\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^6} \, dx\) [29]

Optimal result

Integrand size = 40, antiderivative size = 181 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\frac {B d i^3 (c+d x)^4}{16 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b B i^3 (c+d x)^5}{25 (b c-a d)^2 g^6 (a+b x)^5}+\frac {d i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 (b c-a d)^2 g^6 (a+b x)^5} \] Output:

1/16*B*d*i^3*(d*x+c)^4/(-a*d+b*c)^2/g^6/(b*x+a)^4-1/25*b*B*i^3*(d*x+c)^5/( 
-a*d+b*c)^2/g^6/(b*x+a)^5+1/4*d*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/ 
(-a*d+b*c)^2/g^6/(b*x+a)^4-1/5*b*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c))) 
/(-a*d+b*c)^2/g^6/(b*x+a)^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(181)=362\).

Time = 0.62 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.36 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {i^3 \left (80 A b^5 c^5+16 b^5 B c^5-100 a A b^4 c^4 d-25 a b^4 B c^4 d+20 a^5 A d^5+9 a^5 B d^5+300 A b^5 c^4 d x+55 b^5 B c^4 d x-400 a A b^4 c^3 d^2 x-100 a b^4 B c^3 d^2 x+100 a^4 A b d^5 x+45 a^4 b B d^5 x+400 A b^5 c^3 d^2 x^2+60 b^5 B c^3 d^2 x^2-600 a A b^4 c^2 d^3 x^2-150 a b^4 B c^2 d^3 x^2+200 a^3 A b^2 d^5 x^2+90 a^3 b^2 B d^5 x^2+200 A b^5 c^2 d^3 x^3+10 b^5 B c^2 d^3 x^3-400 a A b^4 c d^4 x^3-100 a b^4 B c d^4 x^3+200 a^2 A b^3 d^5 x^3+90 a^2 b^3 B d^5 x^3-20 b^5 B c d^4 x^4+20 a b^4 B d^5 x^4-20 B d^5 (a+b x)^5 \log (a+b x)+20 B (b c-a d)^2 \left (a^3 d^3+a^2 b d^2 (2 c+5 d x)+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (4 c^3+15 c^2 d x+20 c d^2 x^2+10 d^3 x^3\right )\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+20 a^5 B d^5 \log (c+d x)+100 a^4 b B d^5 x \log (c+d x)+200 a^3 b^2 B d^5 x^2 \log (c+d x)+200 a^2 b^3 B d^5 x^3 \log (c+d x)+100 a b^4 B d^5 x^4 \log (c+d x)+20 b^5 B d^5 x^5 \log (c+d x)\right )}{400 b^4 (b c-a d)^2 g^6 (a+b x)^5} \] Input:

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

Output:

-1/400*(i^3*(80*A*b^5*c^5 + 16*b^5*B*c^5 - 100*a*A*b^4*c^4*d - 25*a*b^4*B* 
c^4*d + 20*a^5*A*d^5 + 9*a^5*B*d^5 + 300*A*b^5*c^4*d*x + 55*b^5*B*c^4*d*x 
- 400*a*A*b^4*c^3*d^2*x - 100*a*b^4*B*c^3*d^2*x + 100*a^4*A*b*d^5*x + 45*a 
^4*b*B*d^5*x + 400*A*b^5*c^3*d^2*x^2 + 60*b^5*B*c^3*d^2*x^2 - 600*a*A*b^4* 
c^2*d^3*x^2 - 150*a*b^4*B*c^2*d^3*x^2 + 200*a^3*A*b^2*d^5*x^2 + 90*a^3*b^2 
*B*d^5*x^2 + 200*A*b^5*c^2*d^3*x^3 + 10*b^5*B*c^2*d^3*x^3 - 400*a*A*b^4*c* 
d^4*x^3 - 100*a*b^4*B*c*d^4*x^3 + 200*a^2*A*b^3*d^5*x^3 + 90*a^2*b^3*B*d^5 
*x^3 - 20*b^5*B*c*d^4*x^4 + 20*a*b^4*B*d^5*x^4 - 20*B*d^5*(a + b*x)^5*Log[ 
a + b*x] + 20*B*(b*c - a*d)^2*(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d 
*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 
+ 10*d^3*x^3))*Log[(e*(a + b*x))/(c + d*x)] + 20*a^5*B*d^5*Log[c + d*x] + 
100*a^4*b*B*d^5*x*Log[c + d*x] + 200*a^3*b^2*B*d^5*x^2*Log[c + d*x] + 200* 
a^2*b^3*B*d^5*x^3*Log[c + d*x] + 100*a*b^4*B*d^5*x^4*Log[c + d*x] + 20*b^5 
*B*d^5*x^5*Log[c + d*x]))/(b^4*(b*c - a*d)^2*g^6*(a + b*x)^5)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.76, 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, 53, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2772

\(\displaystyle \frac {i^3 \left (-B \int -\frac {(c+d x)^6 \left (4 b-\frac {5 d (a+b x)}{c+d x}\right )}{20 (a+b x)^6}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}+\frac {d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}\right )}{g^6 (b c-a d)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i^3 \left (\frac {1}{20} B \int \frac {(c+d x)^6 \left (4 b-\frac {5 d (a+b x)}{c+d x}\right )}{(a+b x)^6}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}+\frac {d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}\right )}{g^6 (b c-a d)^2}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {i^3 \left (\frac {1}{20} B \int \left (\frac {4 b (c+d x)^6}{(a+b x)^6}-\frac {5 d (c+d x)^5}{(a+b x)^5}\right )d\frac {a+b x}{c+d x}-\frac {b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}+\frac {d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}\right )}{g^6 (b c-a d)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^3 \left (-\frac {b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 (a+b x)^5}+\frac {d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}+\frac {1}{20} B \left (\frac {5 d (c+d x)^4}{4 (a+b x)^4}-\frac {4 b (c+d x)^5}{5 (a+b x)^5}\right )\right )}{g^6 (b c-a d)^2}\)

Input:

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

Output:

(i^3*((B*((5*d*(c + d*x)^4)/(4*(a + b*x)^4) - (4*b*(c + d*x)^5)/(5*(a + b* 
x)^5)))/20 + (d*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + 
b*x)^4) - (b*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(a + b*x 
)^5)))/((b*c - a*d)^2*g^6)
 

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 2009

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs. \(2(173)=346\).

Time = 2.26 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.97

method	result	size
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{4} A b}{5 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{3} e^{3} A}{4 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{3} d^{2} e^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {i^{3} d^{3} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{6}}\right )}{d^{2}}\)	\(357\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{4} A b}{5 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{3} e^{3} A}{4 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{3} d^{2} e^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {i^{3} d^{3} e^{3} B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{6}}\right )}{d^{2}}\)	\(357\)
parts	\(\frac {i^{3} A \left (-\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{4 b^{4} \left (b x +a \right )^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{5 b^{4} \left (b x +a \right )^{5}}-\frac {d^{3}}{2 b^{4} \left (b x +a \right )^{2}}+\frac {d^{2} \left (d a -b c \right )}{b^{4} \left (b x +a \right )^{3}}\right )}{g^{6}}-\frac {i^{3} B \left (d a -b c \right )^{4} e^{4} \left (\frac {d^{6} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{6}}-\frac {d^{5} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{6}}\right )}{g^{6} d^{5}}\)	\(369\)
risch	\(-\frac {i^{3} B \left (10 b^{3} d^{3} x^{3}+10 a \,b^{2} d^{3} x^{2}+20 b^{3} c \,d^{2} x^{2}+5 a^{2} b \,d^{3} x +10 a \,b^{2} c \,d^{2} x +15 b^{3} c^{2} d x +a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{20 \left (b x +a \right )^{5} g^{6} b^{4}}-\frac {\left (-100 B a \,b^{4} c \,d^{4} x^{3}-150 B a \,b^{4} c^{2} d^{3} x^{2}+400 A \,b^{5} c^{3} d^{2} x^{2}+300 A \,b^{5} c^{4} d x +20 B \ln \left (d x +c \right ) b^{5} d^{5} x^{5}-20 B \ln \left (-b x -a \right ) b^{5} d^{5} x^{5}+20 B \ln \left (d x +c \right ) a^{5} d^{5}-100 B a \,b^{4} c^{3} d^{2} x -25 B a \,b^{4} c^{4} d +20 B a \,b^{4} d^{5} x^{4}-20 B \,b^{5} c \,d^{4} x^{4}+200 A \,a^{2} b^{3} d^{5} x^{3}+90 B \,a^{2} b^{3} d^{5} x^{3}+10 B \,b^{5} c^{2} d^{3} x^{3}+200 A \,a^{3} b^{2} d^{5} x^{2}+90 B \,a^{3} b^{2} d^{5} x^{2}+60 B \,b^{5} c^{3} d^{2} x^{2}+100 A \,a^{4} b \,d^{5} x +45 B \,a^{4} b \,d^{5} x +55 B \,b^{5} c^{4} d x -20 B \ln \left (-b x -a \right ) a^{5} d^{5}+200 A \,b^{5} c^{2} d^{3} x^{3}-100 A a \,b^{4} c^{4} d -600 A a \,b^{4} c^{2} d^{3} x^{2}-400 A a \,b^{4} c^{3} d^{2} x +100 B \ln \left (d x +c \right ) a \,b^{4} d^{5} x^{4}-100 B \ln \left (-b x -a \right ) a \,b^{4} d^{5} x^{4}+200 B \ln \left (d x +c \right ) a^{2} b^{3} d^{5} x^{3}-200 B \ln \left (-b x -a \right ) a^{2} b^{3} d^{5} x^{3}+200 B \ln \left (d x +c \right ) a^{3} b^{2} d^{5} x^{2}-200 B \ln \left (-b x -a \right ) a^{3} b^{2} d^{5} x^{2}+100 B \ln \left (d x +c \right ) a^{4} b \,d^{5} x -100 B \ln \left (-b x -a \right ) a^{4} b \,d^{5} x +20 A \,a^{5} d^{5}-400 A a \,b^{4} c \,d^{4} x^{3}+80 A \,b^{5} c^{5}+9 B \,a^{5} d^{5}+16 B \,b^{5} c^{5}\right ) i^{3}}{400 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b x +a \right )^{5} g^{6} b^{4}}\)	\(756\)
parallelrisch	\(-\frac {-400 A \,x^{3} a \,b^{7} c \,d^{5} i^{3}-100 B \,x^{3} a \,b^{7} c \,d^{5} i^{3}+400 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{8} c^{3} d^{3} i^{3}+20 A \,a^{5} b^{3} d^{6} i^{3}+80 A \,b^{8} c^{5} d \,i^{3}+9 B \,a^{5} b^{3} d^{6} i^{3}+16 B \,b^{8} c^{5} d \,i^{3}+400 A \,x^{2} b^{8} c^{3} d^{3} i^{3}+90 B \,x^{2} a^{3} b^{5} d^{6} i^{3}+60 B \,x^{2} b^{8} c^{3} d^{3} i^{3}+100 A x \,a^{4} b^{4} d^{6} i^{3}+300 A x \,b^{8} c^{4} d^{2} i^{3}+45 B x \,a^{4} b^{4} d^{6} i^{3}+55 B x \,b^{8} c^{4} d^{2} i^{3}+80 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{8} c^{5} d \,i^{3}-20 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{8} d^{6} i^{3}+20 B \,x^{4} a \,b^{7} d^{6} i^{3}-20 B \,x^{4} b^{8} c \,d^{5} i^{3}+200 A \,x^{3} a^{2} b^{6} d^{6} i^{3}+200 A \,x^{3} b^{8} c^{2} d^{4} i^{3}+90 B \,x^{3} a^{2} b^{6} d^{6} i^{3}+10 B \,x^{3} b^{8} c^{2} d^{4} i^{3}+200 A \,x^{2} a^{3} b^{5} d^{6} i^{3}-100 A a \,b^{7} c^{4} d^{2} i^{3}-25 B a \,b^{7} c^{4} d^{2} i^{3}-600 A \,x^{2} a \,b^{7} c^{2} d^{4} i^{3}-100 B x a \,b^{7} c^{3} d^{3} i^{3}-100 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{7} c^{4} d^{2} i^{3}-100 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{7} d^{6} i^{3}+200 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{8} c^{2} d^{4} i^{3}-150 B \,x^{2} a \,b^{7} c^{2} d^{4} i^{3}+300 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{8} c^{4} d^{2} i^{3}-400 A x a \,b^{7} c^{3} d^{3} i^{3}-400 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{7} c \,d^{5} i^{3}-600 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{7} c^{2} d^{4} i^{3}-400 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{7} c^{3} d^{3} i^{3}}{400 g^{6} \left (b x +a \right )^{5} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{7} d}\)	\(792\)
orering	\(\frac {\left (b x +a \right ) \left (-27 a^{4} b^{2} d^{5} x^{6}-27 a^{3} b^{3} c \,d^{4} x^{6}-27 a^{2} b^{4} c^{2} d^{3} x^{6}-27 a \,b^{5} c^{3} d^{2} x^{6}+48 b^{6} c^{4} d \,x^{6}-28 a^{5} b \,d^{5} x^{5}-162 a^{4} b^{2} c \,d^{4} x^{5}-162 a^{3} b^{3} c^{2} d^{3} x^{5}-162 a^{2} b^{4} c^{3} d^{2} x^{5}+138 a \,b^{5} c^{4} d \,x^{5}+96 b^{6} c^{5} x^{5}+175 a^{6} d^{5} x^{4}-315 a^{5} b c \,d^{4} x^{4}-405 a^{4} b^{2} c^{2} d^{3} x^{4}-405 a^{3} b^{3} c^{3} d^{2} x^{4}-30 a^{2} b^{4} c^{4} d \,x^{4}+480 a \,b^{5} c^{5} x^{4}+700 a^{6} c \,d^{4} x^{3}-980 a^{5} b \,c^{2} d^{3} x^{3}-540 a^{4} b^{2} c^{3} d^{2} x^{3}-540 a^{3} b^{3} c^{4} d \,x^{3}+960 a^{2} b^{4} c^{5} x^{3}+1050 a^{6} c^{2} d^{3} x^{2}-1330 a^{5} b \,c^{3} d^{2} x^{2}-780 a^{4} b^{2} c^{4} d \,x^{2}+960 a^{3} b^{3} c^{5} x^{2}+700 a^{6} c^{3} d^{2} x -1140 a^{5} b \,c^{4} d x +480 a^{4} b^{2} c^{5} x +100 a^{6} c^{4} d -80 a^{5} b \,c^{5}\right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{400 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) a^{5} \left (d x +c \right )^{3} \left (b g x +a g \right )^{6}}-\frac {x \left (9 a^{4} b \,d^{4} x^{4}+9 a^{3} b^{2} c \,d^{3} x^{4}+9 a^{2} b^{3} c^{2} d^{2} x^{4}+9 a \,b^{4} c^{3} d \,x^{4}-16 b^{5} c^{4} x^{4}+25 a^{5} d^{4} x^{3}+45 a^{4} b c \,d^{3} x^{3}+45 a^{3} b^{2} c^{2} d^{2} x^{3}+45 a^{2} b^{3} c^{3} d \,x^{3}-80 a \,b^{4} c^{4} x^{3}+100 a^{5} c \,d^{3} x^{2}+90 a^{4} b \,c^{2} d^{2} x^{2}+90 a^{3} b^{2} c^{3} d \,x^{2}-160 a^{2} b^{3} c^{4} x^{2}+150 a^{5} c^{2} d^{2} x +90 a^{4} b \,c^{3} d x -160 a^{3} b^{2} c^{4} x +100 a^{5} c^{3} d -80 a^{4} b \,c^{4}\right ) \left (b x +a \right )^{2} \left (\frac {3 \left (d i x +c i \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) d i}{\left (b g x +a g \right )^{6}}+\frac {\left (d i x +c i \right )^{3} B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{e \left (b x +a \right ) \left (b g x +a g \right )^{6}}-\frac {6 \left (d i x +c i \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) b g}{\left (b g x +a g \right )^{7}}\right )}{400 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) a^{5} \left (d x +c \right )^{2}}\)	\(949\)
norman	\(\frac {\frac {\left (20 A a \,c^{3} d \,i^{3}-20 A b \,c^{4} i^{3}+5 B a \,c^{3} d \,i^{3}-4 B b \,c^{4} i^{3}\right ) x}{20 g a \left (d a -b c \right )}+\frac {\left (60 A \,a^{2} c^{2} d^{2} i^{3}+20 A a b \,c^{3} d \,i^{3}-80 A \,b^{2} c^{4} i^{3}+15 B \,a^{2} c^{2} d^{2} i^{3}+9 B a b \,c^{3} d \,i^{3}-16 B \,b^{2} c^{4} i^{3}\right ) x^{2}}{40 g \,a^{2} \left (d a -b c \right )}+\frac {\left (40 A \,a^{3} c \,d^{3} i^{3}+20 A \,a^{2} b \,c^{2} d^{2} i^{3}+20 A a \,b^{2} c^{3} d \,i^{3}-80 A \,b^{3} c^{4} i^{3}+10 B \,a^{3} c \,d^{3} i^{3}+9 B \,a^{2} b \,c^{2} d^{2} i^{3}+9 B a \,b^{2} c^{3} d \,i^{3}-16 B \,b^{3} c^{4} i^{3}\right ) x^{3}}{40 g \,a^{3} \left (d a -b c \right )}+\frac {\left (20 A \,a^{4} d^{4} i^{3}+20 A \,a^{3} b c \,d^{3} i^{3}+20 A \,a^{2} b^{2} c^{2} d^{2} i^{3}+20 A a \,b^{3} c^{3} d \,i^{3}-80 A \,b^{4} c^{4} i^{3}+5 B \,a^{4} d^{4} i^{3}+9 B \,a^{3} b c \,d^{3} i^{3}+9 B \,a^{2} b^{2} c^{2} d^{2} i^{3}+9 B a \,b^{3} c^{3} d \,i^{3}-16 B \,b^{4} c^{4} i^{3}\right ) x^{4}}{80 g \,a^{4} \left (d a -b c \right )}+\frac {\left (20 A \,a^{4} d^{4} i^{3}+20 A \,a^{3} b c \,d^{3} i^{3}+20 A \,a^{2} b^{2} c^{2} d^{2} i^{3}+20 A a \,b^{3} c^{3} d \,i^{3}-80 A \,b^{4} c^{4} i^{3}+9 B \,a^{4} d^{4} i^{3}+9 B \,a^{3} b c \,d^{3} i^{3}+9 B \,a^{2} b^{2} c^{2} d^{2} i^{3}+9 B a \,b^{3} c^{3} d \,i^{3}-16 B \,b^{4} c^{4} i^{3}\right ) b \,x^{5}}{400 a^{5} g \left (d a -b c \right )}+\frac {i^{3} B \,c^{4} \left (5 d a -4 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{20 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B a \,d^{5} i^{3} x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B \,d^{5} i^{3} b \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{20 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {i^{3} B c \,d^{3} \left (2 d a -b c \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {i^{3} B \,c^{2} d^{2} \left (3 d a -2 b c \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {i^{3} B \,c^{3} d \left (4 d a -3 b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{g^{5} \left (b x +a \right )^{5}}\)	\(988\)

Input:

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x,method=_RETU 
RNVERBOSE)
 

Output:

-1/d^2*e*(a*d-b*c)*(1/5*i^3*d^2*e^4/(a*d-b*c)^3/g^6*A*b/(b*e/d+(a*d-b*c)*e 
/d/(d*x+c))^5-1/4*i^3*d^3*e^3/(a*d-b*c)^3/g^6*A/(b*e/d+(a*d-b*c)*e/d/(d*x+ 
c))^4-i^3*d^2*e^4/(a*d-b*c)^3/g^6*B*b*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^ 
5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5)+i^ 
3*d^3*e^3/(a*d-b*c)^3/g^6*B*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d 
+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (173) = 346\).

Time = 0.14 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.56 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\frac {20 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{3} x^{4} - 10 \, {\left ({\left (20 \, A + B\right )} b^{5} c^{2} d^{3} - 10 \, {\left (4 \, A + B\right )} a b^{4} c d^{4} + {\left (20 \, A + 9 \, B\right )} a^{2} b^{3} d^{5}\right )} i^{3} x^{3} - 10 \, {\left (2 \, {\left (20 \, A + 3 \, B\right )} b^{5} c^{3} d^{2} - 15 \, {\left (4 \, A + B\right )} a b^{4} c^{2} d^{3} + {\left (20 \, A + 9 \, B\right )} a^{3} b^{2} d^{5}\right )} i^{3} x^{2} - 5 \, {\left ({\left (60 \, A + 11 \, B\right )} b^{5} c^{4} d - 20 \, {\left (4 \, A + B\right )} a b^{4} c^{3} d^{2} + {\left (20 \, A + 9 \, B\right )} a^{4} b d^{5}\right )} i^{3} x - {\left (16 \, {\left (5 \, A + B\right )} b^{5} c^{5} - 25 \, {\left (4 \, A + B\right )} a b^{4} c^{4} d + {\left (20 \, A + 9 \, B\right )} a^{5} d^{5}\right )} i^{3} + 20 \, {\left (B b^{5} d^{5} i^{3} x^{5} + 5 \, B a b^{4} d^{5} i^{3} x^{4} - 10 \, {\left (B b^{5} c^{2} d^{3} - 2 \, B a b^{4} c d^{4}\right )} i^{3} x^{3} - 10 \, {\left (2 \, B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3}\right )} i^{3} x^{2} - 5 \, {\left (3 \, B b^{5} c^{4} d - 4 \, B a b^{4} c^{3} d^{2}\right )} i^{3} x - {\left (4 \, B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} i^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{400 \, {\left ({\left (b^{11} c^{2} - 2 \, a b^{10} c d + a^{2} b^{9} d^{2}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{2} - 2 \, a^{2} b^{9} c d + a^{3} b^{8} d^{2}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{2} - 2 \, a^{3} b^{8} c d + a^{4} b^{7} d^{2}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{2} - 2 \, a^{4} b^{7} c d + a^{5} b^{6} d^{2}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{2} - 2 \, a^{5} b^{6} c d + a^{6} b^{5} d^{2}\right )} g^{6} x + {\left (a^{5} b^{6} c^{2} - 2 \, a^{6} b^{5} c d + a^{7} b^{4} d^{2}\right )} g^{6}\right )}} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algo 
rithm="fricas")
 

Output:

1/400*(20*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^3*x^4 - 10*((20*A + B)*b^5*c^2*d^3 
 - 10*(4*A + B)*a*b^4*c*d^4 + (20*A + 9*B)*a^2*b^3*d^5)*i^3*x^3 - 10*(2*(2 
0*A + 3*B)*b^5*c^3*d^2 - 15*(4*A + B)*a*b^4*c^2*d^3 + (20*A + 9*B)*a^3*b^2 
*d^5)*i^3*x^2 - 5*((60*A + 11*B)*b^5*c^4*d - 20*(4*A + B)*a*b^4*c^3*d^2 + 
(20*A + 9*B)*a^4*b*d^5)*i^3*x - (16*(5*A + B)*b^5*c^5 - 25*(4*A + B)*a*b^4 
*c^4*d + (20*A + 9*B)*a^5*d^5)*i^3 + 20*(B*b^5*d^5*i^3*x^5 + 5*B*a*b^4*d^5 
*i^3*x^4 - 10*(B*b^5*c^2*d^3 - 2*B*a*b^4*c*d^4)*i^3*x^3 - 10*(2*B*b^5*c^3* 
d^2 - 3*B*a*b^4*c^2*d^3)*i^3*x^2 - 5*(3*B*b^5*c^4*d - 4*B*a*b^4*c^3*d^2)*i 
^3*x - (4*B*b^5*c^5 - 5*B*a*b^4*c^4*d)*i^3)*log((b*e*x + a*e)/(d*x + c)))/ 
((b^11*c^2 - 2*a*b^10*c*d + a^2*b^9*d^2)*g^6*x^5 + 5*(a*b^10*c^2 - 2*a^2*b 
^9*c*d + a^3*b^8*d^2)*g^6*x^4 + 10*(a^2*b^9*c^2 - 2*a^3*b^8*c*d + a^4*b^7* 
d^2)*g^6*x^3 + 10*(a^3*b^8*c^2 - 2*a^4*b^7*c*d + a^5*b^6*d^2)*g^6*x^2 + 5* 
(a^4*b^7*c^2 - 2*a^5*b^6*c*d + a^6*b^5*d^2)*g^6*x + (a^5*b^6*c^2 - 2*a^6*b 
^5*c*d + a^7*b^4*d^2)*g^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4218 vs. \(2 (173) = 346\).

Time = 0.25 (sec) , antiderivative size = 4218, normalized size of antiderivative = 23.30 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algo 
rithm="maxima")
 

Output:

-1/1200*B*d^3*i^3*(60*(10*b^3*x^3 + 10*a*b^2*x^2 + 5*a^2*b*x + a^3)*log(b* 
e*x/(d*x + c) + a*e/(d*x + c))/(b^9*g^6*x^5 + 5*a*b^8*g^6*x^4 + 10*a^2*b^7 
*g^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) + (77*a^3*b 
^4*c^4 - 548*a^4*b^3*c^3*d + 352*a^5*b^2*c^2*d^2 - 148*a^6*b*c*d^3 + 27*a^ 
7*d^4 - 60*(10*b^7*c^3*d - 10*a*b^6*c^2*d^2 + 5*a^2*b^5*c*d^3 - a^3*b^4*d^ 
4)*x^4 + 30*(10*b^7*c^4 - 100*a*b^6*c^3*d + 95*a^2*b^5*c^2*d^2 - 46*a^3*b^ 
4*c*d^3 + 9*a^4*b^3*d^4)*x^3 + 10*(50*a*b^6*c^4 - 410*a^2*b^5*c^3*d + 337* 
a^3*b^4*c^2*d^2 - 148*a^4*b^3*c*d^3 + 27*a^5*b^2*d^4)*x^2 + 5*(65*a^2*b^5* 
c^4 - 488*a^3*b^4*c^3*d + 352*a^4*b^3*c^2*d^2 - 148*a^5*b^2*c*d^3 + 27*a^6 
*b*d^4)*x)/((b^13*c^4 - 4*a*b^12*c^3*d + 6*a^2*b^11*c^2*d^2 - 4*a^3*b^10*c 
*d^3 + a^4*b^9*d^4)*g^6*x^5 + 5*(a*b^12*c^4 - 4*a^2*b^11*c^3*d + 6*a^3*b^1 
0*c^2*d^2 - 4*a^4*b^9*c*d^3 + a^5*b^8*d^4)*g^6*x^4 + 10*(a^2*b^11*c^4 - 4* 
a^3*b^10*c^3*d + 6*a^4*b^9*c^2*d^2 - 4*a^5*b^8*c*d^3 + a^6*b^7*d^4)*g^6*x^ 
3 + 10*(a^3*b^10*c^4 - 4*a^4*b^9*c^3*d + 6*a^5*b^8*c^2*d^2 - 4*a^6*b^7*c*d 
^3 + a^7*b^6*d^4)*g^6*x^2 + 5*(a^4*b^9*c^4 - 4*a^5*b^8*c^3*d + 6*a^6*b^7*c 
^2*d^2 - 4*a^7*b^6*c*d^3 + a^8*b^5*d^4)*g^6*x + (a^5*b^8*c^4 - 4*a^6*b^7*c 
^3*d + 6*a^7*b^6*c^2*d^2 - 4*a^8*b^5*c*d^3 + a^9*b^4*d^4)*g^6) - 60*(10*b^ 
3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 - a^3*d^5)*log(b*x + a)/((b^9 
*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5 
*c*d^4 - a^5*b^4*d^5)*g^6) + 60*(10*b^3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*...
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.56 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx=-\frac {1}{400} \, {\left (\frac {20 \, {\left (4 \, B b e^{6} i^{3} - \frac {5 \, {\left (b e x + a e\right )} B d e^{5} i^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b e x + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {80 \, A b e^{6} i^{3} + 16 \, B b e^{6} i^{3} - \frac {100 \, {\left (b e x + a e\right )} A d e^{5} i^{3}}{d x + c} - \frac {25 \, {\left (b e x + a e\right )} B d e^{5} i^{3}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b e x + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algo 
rithm="giac")
 

Output:

-1/400*(20*(4*B*b*e^6*i^3 - 5*(b*e*x + a*e)*B*d*e^5*i^3/(d*x + c))*log((b* 
e*x + a*e)/(d*x + c))/((b*e*x + a*e)^5*b*c*g^6/(d*x + c)^5 - (b*e*x + a*e) 
^5*a*d*g^6/(d*x + c)^5) + (80*A*b*e^6*i^3 + 16*B*b*e^6*i^3 - 100*(b*e*x + 
a*e)*A*d*e^5*i^3/(d*x + c) - 25*(b*e*x + a*e)*B*d*e^5*i^3/(d*x + c))/((b*e 
*x + a*e)^5*b*c*g^6/(d*x + c)^5 - (b*e*x + a*e)^5*a*d*g^6/(d*x + c)^5))*(b 
*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
 

Mupad [B] (verification not implemented)

Time = 30.18 (sec) , antiderivative size = 1053, normalized size of antiderivative = 5.82 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^6 
,x)
 

Output:

- ((20*A*a^4*d^4*i^3 - 80*A*b^4*c^4*i^3 + 9*B*a^4*d^4*i^3 - 16*B*b^4*c^4*i 
^3 + 20*A*a^2*b^2*c^2*d^2*i^3 + 9*B*a^2*b^2*c^2*d^2*i^3 + 20*A*a*b^3*c^3*d 
*i^3 + 20*A*a^3*b*c*d^3*i^3 + 9*B*a*b^3*c^3*d*i^3 + 9*B*a^3*b*c*d^3*i^3)/( 
20*(a*d - b*c)) + (x^2*(20*A*a^2*b^2*d^4*i^3 + 9*B*a^2*b^2*d^4*i^3 - 40*A* 
b^4*c^2*d^2*i^3 - 6*B*b^4*c^2*d^2*i^3 + 20*A*a*b^3*c*d^3*i^3 + 9*B*a*b^3*c 
*d^3*i^3))/(2*(a*d - b*c)) + (x*(20*A*a^3*b*d^4*i^3 + 9*B*a^3*b*d^4*i^3 - 
60*A*b^4*c^3*d*i^3 - 11*B*b^4*c^3*d*i^3 + 20*A*a*b^3*c^2*d^2*i^3 + 20*A*a^ 
2*b^2*c*d^3*i^3 + 9*B*a*b^3*c^2*d^2*i^3 + 9*B*a^2*b^2*c*d^3*i^3))/(4*(a*d 
- b*c)) + (x^3*(20*A*a*b^3*d^4*i^3 + 9*B*a*b^3*d^4*i^3 - 20*A*b^4*c*d^3*i^ 
3 - B*b^4*c*d^3*i^3))/(2*(a*d - b*c)) + (B*b^4*d^4*i^3*x^4)/(a*d - b*c))/( 
20*a^5*b^4*g^6 + 20*b^9*g^6*x^5 + 100*a^4*b^5*g^6*x + 100*a*b^8*g^6*x^4 + 
200*a^3*b^6*g^6*x^2 + 200*a^2*b^7*g^6*x^3) - (log((e*(a + b*x))/(c + d*x)) 
*(x^2*(b*(b*((B*a*d^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3 
*B*a*d^3*i^3)/(20*b^4*g^6) + (3*B*c*d^2*i^3)/(10*b^3*g^6)) + (3*B*a*d^3*i^ 
3)/(10*b^3*g^6) + (3*B*c*d^2*i^3)/(5*b^2*g^6)) + x*(b*(a*((B*a*d^3*i^3)/(2 
0*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*c^2*d*i^3)/(20*b^3*g^6)) + 
 a*(b*((B*a*d^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) + (3*B*a*d 
^3*i^3)/(20*b^4*g^6) + (3*B*c*d^2*i^3)/(10*b^3*g^6)) + (3*B*c^2*d*i^3)/(5* 
b^2*g^6)) + a*(a*((B*a*d^3*i^3)/(20*b^5*g^6) + (B*c*d^2*i^3)/(10*b^4*g^6)) 
 + (3*B*c^2*d*i^3)/(20*b^3*g^6)) + (B*c^3*i^3)/(5*b^2*g^6) + (B*d^3*i^3...
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1647, normalized size of antiderivative = 9.10 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x)
 

Output:

(i*( - 40*log(a + b*x)*a**6*b**2*c*d**4 + 20*log(a + b*x)*a**5*b**3*c**2*d 
**3 - 200*log(a + b*x)*a**5*b**3*c*d**4*x + 100*log(a + b*x)*a**4*b**4*c** 
2*d**3*x - 400*log(a + b*x)*a**4*b**4*c*d**4*x**2 + 200*log(a + b*x)*a**3* 
b**5*c**2*d**3*x**2 - 400*log(a + b*x)*a**3*b**5*c*d**4*x**3 + 200*log(a + 
 b*x)*a**2*b**6*c**2*d**3*x**3 - 200*log(a + b*x)*a**2*b**6*c*d**4*x**4 + 
100*log(a + b*x)*a*b**7*c**2*d**3*x**4 - 40*log(a + b*x)*a*b**7*c*d**4*x** 
5 + 20*log(a + b*x)*b**8*c**2*d**3*x**5 + 40*log(c + d*x)*a**6*b**2*c*d**4 
 - 20*log(c + d*x)*a**5*b**3*c**2*d**3 + 200*log(c + d*x)*a**5*b**3*c*d**4 
*x - 100*log(c + d*x)*a**4*b**4*c**2*d**3*x + 400*log(c + d*x)*a**4*b**4*c 
*d**4*x**2 - 200*log(c + d*x)*a**3*b**5*c**2*d**3*x**2 + 400*log(c + d*x)* 
a**3*b**5*c*d**4*x**3 - 200*log(c + d*x)*a**2*b**6*c**2*d**3*x**3 + 200*lo 
g(c + d*x)*a**2*b**6*c*d**4*x**4 - 100*log(c + d*x)*a*b**7*c**2*d**3*x**4 
+ 40*log(c + d*x)*a*b**7*c*d**4*x**5 - 20*log(c + d*x)*b**8*c**2*d**3*x**5 
 + 40*log((a*e + b*e*x)/(c + d*x))*a**6*b**2*c*d**4 - 20*log((a*e + b*e*x) 
/(c + d*x))*a**5*b**3*c**2*d**3 + 200*log((a*e + b*e*x)/(c + d*x))*a**5*b* 
*3*c*d**4*x - 100*log((a*e + b*e*x)/(c + d*x))*a**4*b**4*c**2*d**3*x + 400 
*log((a*e + b*e*x)/(c + d*x))*a**4*b**4*c*d**4*x**2 - 100*log((a*e + b*e*x 
)/(c + d*x))*a**3*b**5*c**4*d - 400*log((a*e + b*e*x)/(c + d*x))*a**3*b**5 
*c**3*d**2*x - 800*log((a*e + b*e*x)/(c + d*x))*a**3*b**5*c**2*d**3*x**2 - 
 100*log((a*e + b*e*x)/(c + d*x))*a**3*b**5*d**5*x**4 + 80*log((a*e + b...