Integrand size = 30, antiderivative size = 81 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{b g}-\frac {B \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b g} \]
-ln((a*d-b*c)/d/(b*x+a))*(A+B*ln(e*(d*x+c)/(b*x+a)))/b/g-B*polylog(2,1+(-a *d+b*c)/d/(b*x+a))/b/g
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=\frac {\log (g (a+b x)) \left (B \log (g (a+b x))+2 \left (A-B \log \left (\frac {b (c+d x)}{b c-a d}\right )+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )\right )-2 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{2 b g} \]
(Log[g*(a + b*x)]*(B*Log[g*(a + b*x)] + 2*(A - B*Log[(b*(c + d*x))/(b*c - a*d)] + B*Log[(e*(c + d*x))/(a + b*x)])) - 2*B*PolyLog[2, (d*(a + b*x))/(- (b*c) + a*d)])/(2*b*g)
Time = 0.49 (sec) , antiderivative size = 81, 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 = {2944, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{a g+b g x} \, dx\) |
| \(\Big \downarrow \) 2944 |
| \(\displaystyle -\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)}dx}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -\frac {B (b c-a d) \int \frac {b \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) \left (b \left (c-\frac {a d}{b}\right )+d (a+b x)\right )}d(a+b x)}{b^2 g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (b c-a d+d (a+b x))}d(a+b x)}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2778 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {(a+b x) \log \left (-\frac {b c-a d}{d (a+b x)}\right )}{b c-a d+d (a+b x)}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right )}{d+\frac {b c-a d}{a+b x}}d\frac {1}{a+b x}}{b g}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{b g}-\frac {B \operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b g}\) |
-((Log[-((b*c - a*d)/(d*(a + b*x)))]*(A + B*Log[(e*(c + d*x))/(a + b*x)])) /(b*g)) - (B*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*g)
3.2.77.3.1 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)/(b*(c + d*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)/(b*(c + d*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[d*f - c*g, 0]
Time = 1.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.84
| method | result | size |
| parts | \(\frac {A \ln \left (b x +a \right )}{g b}+\frac {B \left (-\frac {\operatorname {dilog}\left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}-\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}\right )}{g}\) | \(149\) |
| risch | \(\frac {A \ln \left (b x +a \right )}{g b}-\frac {B \operatorname {dilog}\left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{g b}-\frac {B \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{g b}\) | \(151\) |
| derivativedivides | \(\frac {e \left (a d -c b \right ) \left (-\frac {b A \ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{g e \left (a d -c b \right )}-\frac {b^{2} B \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}+\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}\right )}{g e \left (a d -c b \right )}\right )}{b^{2}}\) | \(219\) |
| default | \(\frac {e \left (a d -c b \right ) \left (-\frac {b A \ln \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e \right )}{g e \left (a d -c b \right )}-\frac {b^{2} B \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}+\frac {\ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) b -d e}{d e}\right )}{b}\right )}{g e \left (a d -c b \right )}\right )}{b^{2}}\) | \(219\) |
A/g*ln(b*x+a)/b+B/g*(-dilog(-((d*e/b-e*(a*d-b*c)/b/(b*x+a))*b-d*e)/d/e)/b- ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(-((d*e/b-e*(a*d-b*c)/b/(b*x+a))*b-d*e)/ d/e)/b)
\[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A}{b g x + a g} \,d x } \]
\[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=\frac {\int \frac {A}{a + b x}\, dx + \int \frac {B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}{a + b x}\, dx}{g} \]
(Integral(A/(a + b*x), x) + Integral(B*log(c*e/(a + b*x) + d*e*x/(a + b*x) )/(a + b*x), x))/g
\[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=\int { \frac {B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A}{b g x + a g} \,d x } \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (80) = 160\).
Time = 37.79 (sec) , antiderivative size = 617, normalized size of antiderivative = 7.62 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=-\frac {1}{2} \, {\left (\frac {{\left (B b^{3} c^{3} e^{3} - 3 \, B a b^{2} c^{2} d e^{3} + 3 \, B a^{2} b c d^{2} e^{3} - B a^{3} d^{3} e^{3}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{b d^{2} e^{2} g - \frac {2 \, {\left (d e x + c e\right )} b^{2} d e g}{b x + a} + \frac {{\left (d e x + c e\right )}^{2} b^{3} g}{{\left (b x + a\right )}^{2}}} + \frac {A b^{3} c^{3} d e^{3} - B b^{3} c^{3} d e^{3} - 3 \, A a b^{2} c^{2} d^{2} e^{3} + 3 \, B a b^{2} c^{2} d^{2} e^{3} + 3 \, A a^{2} b c d^{3} e^{3} - 3 \, B a^{2} b c d^{3} e^{3} - A a^{3} d^{4} e^{3} + B a^{3} d^{4} e^{3} + \frac {{\left (d e x + c e\right )} B b^{4} c^{3} e^{2}}{b x + a} - \frac {3 \, {\left (d e x + c e\right )} B a b^{3} c^{2} d e^{2}}{b x + a} + \frac {3 \, {\left (d e x + c e\right )} B a^{2} b^{2} c d^{2} e^{2}}{b x + a} - \frac {{\left (d e x + c e\right )} B a^{3} b d^{3} e^{2}}{b x + a}}{b d^{3} e^{2} g - \frac {2 \, {\left (d e x + c e\right )} b^{2} d^{2} e g}{b x + a} + \frac {{\left (d e x + c e\right )}^{2} b^{3} d g}{{\left (b x + a\right )}^{2}}} + \frac {{\left (B b^{3} c^{3} e - 3 \, B a b^{2} c^{2} d e + 3 \, B a^{2} b c d^{2} e - B a^{3} d^{3} e\right )} \log \left (-d e + \frac {{\left (d e x + c e\right )} b}{b x + a}\right )}{b d^{2} g} - \frac {{\left (B b^{3} c^{3} e - 3 \, B a b^{2} c^{2} d e + 3 \, B a^{2} b c d^{2} e - B a^{3} d^{3} e\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{b d^{2} g}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \]
-1/2*((B*b^3*c^3*e^3 - 3*B*a*b^2*c^2*d*e^3 + 3*B*a^2*b*c*d^2*e^3 - B*a^3*d ^3*e^3)*log((d*e*x + c*e)/(b*x + a))/(b*d^2*e^2*g - 2*(d*e*x + c*e)*b^2*d* e*g/(b*x + a) + (d*e*x + c*e)^2*b^3*g/(b*x + a)^2) + (A*b^3*c^3*d*e^3 - B* b^3*c^3*d*e^3 - 3*A*a*b^2*c^2*d^2*e^3 + 3*B*a*b^2*c^2*d^2*e^3 + 3*A*a^2*b* c*d^3*e^3 - 3*B*a^2*b*c*d^3*e^3 - A*a^3*d^4*e^3 + B*a^3*d^4*e^3 + (d*e*x + c*e)*B*b^4*c^3*e^2/(b*x + a) - 3*(d*e*x + c*e)*B*a*b^3*c^2*d*e^2/(b*x + a ) + 3*(d*e*x + c*e)*B*a^2*b^2*c*d^2*e^2/(b*x + a) - (d*e*x + c*e)*B*a^3*b* d^3*e^2/(b*x + a))/(b*d^3*e^2*g - 2*(d*e*x + c*e)*b^2*d^2*e*g/(b*x + a) + (d*e*x + c*e)^2*b^3*d*g/(b*x + a)^2) + (B*b^3*c^3*e - 3*B*a*b^2*c^2*d*e + 3*B*a^2*b*c*d^2*e - B*a^3*d^3*e)*log(-d*e + (d*e*x + c*e)*b/(b*x + a))/(b* d^2*g) - (B*b^3*c^3*e - 3*B*a*b^2*c^2*d*e + 3*B*a^2*b*c*d^2*e - B*a^3*d^3* e)*log((d*e*x + c*e)/(b*x + a))/(b*d^2*g))*(b*c/((b*c*e - a*d*e)*(b*c - a* d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2
Timed out. \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a g+b g x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}{a\,g+b\,g\,x} \,d x \]