Integrand size = 30, antiderivative size = 80 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g}+\frac {B \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b g} \]
-ln((a*d-b*c)/d/(b*x+a))*(A+B*ln(e*(b*x+a)/(d*x+c)))/b/g+B*polylog(2,1+(-a *d+b*c)/d/(b*x+a))/b/g
Time = 0.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\frac {\log (g (a+b x)) \left (-B \log (g (a+b x))+2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{2 b g} \]
(Log[g*(a + b*x)]*(-(B*Log[g*(a + b*x)]) + 2*(A + B*Log[(e*(a + b*x))/(c + d*x)] + B*Log[(b*(c + d*x))/(b*c - a*d)])) + 2*B*PolyLog[2, (d*(a + b*x)) /(-(b*c) + a*d)])/(2*b*g)
Time = 0.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2942, 2858, 27, 2778, 2005, 2752}
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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{a g+b g x} \, dx\) |
| \(\Big \downarrow \) 2942 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)}dx}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {b \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}d(a+b x)}{b^2 g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (b c-a d+d (a+b x))}d(a+b x)}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2778 |
| \(\displaystyle -\frac {B (b c-a d) \int \frac {(a+b x) \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{b c-a d+d (a+b x)}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{d+\frac {b c-a d}{a+b x}}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {B \operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g}\) |
-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])) /(b*g)) + (B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)
3.1.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
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_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[1/n Subst[Int[(a + b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(-Log[-(b*c - a*d)/(d*(a + b*x))])*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/g), x] + Simp[B*n*((b*c - a*d)/g) Int[Log[-(b*c - a*d)/(d*(a + b*x))]/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && NeQ[b* c - a*d, 0] && EqQ[b*f - a*g, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(79)=158\).
Time = 1.15 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.86
| method | result | size |
| parts | \(\frac {A \ln \left (b x +a \right )}{g b}-\frac {B \left (a d -c b \right ) e \left (-\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right ) b e}+\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{\left (a d -c b \right ) b e}\right )}{g \,d^{2}}\) | \(229\) |
| derivativedivides | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{g \left (a d -c b \right )}-\frac {d^{2} B \left (-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b e}\right )}{g \left (a d -c b \right )}\right )}{d^{2}}\) | \(304\) |
| default | \(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e}\right )}{g \left (a d -c b \right )}-\frac {d^{2} B \left (-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b e}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b e}\right )}{g \left (a d -c b \right )}\right )}{d^{2}}\) | \(304\) |
| risch | \(\frac {A \ln \left (b x +a \right )}{g b}+\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} a}{2 g \left (a d -c b \right ) b}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} c}{2 g \left (a d -c b \right )}-\frac {B d \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{g \left (a d -c b \right ) b}+\frac {B \operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c}{g \left (a d -c b \right )}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) a}{g \left (a d -c b \right ) b}+\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right ) c}{g \left (a d -c b \right )}\) | \(416\) |
A/g*ln(b*x+a)/b-B/g/d^2*(a*d-b*c)*e*(-1/2/(a*d-b*c)*d^2/b/e*ln(b*e/d+(a*d- b*c)*e/d/(d*x+c))^2+1/(a*d-b*c)*d^3/b/e*(dilog(-((b*e/d+(a*d-b*c)*e/d/(d*x +c))*d-b*e)/b/e)/d+ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-((b*e/d+(a*d-b*c)*e /d/(d*x+c))*d-b*e)/b/e)/d))
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{b g x + a g} \,d x } \]
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\frac {\int \frac {A}{a + b x}\, dx + \int \frac {B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{a + b x}\, dx}{g} \]
(Integral(A/(a + b*x), x) + Integral(B*log(a*e/(c + d*x) + b*e*x/(c + d*x) )/(a + b*x), x))/g
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{b g x + a g} \,d x } \]
-B*(log(b*x + a)*log(d*x + c)/(b*g) - integrate((b*d*x*log(e) + b*c*log(e) + (2*b*d*x + b*c + a*d)*log(b*x + a))/(b^2*d*g*x^2 + a*b*c*g + (b^2*c*g + a*b*d*g)*x), x)) + A*log(b*g*x + a*g)/(b*g)
\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{b g x + a g} \,d x } \]
Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a g+b g x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{a\,g+b\,g\,x} \,d x \]