\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)} \, dx\) [38]

Optimal result

Integrand size = 40, antiderivative size = 373 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {3 b B d^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 B (c+d x)^3}{9 (b c-a d)^4 g^4 i (a+b x)^3}+\frac {B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^4 i}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i} \] Output:

-3*b*B*d^2*(d*x+c)/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/4*b^2*B*d*(d*x+c)^2/(-a*d+ 
b*c)^4/g^4/i/(b*x+a)^2-1/9*b^3*B*(d*x+c)^3/(-a*d+b*c)^4/g^4/i/(b*x+a)^3+1/ 
2*B*d^3*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^4/g^4/i-3*b*d^2*(d*x+c)*(A+B*ln(e 
*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/2*b^2*d*(d*x+c)^2*(A+B*ln( 
e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-1/3*b^3*(d*x+c)^3*(A+B*ln 
(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-d^3*ln((b*x+a)/(d*x+c))* 
(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^4/i
 

Mathematica [C] (verified)

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

Time = 0.73 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\frac {-\frac {12 A (b c-a d)^3}{(a+b x)^3}-\frac {4 B (b c-a d)^3}{(a+b x)^3}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}+\frac {15 B d (b c-a d)^2}{(a+b x)^2}+\frac {36 A d^2 (-b c+a d)}{a+b x}+\frac {66 B d^2 (-b c+a d)}{a+b x}-36 A d^3 \log (a+b x)-66 B d^3 \log (a+b x)+18 B d^3 \log ^2(a+b x)-\frac {12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {18 B d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2}+\frac {36 B d^2 (-b c+a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x}-36 B d^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 A d^3 \log (c+d x)+66 B d^3 \log (c+d x)-36 B d^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+36 B d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)+18 B d^3 \log ^2(c+d x)-36 B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^4 g^4 i} \] Input:

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^4*(c*i + d*i 
*x)),x]
 

Output:

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3)/(a + b*x)^3 + (18 
*A*d*(b*c - a*d)^2)/(a + b*x)^2 + (15*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (36 
*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d))/(a + b*x) - 3 
6*A*d^3*Log[a + b*x] - 66*B*d^3*Log[a + b*x] + 18*B*d^3*Log[a + b*x]^2 - ( 
12*B*(b*c - a*d)^3*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^3 + (18*B*d*(b* 
c - a*d)^2*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^2 + (36*B*d^2*(-(b*c) + 
 a*d)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x) - 36*B*d^3*Log[a + b*x]*Log[ 
(e*(a + b*x))/(c + d*x)] + 36*A*d^3*Log[c + d*x] + 66*B*d^3*Log[c + d*x] - 
 36*B*d^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 36*B*d^3*Log[(e 
*(a + b*x))/(c + d*x)]*Log[c + d*x] + 18*B*d^3*Log[c + d*x]^2 - 36*B*d^3*L 
og[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 36*B*d^3*PolyLog[2, (d*(a + b 
*x))/(-(b*c) + a*d)] - 36*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(36 
*(b*c - a*d)^4*g^4*i)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.70, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B \int -\frac {(c+d x)^4 \left (2 b^3-\frac {9 d (a+b x) b^2}{c+d x}+\frac {18 d^2 (a+b x)^2 b}{(c+d x)^2}+\frac {6 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^3}\right )}{6 (a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^3 \left (-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} B \int \frac {(c+d x)^4 \left (2 b^3-\frac {9 d (a+b x) b^2}{c+d x}+\frac {18 d^2 (a+b x)^2 b}{(c+d x)^2}+\frac {6 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^3}\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^3 \left (-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {\frac {1}{6} B \int \left (\frac {b \left (2 b^2-\frac {9 d (a+b x) b}{c+d x}+\frac {18 d^2 (a+b x)^2}{(c+d x)^2}\right ) (c+d x)^4}{(a+b x)^4}+\frac {6 d^3 \log \left (\frac {a+b x}{c+d x}\right ) (c+d x)}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^3 \left (-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}+d^3 \left (-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac {1}{6} B \left (-\frac {2 b^3 (c+d x)^3}{3 (a+b x)^3}+\frac {9 b^2 d (c+d x)^2}{2 (a+b x)^2}+3 d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {18 b d^2 (c+d x)}{a+b x}\right )}{g^4 i (b c-a d)^4}\)

Input:

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^4*(c*i + d*i*x)),x 
]
 

Output:

((B*((-18*b*d^2*(c + d*x))/(a + b*x) + (9*b^2*d*(c + d*x)^2)/(2*(a + b*x)^ 
2) - (2*b^3*(c + d*x)^3)/(3*(a + b*x)^3) + 3*d^3*Log[(a + b*x)/(c + d*x)]^ 
2))/6 - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) 
 + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x) 
^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x)^ 
3) - d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(( 
b*c - a*d)^4*g^4*i)
 

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 = 2.43 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.34

method	result	size
parts	\(\frac {A \left (\frac {d^{3} \ln \left (d x +c \right )}{\left (d a -b c \right )^{4}}+\frac {1}{3 \left (d a -b c \right ) \left (b x +a \right )^{3}}+\frac {d}{2 \left (d a -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {d^{2}}{\left (d a -b c \right )^{3} \left (b x +a \right )}-\frac {d^{3} \ln \left (b x +a \right )}{\left (d a -b c \right )^{4}}\right )}{g^{4} i}-\frac {B \left (\frac {d^{4} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )^{4}}-\frac {3 d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4}}+\frac {3 d^{2} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4}}-\frac {d \,b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{4}}\right )}{g^{4} i d}\)	\(500\)
risch	\(\frac {A \,d^{3} \ln \left (d x +c \right )}{g^{4} i \left (d a -b c \right )^{4}}+\frac {A}{3 g^{4} i \left (d a -b c \right ) \left (b x +a \right )^{3}}+\frac {A d}{2 g^{4} i \left (d a -b c \right )^{2} \left (b x +a \right )^{2}}+\frac {A \,d^{2}}{g^{4} i \left (d a -b c \right )^{3} \left (b x +a \right )}-\frac {A \,d^{3} \ln \left (b x +a \right )}{g^{4} i \left (d a -b c \right )^{4}}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{4} i \left (d a -b c \right )^{4}}-\frac {3 B \,d^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}-\frac {3 B \,d^{2} b e}{g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )}+\frac {3 B d \,b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}+\frac {3 B d \,b^{2} e^{2}}{4 g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{2}}-\frac {B \,b^{3} e^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}-\frac {B \,b^{3} e^{3}}{9 g^{4} i \left (d a -b c \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}\)	\(628\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e^{2} A \,b^{3}}{3 i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {3 d^{3} e A \,b^{2}}{2 i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{4} A b}{i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{5} A \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{5} g^{4}}-\frac {d^{2} e^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i \left (d a -b c \right )^{5} g^{4}}+\frac {3 d^{3} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{5} g^{4}}-\frac {3 d^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{5} g^{4}}+\frac {d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (d a -b c \right )^{5} g^{4}}\right )}{d^{2}}\)	\(637\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e^{2} A \,b^{3}}{3 i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {3 d^{3} e A \,b^{2}}{2 i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{4} A b}{i \left (d a -b c \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{5} A \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e i \left (d a -b c \right )^{5} g^{4}}-\frac {d^{2} e^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i \left (d a -b c \right )^{5} g^{4}}+\frac {3 d^{3} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (d a -b c \right )^{5} g^{4}}-\frac {3 d^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i \left (d a -b c \right )^{5} g^{4}}+\frac {d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (d a -b c \right )^{5} g^{4}}\right )}{d^{2}}\)	\(637\)
parallelrisch	\(-\frac {36 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{3} d^{2}+54 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{7} b \,c^{2} d^{3}+108 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{2} d^{3}+108 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{2} d^{3}+108 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{3} d^{2}-18 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{4} d +18 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{5} b^{3} c^{2} d^{3}+36 A \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{2} d^{3}+66 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{2} d^{3}+54 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} b^{2} c^{2} d^{3}+108 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{2} d^{3}+162 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{2} d^{3}-12 A \,x^{3} a^{2} b^{6} c^{5}-4 B \,x^{3} a^{2} b^{6} c^{5}-36 A \,x^{2} a^{3} b^{5} c^{5}-12 B \,x^{2} a^{3} b^{5} c^{5}+18 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{8} c^{2} d^{3}-36 A x \,a^{4} b^{4} c^{5}+36 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} c^{2} d^{3}-12 B x \,a^{4} b^{4} c^{5}+12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{5}+66 A \,x^{3} a^{5} b^{3} c^{2} d^{3}-108 A \,x^{3} a^{4} b^{4} c^{3} d^{2}+54 A \,x^{3} a^{3} b^{5} c^{4} d +85 B \,x^{3} a^{5} b^{3} c^{2} d^{3}-108 B \,x^{3} a^{4} b^{4} c^{3} d^{2}+27 B \,x^{3} a^{3} b^{5} c^{4} d +162 A \,x^{2} a^{6} b^{2} c^{2} d^{3}-288 A \,x^{2} a^{5} b^{3} c^{3} d^{2}+162 A \,x^{2} a^{4} b^{4} c^{4} d +189 B \,x^{2} a^{6} b^{2} c^{2} d^{3}-258 B \,x^{2} a^{5} b^{3} c^{3} d^{2}+81 B \,x^{2} a^{4} b^{4} c^{4} d +108 A x \,a^{7} b \,c^{2} d^{3}-216 A x \,a^{6} b^{2} c^{3} d^{2}+144 A x \,a^{5} b^{3} c^{4} d +108 B x \,a^{7} b \,c^{2} d^{3}-162 B x \,a^{6} b^{2} c^{3} d^{2}+66 B x \,a^{5} b^{3} c^{4} d +108 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{3} d^{2}-54 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{4} d}{36 i \,g^{4} \left (b x +a \right )^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d a -b c \right ) a^{5} c^{2}}\)	\(976\)
norman	\(\text {Expression too large to display}\)	\(1038\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x,method=_RETURN 
VERBOSE)
 

Output:

A/g^4/i*(d^3/(a*d-b*c)^4*ln(d*x+c)+1/3/(a*d-b*c)/(b*x+a)^3+1/2*d/(a*d-b*c) 
^2/(b*x+a)^2+d^2/(a*d-b*c)^3/(b*x+a)-d^3/(a*d-b*c)^4*ln(b*x+a))-B/g^4/i/d* 
(1/2*d^4/(a*d-b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-3*d^3/(a*d-b*c)^4*b 
*e*(-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)))+3*d^2/(a*d-b*c)^4*b^2*e^2*(-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)-d/(a*d-b*c)^4*b^3*e^3*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*l 
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {4 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 27 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + 108 \, {\left (A + B\right )} a^{2} b c d^{2} - {\left (66 \, A + 85 \, B\right )} a^{3} d^{3} + 6 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{3} c d^{2} - {\left (6 \, A + 11 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B a^{3} d^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (6 \, A + 5 \, B\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A + 3 \, B\right )} a b^{2} c d^{2} + {\left (30 \, A + 49 \, B\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} + 6 \, A a^{3} d^{3} + 3 \, {\left (2 \, B b^{3} c d^{2} + 3 \, {\left (2 \, A + 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} - 6 \, {\left (A + B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} i x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} i x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} i x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4} i\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algori 
thm="fricas")
 

Output:

-1/36*(4*(3*A + B)*b^3*c^3 - 27*(2*A + B)*a*b^2*c^2*d + 108*(A + B)*a^2*b* 
c*d^2 - (66*A + 85*B)*a^3*d^3 + 6*((6*A + 11*B)*b^3*c*d^2 - (6*A + 11*B)*a 
*b^2*d^3)*x^2 + 18*(B*b^3*d^3*x^3 + 3*B*a*b^2*d^3*x^2 + 3*B*a^2*b*d^3*x + 
B*a^3*d^3)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((6*A + 5*B)*b^3*c^2*d - 18* 
(2*A + 3*B)*a*b^2*c*d^2 + (30*A + 49*B)*a^2*b*d^3)*x + 6*((6*A + 11*B)*b^3 
*d^3*x^3 + 2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 + 6*A*a^3*d^3 
+ 3*(2*B*b^3*c*d^2 + 3*(2*A + 3*B)*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d - 6*B*a 
*b^2*c*d^2 - 6*(A + B)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c 
^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^ 
4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c 
*d^3 + a^5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b 
^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3*b^4*c^4 - 4*a^4*b 
^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4*i)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1392 vs. \(2 (332) = 664\).

Time = 7.73 (sec) , antiderivative size = 1392, normalized size of antiderivative = 3.73 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4/(d*i*x+c*i),x)
 

Output:

-B*d**3*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**4*i - 8*a**3*b*c*d** 
3*g**4*i + 12*a**2*b**2*c**2*d**2*g**4*i - 8*a*b**3*c**3*d*g**4*i + 2*b**4 
*c**4*g**4*i) + d**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 11* 
B*a*d**4 + 11*B*b*c*d**3 - a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 + 5*a**4* 
b*c*d**7*(6*A + 11*B)/(a*d - b*c)**4 - 10*a**3*b**2*c**2*d**6*(6*A + 11*B) 
/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 - 5*a 
*b**4*c**4*d**4*(6*A + 11*B)/(a*d - b*c)**4 + b**5*c**5*d**3*(6*A + 11*B)/ 
(a*d - b*c)**4)/(12*A*b*d**4 + 22*B*b*d**4))/(6*g**4*i*(a*d - b*c)**4) - d 
**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 11*B*a*d**4 + 11*B*b 
*c*d**3 + a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 - 5*a**4*b*c*d**7*(6*A + 1 
1*B)/(a*d - b*c)**4 + 10*a**3*b**2*c**2*d**6*(6*A + 11*B)/(a*d - b*c)**4 - 
 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 + 5*a*b**4*c**4*d**4*( 
6*A + 11*B)/(a*d - b*c)**4 - b**5*c**5*d**3*(6*A + 11*B)/(a*d - b*c)**4)/( 
12*A*b*d**4 + 22*B*b*d**4))/(6*g**4*i*(a*d - b*c)**4) + (11*B*a**2*d**2 - 
7*B*a*b*c*d + 15*B*a*b*d**2*x + 2*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))/(6*a**6*d**3*g**4*i - 18*a**5*b*c*d* 
*2*g**4*i + 18*a**5*b*d**3*g**4*i*x + 18*a**4*b**2*c**2*d*g**4*i - 54*a**4 
*b**2*c*d**2*g**4*i*x + 18*a**4*b**2*d**3*g**4*i*x**2 - 6*a**3*b**3*c**3*g 
**4*i + 54*a**3*b**3*c**2*d*g**4*i*x - 54*a**3*b**3*c*d**2*g**4*i*x**2 + 6 
*a**3*b**3*d**3*g**4*i*x**3 - 18*a**2*b**4*c**3*g**4*i*x + 54*a**2*b**4...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (363) = 726\).

Time = 0.15 (sec) , antiderivative size = 1469, normalized size of antiderivative = 3.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algori 
thm="maxima")
 

Output:

-1/6*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 
 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)* 
g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3 
)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d 
^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^ 
4*i) + 6*d^3*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^4*i) - 6*d^3*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^4*i))*log(b*e*x/(d*x 
 + c) + a*e/(d*x + c)) - 1/6*A*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 1 
1*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b 
^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3 
*b^3*c*d^2 - a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3 
*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a 
^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*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^4*i) - 6*d^3*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^4*i)) - 1/36*(4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^ 
3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3 
*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 
 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*...
 

Giac [A] (verification not implemented)

Time = 53.52 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b e^{4} - \frac {3 \, {\left (b e x + a e\right )} B d e^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}} + \frac {12 \, A b e^{4} + 4 \, B b e^{4} - \frac {18 \, {\left (b e x + a e\right )} A d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B d e^{3}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algori 
thm="giac")
 

Output:

-1/36*(6*(2*B*b*e^4 - 3*(b*e*x + a*e)*B*d*e^3/(d*x + c))*log((b*e*x + a*e) 
/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4*i/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g 
^4*i/(d*x + c)^3) + (12*A*b*e^4 + 4*B*b*e^4 - 18*(b*e*x + a*e)*A*d*e^3/(d* 
x + c) - 9*(b*e*x + a*e)*B*d*e^3/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4*i/(d* 
x + c)^3 - (b*e*x + a*e)^3*a*d*g^4*i/(d*x + c)^3))*(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 = 31.78 (sec) , antiderivative size = 970, normalized size of antiderivative = 2.60 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^4*(c*i + d*i*x)),x 
)
 

Output:

(A*d^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(g^4*i*(a*d - b* 
c)^4) + (B*d^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*11i)/(3*g^4* 
i*(a*d - b*c)^4) - (B*d^3*log((e*(a + b*x))/(c + d*x))^2)/(2*g^4*i*(a*d - 
b*c)^4) + (11*A*a^2*d^2)/(6*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (A*b^2*c^2) 
/(3*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (85*B*a^2*d^2)/(36*g^4*i*(a*d - b*c 
)^3*(a + b*x)^3) + (B*b^2*c^2)/(9*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (11*B 
*a^3*d^3*log((e*(a + b*x))/(c + d*x)))/(6*g^4*i*(a*d - b*c)^4*(a + b*x)^3) 
 - (B*b^3*c^3*log((e*(a + b*x))/(c + d*x)))/(3*g^4*i*(a*d - b*c)^4*(a + b* 
x)^3) + (A*b^2*d^2*x^2)/(g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (11*B*b^2*d^2* 
x^2)/(6*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (7*A*a*b*c*d)/(6*g^4*i*(a*d - b 
*c)^3*(a + b*x)^3) - (23*B*a*b*c*d)/(36*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + 
 (5*A*a*b*d^2*x)/(2*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (49*B*a*b*d^2*x)/(1 
2*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (A*b^2*c*d*x)/(2*g^4*i*(a*d - b*c)^3* 
(a + b*x)^3) - (5*B*b^2*c*d*x)/(12*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (3*B 
*a*b^2*c^2*d*log((e*(a + b*x))/(c + d*x)))/(2*g^4*i*(a*d - b*c)^4*(a + b*x 
)^3) - (3*B*a^2*b*c*d^2*log((e*(a + b*x))/(c + d*x)))/(g^4*i*(a*d - b*c)^4 
*(a + b*x)^3) + (5*B*a^2*b*d^3*x*log((e*(a + b*x))/(c + d*x)))/(2*g^4*i*(a 
*d - b*c)^4*(a + b*x)^3) + (B*b^3*c^2*d*x*log((e*(a + b*x))/(c + d*x)))/(2 
*g^4*i*(a*d - b*c)^4*(a + b*x)^3) + (B*a*b^2*d^3*x^2*log((e*(a + b*x))/(c 
+ d*x)))/(g^4*i*(a*d - b*c)^4*(a + b*x)^3) - (B*b^3*c*d^2*x^2*log((e*(a...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1284, normalized size of antiderivative = 3.44 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x)
 

Output:

(i*(36*log(a + b*x)*a**5*d**3 + 108*log(a + b*x)*a**4*b*d**3*x + 54*log(a 
+ b*x)*a**4*b*d**3 + 12*log(a + b*x)*a**3*b**2*c*d**2 + 108*log(a + b*x)*a 
**3*b**2*d**3*x**2 + 162*log(a + b*x)*a**3*b**2*d**3*x + 36*log(a + b*x)*a 
**2*b**3*c*d**2*x + 36*log(a + b*x)*a**2*b**3*d**3*x**3 + 162*log(a + b*x) 
*a**2*b**3*d**3*x**2 + 36*log(a + b*x)*a*b**4*c*d**2*x**2 + 54*log(a + b*x 
)*a*b**4*d**3*x**3 + 12*log(a + b*x)*b**5*c*d**2*x**3 - 36*log(c + d*x)*a* 
*5*d**3 - 108*log(c + d*x)*a**4*b*d**3*x - 54*log(c + d*x)*a**4*b*d**3 - 1 
2*log(c + d*x)*a**3*b**2*c*d**2 - 108*log(c + d*x)*a**3*b**2*d**3*x**2 - 1 
62*log(c + d*x)*a**3*b**2*d**3*x - 36*log(c + d*x)*a**2*b**3*c*d**2*x - 36 
*log(c + d*x)*a**2*b**3*d**3*x**3 - 162*log(c + d*x)*a**2*b**3*d**3*x**2 - 
 36*log(c + d*x)*a*b**4*c*d**2*x**2 - 54*log(c + d*x)*a*b**4*d**3*x**3 - 1 
2*log(c + d*x)*b**5*c*d**2*x**3 + 18*log((a*e + b*e*x)/(c + d*x))**2*a**4* 
b*d**3 + 54*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**2*d**3*x + 54*log((a*e 
 + b*e*x)/(c + d*x))**2*a**2*b**3*d**3*x**2 + 18*log((a*e + b*e*x)/(c + d* 
x))**2*a*b**4*d**3*x**3 - 54*log((a*e + b*e*x)/(c + d*x))*a**4*b*d**3 + 96 
*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d**2 - 54*log((a*e + b*e*x)/(c + 
 d*x))*a**3*b**2*d**3*x - 54*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c**2*d 
 + 72*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c*d**2*x + 12*log((a*e + b*e* 
x)/(c + d*x))*a*b**4*c**3 - 18*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**2*d* 
x + 12*log((a*e + b*e*x)/(c + d*x))*a*b**4*d**3*x**3 - 12*log((a*e + b*...