\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^3 (c i+d i x)^2} \, dx\) [45]

Optimal result

Integrand size = 40, antiderivative size = 346 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {B d^3 (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}-\frac {b B (c+d x)^2 \left (b-\frac {6 d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {3 b B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^3 i^2}-\frac {d^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^3 i^2} \] Output:

B*d^3*(b*x+a)/(-a*d+b*c)^4/g^3/i^2/(d*x+c)-1/4*b*B*(d*x+c)^2*(b-6*d*(b*x+a 
)/(d*x+c))^2/(-a*d+b*c)^4/g^3/i^2/(b*x+a)^2-3/2*b*B*d^2*ln((b*x+a)/(d*x+c) 
)^2/(-a*d+b*c)^4/g^3/i^2-d^3*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c 
)^4/g^3/i^2/(d*x+c)+3*b^2*d*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c) 
^4/g^3/i^2/(b*x+a)-1/2*b^3*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c 
)^4/g^3/i^2/(b*x+a)^2+3*b*d^2*ln((b*x+a)/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c 
)))/(-a*d+b*c)^4/g^3/i^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.65 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {-\frac {b B (b c-a d)^2}{(a+b x)^2}+\frac {8 b^2 B c d}{a+b x}-\frac {8 a b B d^2}{a+b x}+\frac {2 b B d (b c-a d)}{a+b x}-\frac {4 b B c d^2}{c+d x}+\frac {4 a B d^3}{c+d x}+6 b B d^2 \log (a+b x)-\frac {2 b (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {8 b d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+\frac {4 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+12 b d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 b B d^2 \log (c+d x)-12 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-6 b B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+6 b B d^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^3 i^2} \] Input:

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i 
*x)^2),x]
 

Output:

(-((b*B*(b*c - a*d)^2)/(a + b*x)^2) + (8*b^2*B*c*d)/(a + b*x) - (8*a*b*B*d 
^2)/(a + b*x) + (2*b*B*d*(b*c - a*d))/(a + b*x) - (4*b*B*c*d^2)/(c + d*x) 
+ (4*a*B*d^3)/(c + d*x) + 6*b*B*d^2*Log[a + b*x] - (2*b*(b*c - a*d)^2*(A + 
 B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (8*b*d*(b*c - a*d)*(A + B* 
Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + (4*d^2*(b*c - a*d)*(A + B*Log[( 
e*(a + b*x))/(c + d*x)]))/(c + d*x) + 12*b*d^2*Log[a + b*x]*(A + B*Log[(e* 
(a + b*x))/(c + d*x)]) - 6*b*B*d^2*Log[c + d*x] - 12*b*d^2*(A + B*Log[(e*( 
a + b*x))/(c + d*x)])*Log[c + d*x] - 6*b*B*d^2*(Log[a + b*x]*(Log[a + b*x] 
 - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) 
+ a*d)]) + 6*b*B*d^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x]) 
*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(4*(b*c - a*d)^4 
*g^3*i^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2772, 27, 2010, 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.

\(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(a g+b g x)^3 (c i+d i x)^2} \, dx\)

\(\Big \downarrow \) 2962

\(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B \int -\frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}-\frac {6 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right ) b}{(c+d x)^2}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{2 (a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} B \int \frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}-\frac {6 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right ) b}{(c+d x)^2}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {\frac {1}{2} B \int \left (\frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x)^3}-\frac {6 b d^2 (c+d x) \log \left (\frac {a+b x}{c+d x}\right )}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {1}{2} B \left (-\frac {b^3 (c+d x)^2}{2 (a+b x)^2}+\frac {6 b^2 d (c+d x)}{a+b x}+\frac {2 d^3 (a+b x)}{c+d x}-3 b d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{g^3 i^2 (b c-a d)^4}\)

Input:

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)^2) 
,x]
 

Output:

((B*((2*d^3*(a + b*x))/(c + d*x) + (6*b^2*d*(c + d*x))/(a + b*x) - (b^3*(c 
 + d*x)^2)/(2*(a + b*x)^2) - 3*b*d^2*Log[(a + b*x)/(c + d*x)]^2))/2 - (d^3 
*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + (3*b^2*d*(c + 
 d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (b^3*(c + d*x)^2*( 
A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) + 3*b*d^2*Log[(a + b* 
x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^2 
)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

rule 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.46

method	result	size
parts	\(\frac {A \left (-\frac {d^{2}}{\left (d a -b c \right )^{3} \left (d x +c \right )}-\frac {3 d^{2} b \ln \left (d x +c \right )}{\left (d a -b c \right )^{4}}-\frac {b}{2 \left (d a -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {3 d^{2} b \ln \left (b x +a \right )}{\left (d a -b c \right )^{4}}-\frac {2 b d}{\left (d a -b c \right )^{3} \left (b x +a \right )}\right )}{g^{3} i^{2}}-\frac {B \left (\frac {d^{3} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{3}}-\frac {3 b \,d^{2} e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{3}}+\frac {3 b^{2} d \,e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}-\frac {b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} i^{2} \left (d a -b c \right ) e}\)	\(506\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e A \,b^{3}}{2 i^{2} \left (d a -b c \right )^{5} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {3 d^{3} A \,b^{2}}{i^{2} \left (d a -b c \right )^{5} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {3 d^{4} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{5} g^{3}}-\frac {d^{2} e B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {3 d^{3} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{5} g^{3}}-\frac {3 d^{4} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {d^{5} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{5} g^{3}}\right )}{d^{2}}\)	\(628\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e A \,b^{3}}{2 i^{2} \left (d a -b c \right )^{5} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {3 d^{3} A \,b^{2}}{i^{2} \left (d a -b c \right )^{5} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {3 d^{4} A b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {d^{5} A \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{5} g^{3}}-\frac {d^{2} e B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {3 d^{3} B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{2} \left (d a -b c \right )^{5} g^{3}}-\frac {3 d^{4} B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{2} \left (d a -b c \right )^{5} g^{3}}+\frac {d^{5} B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{5} g^{3}}\right )}{d^{2}}\)	\(628\)
risch	\(-\frac {A \,d^{2}}{g^{3} i^{2} \left (d a -b c \right )^{3} \left (d x +c \right )}-\frac {3 A \,d^{2} b \ln \left (d x +c \right )}{g^{3} i^{2} \left (d a -b c \right )^{4}}-\frac {A b}{2 g^{3} i^{2} \left (d a -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {3 A \,d^{2} b \ln \left (b x +a \right )}{g^{3} i^{2} \left (d a -b c \right )^{4}}-\frac {2 A b d}{g^{3} i^{2} \left (d a -b c \right )^{3} \left (b x +a \right )}-\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b}{g^{3} i^{2} \left (d a -b c \right )^{4}}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (d x +c \right )}+\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b c}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (d x +c \right )}+\frac {B \,d^{3} a}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (d x +c \right )}-\frac {B \,d^{2} b c}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (d x +c \right )}+\frac {B b \,d^{2}}{g^{3} i^{2} \left (d a -b c \right )^{4}}+\frac {3 B b \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} i^{2} \left (d a -b c \right )^{4}}+\frac {3 B e \,b^{2} d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {3 B e \,b^{2} d}{g^{3} i^{2} \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {B \,e^{2} b^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} i^{2} \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}-\frac {B \,e^{2} b^{3}}{4 g^{3} i^{2} \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}\)	\(736\)
parallelrisch	\(\frac {6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} c \,d^{5}+24 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{6}+12 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c \,d^{5}+18 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c \,d^{5}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} d^{6}+12 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{6}-12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{6}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{2} d^{4}+6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} c \,d^{5}+12 A x a \,b^{6} c \,d^{5}+12 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{5}-6 B x a \,b^{6} c \,d^{5}+12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{4}+12 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} d^{6}-4 A \,a^{3} b^{4} d^{6}-2 A \,b^{7} c^{3} d^{3}+4 B \,a^{3} b^{4} d^{6}-B \,b^{7} c^{3} d^{3}-18 A x \,a^{2} b^{5} d^{6}+6 A x \,b^{7} c^{2} d^{4}-3 B x \,a^{2} b^{5} d^{6}+9 B x \,b^{7} c^{2} d^{4}-4 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{4} d^{6}-2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d^{3}-6 a^{2} A \,b^{5} c \,d^{5}+12 a A \,b^{6} c^{2} d^{4}-15 a^{2} B \,b^{5} c \,d^{5}+12 a B \,b^{6} c^{2} d^{4}+12 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} c \,d^{5}+24 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c \,d^{5}+24 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c \,d^{5}+6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} d^{6}+12 A \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{6}+6 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{6}-12 A \,x^{2} a \,b^{6} d^{6}+12 A \,x^{2} b^{7} c \,d^{5}-6 B \,x^{2} a \,b^{6} d^{6}+6 B \,x^{2} b^{7} c \,d^{5}}{4 i^{2} g^{3} \left (d x +c \right ) \left (b x +a \right )^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4} d^{3} \left (d a -b c \right )}\)	\(870\)
norman	\(\text {Expression too large to display}\)	\(1179\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x,method=_RETU 
RNVERBOSE)
 

Output:

A/g^3/i^2*(-d^2/(a*d-b*c)^3/(d*x+c)-3*d^2/(a*d-b*c)^4*b*ln(d*x+c)-1/2*b/(a 
*d-b*c)^2/(b*x+a)^2+3*d^2/(a*d-b*c)^4*b*ln(b*x+a)-2*b/(a*d-b*c)^3*d/(b*x+a 
))-B/g^3/i^2/(a*d-b*c)/e*(d^3/(a*d-b*c)^3*((b*e/d+(a*d-b*c)*e/d/(d*x+c))*l 
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))-(a*d-b*c)*e/d/(d*x+c)-b*e/d)-3/2/(a*d-b*c)^ 
3*b*d^2*e*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+3/(a*d-b*c)^3*b^2*d*e^2*(-1/(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))-1/(b*e/d+(a*d- 
b*c)*e/d/(d*x+c)))-1/(a*d-b*c)^3*b^3*e^3*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c 
))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.92 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=-\frac {{\left (2 \, A + B\right )} b^{3} c^{3} - 12 \, {\left (A + B\right )} a b^{2} c^{2} d + 3 \, {\left (2 \, A + 5 \, B\right )} a^{2} b c d^{2} + 4 \, {\left (A - B\right )} a^{3} d^{3} - 6 \, {\left ({\left (2 \, A + B\right )} b^{3} c d^{2} - {\left (2 \, A + B\right )} a b^{2} d^{3}\right )} x^{2} - 6 \, {\left (B b^{3} d^{3} x^{3} + B a^{2} b c d^{2} + {\left (B b^{3} c d^{2} + 2 \, B a b^{2} d^{3}\right )} x^{2} + {\left (2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{3} c^{2} d + 2 \, {\left (2 \, A - B\right )} a b^{2} c d^{2} - {\left (6 \, A + B\right )} a^{2} b d^{3}\right )} x - 2 \, {\left (3 \, {\left (2 \, A + B\right )} b^{3} d^{3} x^{3} - B b^{3} c^{3} + 6 \, B a b^{2} c^{2} d + 6 \, A a^{2} b c d^{2} - 2 \, B a^{3} d^{3} + 3 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{3} c d^{2} + 4 \, A a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (B b^{3} c^{2} d + 4 \, {\left (A + B\right )} a b^{2} c d^{2} + 2 \, {\left (A - B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} g^{3} i^{2} x^{3} + {\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} g^{3} i^{2} x^{2} + {\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} g^{3} i^{2} x + {\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4}\right )} g^{3} i^{2}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, algo 
rithm="fricas")
 

Output:

-1/4*((2*A + B)*b^3*c^3 - 12*(A + B)*a*b^2*c^2*d + 3*(2*A + 5*B)*a^2*b*c*d 
^2 + 4*(A - B)*a^3*d^3 - 6*((2*A + B)*b^3*c*d^2 - (2*A + B)*a*b^2*d^3)*x^2 
 - 6*(B*b^3*d^3*x^3 + B*a^2*b*c*d^2 + (B*b^3*c*d^2 + 2*B*a*b^2*d^3)*x^2 + 
(2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((2* 
A + 3*B)*b^3*c^2*d + 2*(2*A - B)*a*b^2*c*d^2 - (6*A + B)*a^2*b*d^3)*x - 2* 
(3*(2*A + B)*b^3*d^3*x^3 - B*b^3*c^3 + 6*B*a*b^2*c^2*d + 6*A*a^2*b*c*d^2 - 
 2*B*a^3*d^3 + 3*((2*A + 3*B)*b^3*c*d^2 + 4*A*a*b^2*d^3)*x^2 + 3*(B*b^3*c^ 
2*d + 4*(A + B)*a*b^2*c*d^2 + 2*(A - B)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d 
*x + c)))/((b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c* 
d^4 + a^4*b^2*d^5)*g^3*i^2*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3* 
d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*g^3*i^2*x^2 + (2* 
a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 - 2*a^ 
5*b*c*d^4 + a^6*d^5)*g^3*i^2*x + (a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^ 
2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a^6*c*d^4)*g^3*i^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1562 vs. \(2 (316) = 632\).

Time = 12.13 (sec) , antiderivative size = 1562, normalized size of antiderivative = 4.51 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3/(d*i*x+c*i)**2,x)
 

Output:

3*B*b*d**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**3*i**2 - 8*a**3*b 
*c*d**3*g**3*i**2 + 12*a**2*b**2*c**2*d**2*g**3*i**2 - 8*a*b**3*c**3*d*g** 
3*i**2 + 2*b**4*c**4*g**3*i**2) - 3*b*d**2*(2*A + B)*log(x + (6*A*a*b*d**3 
 + 6*A*b**2*c*d**2 + 3*B*a*b*d**3 + 3*B*b**2*c*d**2 - 3*a**5*b*d**7*(2*A + 
 B)/(a*d - b*c)**4 + 15*a**4*b**2*c*d**6*(2*A + B)/(a*d - b*c)**4 - 30*a** 
3*b**3*c**2*d**5*(2*A + B)/(a*d - b*c)**4 + 30*a**2*b**4*c**3*d**4*(2*A + 
B)/(a*d - b*c)**4 - 15*a*b**5*c**4*d**3*(2*A + B)/(a*d - b*c)**4 + 3*b**6* 
c**5*d**2*(2*A + B)/(a*d - b*c)**4)/(12*A*b**2*d**3 + 6*B*b**2*d**3))/(2*g 
**3*i**2*(a*d - b*c)**4) + 3*b*d**2*(2*A + B)*log(x + (6*A*a*b*d**3 + 6*A* 
b**2*c*d**2 + 3*B*a*b*d**3 + 3*B*b**2*c*d**2 + 3*a**5*b*d**7*(2*A + B)/(a* 
d - b*c)**4 - 15*a**4*b**2*c*d**6*(2*A + B)/(a*d - b*c)**4 + 30*a**3*b**3* 
c**2*d**5*(2*A + B)/(a*d - b*c)**4 - 30*a**2*b**4*c**3*d**4*(2*A + B)/(a*d 
 - b*c)**4 + 15*a*b**5*c**4*d**3*(2*A + B)/(a*d - b*c)**4 - 3*b**6*c**5*d* 
*2*(2*A + B)/(a*d - b*c)**4)/(12*A*b**2*d**3 + 6*B*b**2*d**3))/(2*g**3*i** 
2*(a*d - b*c)**4) + (-2*B*a**2*d**2 - 5*B*a*b*c*d - 9*B*a*b*d**2*x + B*b** 
2*c**2 - 3*B*b**2*c*d*x - 6*B*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/( 
2*a**5*c*d**3*g**3*i**2 + 2*a**5*d**4*g**3*i**2*x - 6*a**4*b*c**2*d**2*g** 
3*i**2 - 2*a**4*b*c*d**3*g**3*i**2*x + 4*a**4*b*d**4*g**3*i**2*x**2 + 6*a* 
*3*b**2*c**3*d*g**3*i**2 - 6*a**3*b**2*c**2*d**2*g**3*i**2*x - 10*a**3*b** 
2*c*d**3*g**3*i**2*x**2 + 2*a**3*b**2*d**4*g**3*i**2*x**3 - 2*a**2*b**3...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1721 vs. \(2 (340) = 680\).

Time = 0.16 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.97 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, algo 
rithm="maxima")
 

Output:

1/2*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a 
*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)* 
g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 
 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c 
^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d 
 + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*c^4 
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 
6*b*d^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3 
*b*c*d^3 + a^4*d^4)*g^3*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 1/2*A 
*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^ 
2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*i 
^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 - 2* 
a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^ 
2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3* 
a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*c^4 - 4*a 
*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 6*b*d 
^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c* 
d^3 + a^4*d^4)*g^3*i^2)) - 1/4*(b^3*c^3 - 12*a*b^2*c^2*d + 15*a^2*b*c*d^2 
- 4*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 
 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b...
 

Giac [A] (verification not implemented)

Time = 53.40 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.80 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (B b e^{3} - \frac {2 \, {\left (b e x + a e\right )} B d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, A b e^{3} + B b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A d e^{2}}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} B d e^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, algo 
rithm="giac")
 

Output:

-1/4*(2*(B*b*e^3 - 2*(b*e*x + a*e)*B*d*e^2/(d*x + c))*log((b*e*x + a*e)/(d 
*x + c))/((b*e*x + a*e)^2*b*c*g^3*i^2/(d*x + c)^2 - (b*e*x + a*e)^2*a*d*g^ 
3*i^2/(d*x + c)^2) + (2*A*b*e^3 + B*b*e^3 - 4*(b*e*x + a*e)*A*d*e^2/(d*x + 
 c) - 4*(b*e*x + a*e)*B*d*e^2/(d*x + c))/((b*e*x + a*e)^2*b*c*g^3*i^2/(d*x 
 + c)^2 - (b*e*x + a*e)^2*a*d*g^3*i^2/(d*x + c)^2))*(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
 

Mupad [B] (verification not implemented)

Time = 31.11 (sec) , antiderivative size = 984, normalized size of antiderivative = 2.84 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^3*(c*i + d*i*x)^2) 
,x)
 

Output:

(3*B*b*d^2*log((e*(a + b*x))/(c + d*x))^2)/(2*g^3*i^2*(a*d - b*c)^4) - (A* 
b*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*6i)/(g^3*i^2*(a*d - b 
*c)^4) - (B*b*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*3i)/(g^3* 
i^2*(a*d - b*c)^4) - (A*a^2*d^2)/(g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d 
*x)) + (A*b^2*c^2)/(2*g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) + (B*a^ 
2*d^2)/(g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) + (B*b^2*c^2)/(4*g^3* 
i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (B*a*d*log((e*(a + b*x))/(c + d 
*x)))/(g^3*i^2*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)) - (B*b*c*log((e*(a + b 
*x))/(c + d*x)))/(2*g^3*i^2*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)) - (3*A*b^ 
2*d^2*x^2)/(g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (3*B*b^2*d^2*x^ 
2)/(2*g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (5*A*a*b*c*d)/(2*g^3* 
i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (11*B*a*b*c*d)/(4*g^3*i^2*(a*d 
- b*c)^3*(a + b*x)^2*(c + d*x)) - (3*B*b*d*x*log((e*(a + b*x))/(c + d*x))) 
/(2*g^3*i^2*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)) - (3*B*b^2*d^2*x^2*log((e 
*(a + b*x))/(c + d*x)))/(g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (9 
*A*a*b*d^2*x)/(2*g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (3*B*a*b*d 
^2*x)/(4*g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (3*A*b^2*c*d*x)/(2 
*g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (9*B*b^2*c*d*x)/(4*g^3*i^2 
*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (3*B*a*b*c*d*log((e*(a + b*x))/(c 
+ d*x)))/(g^3*i^2*(a*d - b*c)^3*(a + b*x)^2*(c + d*x)) - (3*B*a*b*d^2*x...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1775, normalized size of antiderivative = 5.13 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x)
 

Output:

( - 24*log(a + b*x)*a**4*b*c*d**3 - 24*log(a + b*x)*a**4*b*d**4*x - 12*log 
(a + b*x)*a**3*b**2*c**2*d**2 - 60*log(a + b*x)*a**3*b**2*c*d**3*x - 48*lo 
g(a + b*x)*a**3*b**2*d**4*x**2 - 24*log(a + b*x)*a**2*b**3*c**2*d**2*x - 1 
8*log(a + b*x)*a**2*b**3*c**2*d**2 - 48*log(a + b*x)*a**2*b**3*c*d**3*x**2 
 - 18*log(a + b*x)*a**2*b**3*c*d**3*x - 24*log(a + b*x)*a**2*b**3*d**4*x** 
3 - 12*log(a + b*x)*a*b**4*c**2*d**2*x**2 - 36*log(a + b*x)*a*b**4*c**2*d* 
*2*x - 12*log(a + b*x)*a*b**4*c*d**3*x**3 - 36*log(a + b*x)*a*b**4*c*d**3* 
x**2 - 18*log(a + b*x)*b**5*c**2*d**2*x**2 - 18*log(a + b*x)*b**5*c*d**3*x 
**3 + 24*log(c + d*x)*a**4*b*c*d**3 + 24*log(c + d*x)*a**4*b*d**4*x + 12*l 
og(c + d*x)*a**3*b**2*c**2*d**2 + 60*log(c + d*x)*a**3*b**2*c*d**3*x + 48* 
log(c + d*x)*a**3*b**2*d**4*x**2 + 24*log(c + d*x)*a**2*b**3*c**2*d**2*x + 
 18*log(c + d*x)*a**2*b**3*c**2*d**2 + 48*log(c + d*x)*a**2*b**3*c*d**3*x* 
*2 + 18*log(c + d*x)*a**2*b**3*c*d**3*x + 24*log(c + d*x)*a**2*b**3*d**4*x 
**3 + 12*log(c + d*x)*a*b**4*c**2*d**2*x**2 + 36*log(c + d*x)*a*b**4*c**2* 
d**2*x + 12*log(c + d*x)*a*b**4*c*d**3*x**3 + 36*log(c + d*x)*a*b**4*c*d** 
3*x**2 + 18*log(c + d*x)*b**5*c**2*d**2*x**2 + 18*log(c + d*x)*b**5*c*d**3 
*x**3 - 12*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**2*c*d**3 - 12*log((a*e 
+ b*e*x)/(c + d*x))**2*a**3*b**2*d**4*x - 6*log((a*e + b*e*x)/(c + d*x))** 
2*a**2*b**3*c**2*d**2 - 30*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c*d** 
3*x - 24*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*d**4*x**2 - 12*log((...