\(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\) [62]

Optimal result

Integrand size = 40, antiderivative size = 287 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {B^2 d i (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B^2 i (c+d x)^3}{27 (b c-a d)^2 g^4 (a+b x)^3}+\frac {B d i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^2 g^4 (a+b x)^3} \] Output:

1/4*B^2*d*i*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-2/27*b*B^2*i*(d*x+c)^3/(- 
a*d+b*c)^2/g^4/(b*x+a)^3+1/2*B*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/( 
-a*d+b*c)^2/g^4/(b*x+a)^2-2/9*b*B*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/ 
(-a*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2 
/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^ 
2/(-a*d+b*c)^2/g^4/(b*x+a)^3
 

Mathematica [C] (verified)

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

Time = 1.06 (sec) , antiderivative size = 1032, normalized size of antiderivative = 3.60 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/108*(i*(36*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*d* 
(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 2*B*(12*A 
*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3 - 18*A*d*(b*c - a*d)^2*(a + b*x) - 15*B 
*d*(b*c - a*d)^2*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*( 
b*c - a*d)*(a + b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*(a + 
 b*x)^3*Log[a + b*x] - 18*B*d^3*(a + b*x)^3*Log[a + b*x]^2 + 12*B*(b*c - a 
*d)^3*Log[(e*(a + b*x))/(c + d*x)] - 18*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e 
*(a + b*x))/(c + d*x)] + 36*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*(a + b*x) 
)/(c + d*x)] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d* 
x)] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3*(a + b*x)^3*Log[c + d*x 
] + 36*B*d^3*(a + b*x)^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 
36*B*d^3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] - 18*B*d^3* 
(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + 
 d*x))/(b*c - a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c 
) + a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 
27*B*d*(a + b*x)*(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4 
*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*d^2*( 
a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d^2*(a + 
b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x) 
*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*...
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 2795, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \left (-\frac {b (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}+\frac {B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {2 b B^2 (c+d x)^3}{27 (a+b x)^3}+\frac {B^2 d (c+d x)^2}{4 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\)

Input:

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

Output:

(i*((B^2*d*(c + d*x)^2)/(4*(a + b*x)^2) - (2*b*B^2*(c + d*x)^3)/(27*(a + b 
*x)^3) + (B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b* 
x)^2) - (2*b*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a + b 
*x)^3) + (d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b* 
x)^2) - (b*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + b*x 
)^3)))/((b*c - a*d)^2*g^4)
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

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. \(642\) vs. \(2(275)=550\).

Time = 1.66 (sec) , antiderivative size = 643, normalized size of antiderivative = 2.24

method	result	size
parts	\(\frac {i \,A^{2} \left (-\frac {d}{2 b^{2} \left (b x +a \right )^{2}}-\frac {-d a +b c}{3 b^{2} \left (b x +a \right )^{3}}\right )}{g^{4}}-\frac {i \,B^{2} \left (d a -b c \right )^{2} e^{2} \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\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^{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 )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \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} d^{3}}-\frac {2 i A B \left (d a -b c \right )^{2} e^{2} \left (\frac {d^{4} \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^{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 )}{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} d^{3}}\)	\(643\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e^{2} A^{2} b}{3 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e \,A^{2}}{2 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {2 i \,d^{2} e^{2} A 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 )}{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 )^{3} g^{4}}+\frac {2 i \,d^{3} e A B \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} g^{4}}-\frac {i \,d^{2} e^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \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 )^{3} g^{4}}+\frac {i \,d^{3} e \,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}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\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} g^{4}}\right )}{d^{2}}\)	\(704\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {i \,d^{2} e^{2} A^{2} b}{3 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e \,A^{2}}{2 \left (d a -b c \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {2 i \,d^{2} e^{2} A 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 )}{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 )^{3} g^{4}}+\frac {2 i \,d^{3} e A B \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} g^{4}}-\frac {i \,d^{2} e^{2} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \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 )^{3} g^{4}}+\frac {i \,d^{3} e \,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}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\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} g^{4}}\right )}{d^{2}}\)	\(704\)
norman	\(\frac {-\frac {18 A^{2} a^{2} b \,d^{2} i +18 A^{2} a \,b^{2} c d i -36 A^{2} b^{3} c^{2} i +30 A B \,a^{2} b \,d^{2} i +30 A B a \,b^{2} c d i -24 A B \,b^{3} c^{2} i +19 B^{2} a^{2} b \,d^{2} i +19 B^{2} a \,b^{2} c d i -8 B^{2} b^{3} c^{2} i}{108 g \,b^{3} \left (d a -b c \right )}-\frac {\left (18 A^{2} a b \,d^{2} i -18 A^{2} b^{2} c d i +30 A B a b \,d^{2} i -6 A B \,b^{2} c d i +19 B^{2} a b \,d^{2} i +B^{2} b^{2} c d i \right ) x}{36 g \left (d a -b c \right ) b^{2}}-\frac {\left (6 A B b \,d^{2} i +5 B^{2} b \,d^{2} i \right ) x^{2}}{18 b \left (d a -b c \right ) g}+\frac {B^{2} i \,c^{2} \left (3 d a -2 b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{6 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {c^{2} B i \left (18 A d a -12 A b c +9 B a d -4 B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{18 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B \,d^{2} i \left (6 A d a +3 B a d +2 B b c \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B^{2} a i \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {b \,d^{3} B^{2} i \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{6 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B^{2} i c d \left (2 d a -b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {b \,d^{3} B i \left (6 A +5 B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{18 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {c B i d \left (12 A d a -6 A b c +6 B a d -B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}}{g^{3} \left (b x +a \right )^{3}}\)	\(762\)
parallelrisch	\(-\frac {-54 A B a \,b^{5} c^{2} d^{2} i +108 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{2} i -108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{3} i -108 A B x a \,b^{5} c \,d^{3} i -108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{2} i +72 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d i -54 B^{2} x a \,b^{5} c \,d^{3} i -36 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{4} i -54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} d^{4} i -54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{4} i -36 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c \,d^{3} i +54 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{2} d^{2} i +36 A B \,x^{2} a \,b^{5} d^{4} i -36 A B \,x^{2} b^{6} c \,d^{3} i +18 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{2} d^{2} i -54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c^{2} d^{2} i -54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c^{2} d^{2} i -108 A^{2} x a \,b^{5} c \,d^{3} i +90 A B x \,a^{2} b^{4} d^{4} i +18 A B x \,b^{6} c^{2} d^{2} i +18 A^{2} a^{3} b^{3} d^{4} i +36 A^{2} b^{6} c^{3} d i +19 B^{2} a^{3} b^{3} d^{4} i +8 B^{2} b^{6} c^{3} d i -108 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} d^{4} i -108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{5} c \,d^{3} i -216 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{5} c \,d^{3} i -54 A^{2} a \,b^{5} c^{2} d^{2} i -30 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} d^{4} i +54 A^{2} x \,a^{2} b^{4} d^{4} i +54 A^{2} x \,b^{6} c^{2} d^{2} i +57 B^{2} x \,a^{2} b^{4} d^{4} i -3 B^{2} x \,b^{6} c^{2} d^{2} i +24 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{6} c^{3} d i -18 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} d^{4} i +36 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{6} c^{3} d i +30 B^{2} x^{2} a \,b^{5} d^{4} i -30 B^{2} x^{2} b^{6} c \,d^{3} i +30 A B \,a^{3} b^{3} d^{4} i +24 A B \,b^{6} c^{3} d i -27 B^{2} a \,b^{5} c^{2} d^{2} i}{108 g^{4} \left (b x +a \right )^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{5} d}\)	\(952\)
orering	\(\text {Expression too large to display}\)	\(1223\)
risch	\(\text {Expression too large to display}\)	\(2435\)

Input:

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x,method=_RETU 
RNVERBOSE)
 

Output:

i*A^2/g^4*(-1/2*d/b^2/(b*x+a)^2-1/3*(-a*d+b*c)/b^2/(b*x+a)^3)-i*B^2/g^4/d^ 
3*(a*d-b*c)^2*e^2*(d^4/(a*d-b*c)^4*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*l 
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))^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^3/(a*d-b 
*c)^4*b*e*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d* 
x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c) 
)-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))-2*i*A*B/g^4/d^3*(a*d-b*c)^2*e^2*( 
d^4/(a*d-b*c)^4*(-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^3/(a*d-b*c)^4*b*e*(-1/3 
/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (275) = 550\).

Time = 0.09 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.09 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B + 5 \, B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c d^{2} + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{2} b d^{3}\right )} i x + 18 \, {\left (B^{2} b^{3} d^{3} i x^{3} + 3 \, B^{2} a b^{2} d^{3} i x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 2 \, B^{2} a b^{2} c d^{2}\right )} i x - {\left (2 \, B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - {\left (4 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + {\left (18 \, A^{2} + 30 \, A B + 19 \, B^{2}\right )} a^{3} d^{3}\right )} i + 6 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} d^{3} i x^{3} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (2 \, A B + B^{2}\right )} a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A B + B^{2}\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c d^{2}\right )} i x - {\left (4 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \, {\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{108 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algo 
rithm="fricas")
 

Output:

1/108*(6*((6*A*B + 5*B^2)*b^3*c*d^2 - (6*A*B + 5*B^2)*a*b^2*d^3)*i*x^2 - 3 
*((18*A^2 + 6*A*B - B^2)*b^3*c^2*d - 18*(2*A^2 + 2*A*B + B^2)*a*b^2*c*d^2 
+ (18*A^2 + 30*A*B + 19*B^2)*a^2*b*d^3)*i*x + 18*(B^2*b^3*d^3*i*x^3 + 3*B^ 
2*a*b^2*d^3*i*x^2 - 3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i*x - (2*B^2*b^3 
*c^3 - 3*B^2*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c))^2 - (4*(9*A^2 + 
6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + (18*A^2 + 
30*A*B + 19*B^2)*a^3*d^3)*i + 6*((6*A*B + 5*B^2)*b^3*d^3*i*x^3 + 3*(2*B^2* 
b^3*c*d^2 + 3*(2*A*B + B^2)*a*b^2*d^3)*i*x^2 - 3*((6*A*B + B^2)*b^3*c^2*d 
- 6*(2*A*B + B^2)*a*b^2*c*d^2)*i*x - (4*(3*A*B + B^2)*b^3*c^3 - 9*(2*A*B + 
 B^2)*a*b^2*c^2*d)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c* 
d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4 
*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 
- 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1387 vs. \(2 (267) = 534\).

Time = 10.72 (sec) , antiderivative size = 1387, normalized size of antiderivative = 4.83 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)
 

Output:

-B*d**3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2* 
a*d**4*i + 5*B**2*b*c*d**3*i - B*a**3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 + 
3*B*a**2*b*c*d**5*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i*(6 
*A + 5*B)/(a*d - b*c)**2 + B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)/ 
(12*A*B*b*d**4*i + 10*B**2*b*d**4*i))/(18*b**2*g**4*(a*d - b*c)**2) + B*d* 
*3*i*(6*A + 5*B)*log(x + (6*A*B*a*d**4*i + 6*A*B*b*c*d**3*i + 5*B**2*a*d** 
4*i + 5*B**2*b*c*d**3*i + B*a**3*d**6*i*(6*A + 5*B)/(a*d - b*c)**2 - 3*B*a 
**2*b*c*d**5*i*(6*A + 5*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**4*i*(6*A + 
5*B)/(a*d - b*c)**2 - B*b**3*c**3*d**3*i*(6*A + 5*B)/(a*d - b*c)**2)/(12*A 
*B*b*d**4*i + 10*B**2*b*d**4*i))/(18*b**2*g**4*(a*d - b*c)**2) + (3*B**2*a 
*c**2*d*i + 6*B**2*a*c*d**2*i*x + 3*B**2*a*d**3*i*x**2 - 2*B**2*b*c**3*i - 
 3*B**2*b*c**2*d*i*x + B**2*b*d**3*i*x**3)*log(e*(a + b*x)/(c + d*x))**2/( 
6*a**5*d**2*g**4 - 12*a**4*b*c*d*g**4 + 18*a**4*b*d**2*g**4*x + 6*a**3*b** 
2*c**2*g**4 - 36*a**3*b**2*c*d*g**4*x + 18*a**3*b**2*d**2*g**4*x**2 + 18*a 
**2*b**3*c**2*g**4*x - 36*a**2*b**3*c*d*g**4*x**2 + 6*a**2*b**3*d**2*g**4* 
x**3 + 18*a*b**4*c**2*g**4*x**2 - 12*a*b**4*c*d*g**4*x**3 + 6*b**5*c**2*g* 
*4*x**3) + (-6*A*B*a**2*d**2*i - 6*A*B*a*b*c*d*i - 18*A*B*a*b*d**2*i*x + 1 
2*A*B*b**2*c**2*i + 18*A*B*b**2*c*d*i*x - 5*B**2*a**2*d**2*i - 5*B**2*a*b* 
c*d*i - 15*B**2*a*b*d**2*i*x + 4*B**2*b**2*c**2*i + 3*B**2*b**2*c*d*i*x - 
6*B**2*b**2*d**2*i*x**2)*log(e*(a + b*x)/(c + d*x))/(18*a**4*b**2*d*g**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3282 vs. \(2 (275) = 550\).

Time = 0.27 (sec) , antiderivative size = 3282, normalized size of antiderivative = 11.44 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algo 
rithm="maxima")
 

Output:

-1/6*(3*b*x + a)*B^2*d*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^4*x 
^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/54*(6*((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^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c* 
d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g 
^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a) 
/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log 
(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*l 
og(b*e*x/(d*x + c) + a*e/(d*x + c)) + (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*c*d^2 + 49*a^2*b*d^3)*x + 66*(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) - 6*(11*b^3*d^3*x^3 + 33*a*b^2 
*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(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))*log(d*x + c))/(a^3*b^4*c^3*g^4 - 
3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 
 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6 
*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^ 
2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^...
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.59 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{108} \, {\left (\frac {18 \, {\left (2 \, B^{2} b e^{4} i - \frac {3 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (12 \, A B b e^{4} i + 4 \, B^{2} b e^{4} i - \frac {18 \, {\left (b e x + a e\right )} A B d e^{3} i}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {36 \, A^{2} b e^{4} i + 24 \, A B b e^{4} i + 8 \, B^{2} b e^{4} i - \frac {54 \, {\left (b e x + a e\right )} A^{2} d e^{3} i}{d x + c} - \frac {54 \, {\left (b e x + a e\right )} A B d e^{3} i}{d x + c} - \frac {27 \, {\left (b e x + a e\right )} B^{2} d e^{3} i}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algo 
rithm="giac")
 

Output:

-1/108*(18*(2*B^2*b*e^4*i - 3*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))*log((b* 
e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*e*x + a* 
e)^3*a*d*g^4/(d*x + c)^3) + 6*(12*A*B*b*e^4*i + 4*B^2*b*e^4*i - 18*(b*e*x 
+ a*e)*A*B*d*e^3*i/(d*x + c) - 9*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))*log( 
(b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4/(d*x + c)^3 - (b*e*x + a 
*e)^3*a*d*g^4/(d*x + c)^3) + (36*A^2*b*e^4*i + 24*A*B*b*e^4*i + 8*B^2*b*e^ 
4*i - 54*(b*e*x + a*e)*A^2*d*e^3*i/(d*x + c) - 54*(b*e*x + a*e)*A*B*d*e^3* 
i/(d*x + c) - 27*(b*e*x + a*e)*B^2*d*e^3*i/(d*x + c))/((b*e*x + a*e)^3*b*c 
*g^4/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g^4/(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)))
 

Mupad [B] (verification not implemented)

Time = 29.66 (sec) , antiderivative size = 955, normalized size of antiderivative = 3.33 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x)^4 
,x)
 

Output:

- log((e*(a + b*x))/(c + d*x))^2*(((B^2*c*i)/(3*b^2*g^4) + (B^2*a*d*i)/(6* 
b^3*g^4) + (B^2*d*i*x)/(2*b^2*g^4))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2 
) - (B^2*d^3*i)/(6*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((18*A^2*a^ 
2*d^2*i - 36*A^2*b^2*c^2*i + 19*B^2*a^2*d^2*i - 8*B^2*b^2*c^2*i + 30*A*B*a 
^2*d^2*i - 24*A*B*b^2*c^2*i + 18*A^2*a*b*c*d*i + 19*B^2*a*b*c*d*i + 30*A*B 
*a*b*c*d*i)/(6*(a*d - b*c)) + (x^2*(5*B^2*b^2*d^2*i + 6*A*B*b^2*d^2*i))/(a 
*d - b*c) + (x*(18*A^2*a*b*d^2*i + 19*B^2*a*b*d^2*i - 18*A^2*b^2*c*d*i + B 
^2*b^2*c*d*i + 30*A*B*a*b*d^2*i - 6*A*B*b^2*c*d*i))/(2*(a*d - b*c)))/(18*a 
^3*b^2*g^4 + 18*b^5*g^4*x^3 + 54*a^2*b^3*g^4*x + 54*a*b^4*g^4*x^2) - (log( 
(e*(a + b*x))/(c + d*x))*(x*((A*B*i)/(b^2*g^4) + (B^2*d^3*i*(b*((3*a^2*d^2 
 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^ 
2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*b^2*g^4* 
(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (A*B*a*i)/(3*b^3*g^4) + (B*i*(2*A*b*c 
- B*a*d + B*b*c))/(3*b^3*d*g^4) + (B^2*d^3*i*(a*((3*a^2*d^2 + b^2*c^2 - 4* 
a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2)) + (3*a^3*d^3 - b^3*c^3 + 4 
*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 
2*a*b*c*d)) - (B^2*d^3*i*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/ 
(3*d^2)))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/((3*a^2*x)/d + a^3 
/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3*i*atan(((2*b*d*x - (18*b^4* 
c^2*g^4 - 18*a^2*b^2*d^2*g^4)/(18*b^2*g^4*(a*d - b*c)))*1i)/(a*d - b*c)...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1481, normalized size of antiderivative = 5.16 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x)
 

Output:

(i*(36*log(a + b*x)*a**5*b*d**3 + 108*log(a + b*x)*a**4*b**2*d**3*x + 18*l 
og(a + b*x)*a**4*b**2*d**3 + 12*log(a + b*x)*a**3*b**3*c*d**2 + 108*log(a 
+ b*x)*a**3*b**3*d**3*x**2 + 54*log(a + b*x)*a**3*b**3*d**3*x + 36*log(a + 
 b*x)*a**2*b**4*c*d**2*x + 36*log(a + b*x)*a**2*b**4*d**3*x**3 + 54*log(a 
+ b*x)*a**2*b**4*d**3*x**2 + 36*log(a + b*x)*a*b**5*c*d**2*x**2 + 18*log(a 
 + b*x)*a*b**5*d**3*x**3 + 12*log(a + b*x)*b**6*c*d**2*x**3 - 36*log(c + d 
*x)*a**5*b*d**3 - 108*log(c + d*x)*a**4*b**2*d**3*x - 18*log(c + d*x)*a**4 
*b**2*d**3 - 12*log(c + d*x)*a**3*b**3*c*d**2 - 108*log(c + d*x)*a**3*b**3 
*d**3*x**2 - 54*log(c + d*x)*a**3*b**3*d**3*x - 36*log(c + d*x)*a**2*b**4* 
c*d**2*x - 36*log(c + d*x)*a**2*b**4*d**3*x**3 - 54*log(c + d*x)*a**2*b**4 
*d**3*x**2 - 36*log(c + d*x)*a*b**5*c*d**2*x**2 - 18*log(c + d*x)*a*b**5*d 
**3*x**3 - 12*log(c + d*x)*b**6*c*d**2*x**3 + 54*log((a*e + b*e*x)/(c + d* 
x))**2*a**2*b**4*c**2*d + 108*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**4*c* 
d**2*x + 54*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**4*d**3*x**2 - 36*log(( 
a*e + b*e*x)/(c + d*x))**2*a*b**5*c**3 - 54*log((a*e + b*e*x)/(c + d*x))** 
2*a*b**5*c**2*d*x + 18*log((a*e + b*e*x)/(c + d*x))**2*a*b**5*d**3*x**3 - 
36*log((a*e + b*e*x)/(c + d*x))*a**5*b*d**3 - 108*log((a*e + b*e*x)/(c + d 
*x))*a**4*b**2*d**3*x - 18*log((a*e + b*e*x)/(c + d*x))*a**4*b**2*d**3 + 1 
08*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*c**2*d + 216*log((a*e + b*e*x)/( 
c + d*x))*a**3*b**3*c*d**2*x - 12*log((a*e + b*e*x)/(c + d*x))*a**3*b**...