Integrand size = 32, antiderivative size = 421 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\frac {b e g j m n \log (x)}{d i}-\frac {b e g j m n \log (d+e x)}{2 d i}-\frac {g j m \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i x}-\frac {g j^2 m \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 i^2}-\frac {b e g j m n \log (i+j x)}{2 d i}+\frac {g j^2 m \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{2 i^2}-\frac {b e n \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d x}-\frac {b e^2 n \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}+\frac {b e^2 n \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 d^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}+\frac {b g j^2 m n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{2 i^2}-\frac {b g j^2 m n \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{2 i^2}+\frac {b e^2 g m n \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{2 d^2}-\frac {b e^2 g m n \operatorname {PolyLog}\left (2,1+\frac {j x}{i}\right )}{2 d^2} \]
b*e*g*j*m*n*ln(x)/d/i-1/2*b*e*g*j*m*n*ln(e*x+d)/d/i-1/2*g*j*m*(a+b*ln(c*(e *x+d)^n))/i/x-1/2*g*j^2*m*ln(-e*x/d)*(a+b*ln(c*(e*x+d)^n))/i^2-1/2*b*e*g*j *m*n*ln(j*x+i)/d/i+1/2*g*j^2*m*(a+b*ln(c*(e*x+d)^n))*ln(e*(j*x+i)/(-d*j+e* i))/i^2-1/2*b*e*n*(f+g*ln(h*(j*x+i)^m))/d/x-1/2*b*e^2*n*ln(-j*x/i)*(f+g*ln (h*(j*x+i)^m))/d^2+1/2*b*e^2*n*ln(-j*(e*x+d)/(-d*j+e*i))*(f+g*ln(h*(j*x+i) ^m))/d^2-1/2*(a+b*ln(c*(e*x+d)^n))*(f+g*ln(h*(j*x+i)^m))/x^2+1/2*b*g*j^2*m *n*polylog(2,-j*(e*x+d)/(-d*j+e*i))/i^2-1/2*b*g*j^2*m*n*polylog(2,1+e*x/d) /i^2+1/2*b*e^2*g*m*n*polylog(2,e*(j*x+i)/(-d*j+e*i))/d^2-1/2*b*e^2*g*m*n*p olylog(2,1+j*x/i)/d^2
Time = 0.21 (sec) , antiderivative size = 765, normalized size of antiderivative = 1.82 \[ \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^3} \, dx=-\frac {b e^2 n \log (x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d^2}+\frac {b e^2 n \log (d+e x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d^2}-\frac {b n \log (d+e x) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 x^2}-\frac {\left (a+b \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )\right ) \left (f+g \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 x^2}-\frac {e \left (b f n+b g n \left (-m \log (i+j x)+\log \left (h (i+j x)^m\right )\right )\right )}{2 d x}+\frac {1}{2} a g m \left (\frac {j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}-\left (\frac {j^2 (i+j x)^2}{i^4 \left (1-\frac {i+j x}{i}\right )^2}+\frac {2 j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}\right ) \log (i+j x)-\frac {j^2 \log \left (1-\frac {i+j x}{i}\right )}{i^2}\right )+\frac {1}{2} b g m \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\frac {j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}-\left (\frac {j^2 (i+j x)^2}{i^4 \left (1-\frac {i+j x}{i}\right )^2}+\frac {2 j^2 (i+j x)}{i^3 \left (1-\frac {i+j x}{i}\right )}\right ) \log (i+j x)-\frac {j^2 \log \left (1-\frac {i+j x}{i}\right )}{i^2}\right )+\frac {1}{2} b g m n \left (-\frac {\log (d+e x) \log (i+j x)}{x^2}+j \left (\frac {\frac {e \log (x)}{d}-\frac {e \log (d+e x)}{d}-\frac {\log (d+e x)}{x}}{i}-\frac {j \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x)+\operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )\right )}{i^2}+\frac {j^2 \left (\frac {\log (d+e x) \log \left (\frac {e (i+j x)}{e i-d j}\right )}{j}+\frac {\operatorname {PolyLog}\left (2,\frac {j (d+e x)}{-e i+d j}\right )}{j}\right )}{i^2}\right )+e \left (\frac {\frac {j \log (x)}{i}-\frac {j \log (i+j x)}{i}-\frac {\log (i+j x)}{x}}{d}-\frac {e \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 )}{d^2}+\frac {e^2 \left (\frac {\log \left (\frac {j (d+e x)}{-e i+d j}\right ) \log (i+j x)}{e}+\frac {\operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{e}\right )}{d^2}\right )\right ) \]
-1/2*(b*e^2*n*Log[x]*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)^m])))/d^2 + (b*e^2*n*Log[d + e*x]*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)^m]))) /(2*d^2) - (b*n*Log[d + e*x]*(f + g*(-(m*Log[i + j*x]) + Log[h*(i + j*x)^m ])))/(2*x^2) - ((a + b*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n]))*(f + g*(- (m*Log[i + j*x]) + Log[h*(i + j*x)^m])))/(2*x^2) - (e*(b*f*n + b*g*n*(-(m* Log[i + j*x]) + Log[h*(i + j*x)^m])))/(2*d*x) + (a*g*m*((j^2*(i + j*x))/(i ^3*(1 - (i + j*x)/i)) - ((j^2*(i + j*x)^2)/(i^4*(1 - (i + j*x)/i)^2) + (2* j^2*(i + j*x))/(i^3*(1 - (i + j*x)/i)))*Log[i + j*x] - (j^2*Log[1 - (i + j *x)/i])/i^2))/2 + (b*g*m*(-(n*Log[d + e*x]) + Log[c*(d + e*x)^n])*((j^2*(i + j*x))/(i^3*(1 - (i + j*x)/i)) - ((j^2*(i + j*x)^2)/(i^4*(1 - (i + j*x)/ i)^2) + (2*j^2*(i + j*x))/(i^3*(1 - (i + j*x)/i)))*Log[i + j*x] - (j^2*Log [1 - (i + j*x)/i])/i^2))/2 + (b*g*m*n*(-((Log[d + e*x]*Log[i + j*x])/x^2) + j*(((e*Log[x])/d - (e*Log[d + e*x])/d - Log[d + e*x]/x)/i - (j*(Log[-((e *x)/d)]*Log[d + e*x] + PolyLog[2, (d + e*x)/d]))/i^2 + (j^2*((Log[d + e*x] *Log[(e*(i + j*x))/(e*i - d*j)])/j + PolyLog[2, (j*(d + e*x))/(-(e*i) + d* j)]/j))/i^2) + e*(((j*Log[x])/i - (j*Log[i + j*x])/i - Log[i + j*x]/x)/d - (e*(Log[x]*(Log[i + j*x] - Log[1 + (j*x)/i]) - PolyLog[2, -((j*x)/i)]))/d ^2 + (e^2*((Log[(j*(d + e*x))/(-(e*i) + d*j)]*Log[i + j*x])/e + PolyLog[2, (e*(i + j*x))/(e*i - d*j)]/e))/d^2)))/2
Time = 0.76 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.89, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2889 |
| \(\displaystyle \frac {1}{2} g j m \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (i+j x)}dx+\frac {1}{2} b e n \int \frac {f+g \log \left (h (i+j x)^m\right )}{x^2 (d+e x)}dx-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} g j m \int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) j^2}{i^2 (i+j x)}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) j}{i^2 x}+\frac {a+b \log \left (c (d+e x)^n\right )}{i x^2}\right )dx+\frac {1}{2} b e n \int \left (\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) e^2}{d^2 (d+e x)}-\frac {\left (f+g \log \left (h (i+j x)^m\right )\right ) e}{d^2 x}+\frac {f+g \log \left (h (i+j x)^m\right )}{d x^2}\right )dx-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{2 x^2}+\frac {1}{2} g j m \left (-\frac {j \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i^2}+\frac {j \log \left (\frac {e (i+j x)}{e i-d j}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{i^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{i x}+\frac {b j n \operatorname {PolyLog}\left (2,-\frac {j (d+e x)}{e i-d j}\right )}{i^2}-\frac {b j n \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{i^2}+\frac {b e n \log (x)}{d i}-\frac {b e n \log (d+e x)}{d i}\right )+\frac {1}{2} b e n \left (-\frac {e \log \left (-\frac {j x}{i}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d^2}+\frac {e \log \left (-\frac {j (d+e x)}{e i-d j}\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{d^2}+\frac {e g m \operatorname {PolyLog}\left (2,\frac {e (i+j x)}{e i-d j}\right )}{d^2}-\frac {e g m \operatorname {PolyLog}\left (2,\frac {j x}{i}+1\right )}{d^2}-\frac {f+g \log \left (h (i+j x)^m\right )}{d x}+\frac {g j m \log (x)}{d i}-\frac {g j m \log (i+j x)}{d i}\right )\) |
-1/2*((a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]))/x^2 + (g*j*m* ((b*e*n*Log[x])/(d*i) - (b*e*n*Log[d + e*x])/(d*i) - (a + b*Log[c*(d + e*x )^n])/(i*x) - (j*Log[-((e*x)/d)]*(a + b*Log[c*(d + e*x)^n]))/i^2 + (j*(a + b*Log[c*(d + e*x)^n])*Log[(e*(i + j*x))/(e*i - d*j)])/i^2 + (b*j*n*PolyLo g[2, -((j*(d + e*x))/(e*i - d*j))])/i^2 - (b*j*n*PolyLog[2, 1 + (e*x)/d])/ i^2))/2 + (b*e*n*((g*j*m*Log[x])/(d*i) - (g*j*m*Log[i + j*x])/(d*i) - (f + g*Log[h*(i + j*x)^m])/(d*x) - (e*Log[-((j*x)/i)]*(f + g*Log[h*(i + j*x)^m ]))/d^2 + (e*Log[-((j*(d + e*x))/(e*i - d*j))]*(f + g*Log[h*(i + j*x)^m])) /d^2 + (e*g*m*PolyLog[2, (e*(i + j*x))/(e*i - d*j)])/d^2 - (e*g*m*PolyLog[ 2, 1 + (j*x)/i])/d^2))/2
3.4.92.3.1 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]
\[\int \frac {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right ) \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{3}} \,d x } \]
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^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{3}} \,d x } \]
1/2*b*e*f*n*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + 1/2*a*g*j*m*(j *log(j*x + i)/i^2 - j*log(x)/i^2 - 1/(i*x)) + 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^3, x) - 1/2*b*f*log((e*x + d)^n*c)/x^2 - 1/2*a*g*log((j*x + i)^m*h)/x^2 - 1/2*a*f/x^2
\[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^3} \,d x \]