g*j*m*n*x*PolyLog[2, (j*(d + e*x))/(-(e*i) + d*j)] - b*d*g*j*m*n*x*PolyLog[2, 1 + (e*x)/d] + b*e*g*i*m* n*x*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/(d*i*x))
Time = 0.61 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2889, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle g j m \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (i+j x)}dx+b e n \int \frac {f+g \log \left (h (i+j x)^m\right )}{x (d+e x)}dx-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle g j m \int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{i x}-\frac {j \left (a+b \log \left (c (d+e x)^n\right )\right )}{i (i+j x)}\right )dx+b e n \int \left (\frac {f+g \log \left (h (i+j x)^m\right )}{d x}-\frac {e \left (f+g \log \left (h (i+j x)^m\right )\right )}{d (d+e x)}\right )dx-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x}+g j m \left (-\frac {\log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{i}+\frac {b n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{i}\right )+b e n \left (-\frac {\log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}-\frac {g m \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d}+\frac {\log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d}+\frac {g m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )}{d}\right )\) |
-(((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x) + g*j*m*((Log [-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/i - ((a + b*Log[c*(d + e*x)^n])*L og[(e*(i + j*x))/(e*i - d*j)])/i - (b*n*PolyLog[2, -((j*(d + e*x))/(e*i - d*j))])/i + (b*n*PolyLog[2, 1 + (e*x)/d])/i) + b*e*n*((Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m]))/d - (Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log [h*(i + j*x)^m]))/d - (g*m*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/d + (g*m *PolyLog[2, 1 + (j*x)/i])/d)
3.4.91.3.1 Defintions of rubi rules used
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*(x_)^(r_.), x_Symbol] :> Simp[x^( r + 1)*(a + b*Log[c*(d + e*x)^n])^p*((f + g*Log[h*(i + j*x)^m])/(r + 1)), x ] + (-Simp[g*j*(m/(r + 1)) Int[x^(r + 1)*((a + b*Log[c*(d + e*x)^n])^p/(i + j*x)), x], x] - Simp[b*e*n*(p/(r + 1)) Int[x^(r + 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1)*((f + g*Log[h*(i + j*x)^m])/(d + e*x)), x], x]) /; FreeQ[ {a, b, c, d, e, f, g, h, i, j, m, n}, x] && IGtQ[p, 0] && IntegerQ[r] && (E qQ[p, 1] || GtQ[r, 0]) && NeQ[r, -1]
\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{2}}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]
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^2, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]
-b*e*f*n*(log(e*x + d)/d - log(x)/d) - a*g*j*m*(log(j*x + i)/i - log(x)/i) + b*g*integrate(((log((e*x + d)^n) + log(c))*log((j*x + i)^m) + log((e*x + d)^n)*log(h) + log(c)*log(h))/x^2, x) - b*f*log((e*x + d)^n*c)/x - a*g*l og((j*x + i)^m*h)/x - a*f/x
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^2} \,d x \]