\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\) [104]

Optimal result

Integrand size = 32, antiderivative size = 418 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \] Output:

-2*B^2*d^2*(d*x+c)/(-a*d+b*c)^3/g^4/(b*x+a)+1/2*b*B^2*d*(d*x+c)^2/(-a*d+b* 
c)^3/g^4/(b*x+a)^2-2/27*b^2*B^2*(d*x+c)^3/(-a*d+b*c)^3/g^4/(b*x+a)^3-2*B*d 
^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^4/(b*x+a)+b*B*d*(d*x 
+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^4/(b*x+a)^2-2/9*b^2*B*(d* 
x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^4/(b*x+a)^3-d^2*(d*x+c)* 
(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^4/(b*x+a)+b*d*(d*x+c)^2*(A+B* 
ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-1/3*b^2*(d*x+c)^3*(A+B 
*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/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 = 0.64 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.39 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {18 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (12 A (b c-a d)^3+4 B (b c-a d)^3-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 (a+b x)^3 \log (a+b x)-18 B d^3 (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-18 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 (a+b x)^3 \log (c+d x)+36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log ^2(c+d x)+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{54 b g^4 (a+b x)^3} \] Input:

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

Output:

-1/54*(18*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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)]))/(b*c - a*d)^3)/( 
b*g^4*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2950, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-2*B^2*d^2*(c + d*x))/(a + b*x) + (b*B^2*d*(c + d*x)^2)/(2*(a + b*x)^2) 
- (2*b^2*B^2*(c + d*x)^3)/(27*(a + b*x)^3) - (2*B*d^2*(c + d*x)*(A + B*Log 
[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + (b*B*d*(c + d*x)^2*(A + B*Log[(e*( 
a + b*x))/(c + d*x)]))/(a + b*x)^2 - (2*b^2*B*(c + d*x)^3*(A + B*Log[(e*(a 
 + b*x))/(c + d*x)]))/(9*(a + b*x)^3) - (d^2*(c + d*x)*(A + B*Log[(e*(a + 
b*x))/(c + d*x)])^2)/(a + b*x) + (b*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x)) 
/(c + d*x)])^2)/(a + b*x)^2 - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c 
 + d*x)])^2)/(3*(a + b*x)^3))/((b*c - a*d)^3*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 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] 
, x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(410)=820\).

Time = 1.39 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.13

method	result	size
parts	\(-\frac {A^{2}}{3 g^{4} \left (b x +a \right )^{3} b}-\frac {B^{2} \left (d a -b c \right ) e \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}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \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 {2}{\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 {2 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}}{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^{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}}{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^{2}}-\frac {2 B A \left (d a -b c \right ) e \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 )}{\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 {2 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 \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^{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 )}{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^{2}}\)	\(892\)
norman	\(\frac {\frac {B^{2} a^{2} d^{3} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {B^{2} a b \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {18 A^{2} a^{2} b^{2} d^{2}-36 A^{2} a \,b^{3} c d +18 A^{2} b^{4} c^{2}+66 A B \,a^{2} b^{2} d^{2}-42 A B a \,b^{3} c d +12 A B \,b^{4} c^{2}+85 B^{2} a^{2} b^{2} d^{2}-23 B^{2} a \,b^{3} c d +4 B^{2} b^{4} c^{2}}{54 g \left (d a -b c \right )^{2} b^{3}}-\frac {\left (30 A B a \,b^{2} d^{2}-6 A B \,b^{3} c d +49 B^{2} a \,b^{2} d^{2}-5 B^{2} b^{3} c d \right ) x}{18 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{2}}-\frac {\left (6 A B \,b^{2} d^{2}+11 B^{2} b^{2} d^{2}\right ) x^{2}}{9 b \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B^{2} c \left (3 a^{2} d^{2}-3 a c d b +c^{2} b^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{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 ) g}+\frac {c B \left (18 A \,a^{2} d^{2}-18 A a b c d +6 A \,b^{2} c^{2}+18 B \,a^{2} d^{2}-9 B a b c d +2 B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B d \left (6 A \,a^{2} d^{2}+6 B \,a^{2} d^{2}+6 B a b c d -B \,b^{2} c^{2}\right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} d^{3} B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{3 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{2} d^{3} B \left (6 A +11 B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (6 A d a +9 B a d +2 B b c \right ) B b \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \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 (b x +a \right )^{3} g^{3}}\)	\(929\)
parallelrisch	\(-\frac {147 B^{2} x \,a^{2} b^{5} d^{4}+15 B^{2} x \,b^{7} c^{2} d^{2}-12 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d -18 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} d^{4}-66 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{4}+66 B^{2} x^{2} a \,b^{6} d^{4}-66 B^{2} x^{2} b^{7} c \,d^{3}-18 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{7} c^{3} d -54 A^{2} a^{2} b^{5} c \,d^{3}+54 A^{2} a \,b^{6} c^{2} d^{2}+18 A^{2} a^{3} b^{4} d^{4}-18 A^{2} b^{7} c^{3} d +85 B^{2} a^{3} b^{4} d^{4}-4 B^{2} b^{7} c^{3} d +66 A B \,a^{3} b^{4} d^{4}-12 A B \,b^{7} c^{3} d -108 B^{2} a^{2} b^{5} c \,d^{3}+27 B^{2} a \,b^{6} c^{2} d^{2}-108 A B \,a^{2} b^{5} c \,d^{3}+54 A B a \,b^{6} c^{2} d^{2}-36 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} d^{4}-54 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} d^{4}-162 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{4}-36 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c \,d^{3}-54 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} d^{4}+36 A B \,x^{2} a \,b^{6} d^{4}-36 A B \,x^{2} b^{7} c \,d^{3}-108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{4}+18 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{2} d^{2}-54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{5} c \,d^{3}+54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{6} c^{2} d^{2}+90 A B x \,a^{2} b^{5} d^{4}+18 A B x \,b^{7} c^{2} d^{2}-36 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{7} c^{3} d -162 B^{2} x a \,b^{6} c \,d^{3}-108 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{3}+54 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{2}-108 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} d^{4}-108 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} d^{4}-108 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c \,d^{3}-108 A B x a \,b^{6} c \,d^{3}-108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{5} c \,d^{3}+108 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{6} c^{2} d^{2}}{54 g^{4} \left (b x +a \right )^{3} \left (d a -b c \right )^{3} b^{5} d}\)	\(960\)
derivativedivides	\(\text {Expression too large to display}\)	\(1039\)
default	\(\text {Expression too large to display}\)	\(1039\)
orering	\(\text {Expression too large to display}\)	\(1049\)
risch	\(\text {Expression too large to display}\)	\(2290\)

Input:

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

Output:

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

Time = 0.09 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.61 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 \, {\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 + 54 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{2} b c d^{2} - {\left (18 \, A^{2} + 66 \, A B + 85 \, B^{2}\right )} a^{3} d^{3} + 6 \, {\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (30 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + 2 \, {\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 + 18 \, {\left (A B + B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 6 \, {\left (A B + B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{54 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="frica 
s")
 

Output:

-1/54*(2*(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 + 54*(A^2 + 2*A*B + 2*B^2)*a^2*b*c*d^2 - (18*A^2 + 66*A*B + 85*B^2)* 
a^3*d^3 + 6*((6*A*B + 11*B^2)*b^3*c*d^2 - (6*A*B + 11*B^2)*a*b^2*d^3)*x^2 
+ 18*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3* 
c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*log((b*e*x + a*e)/(d*x + c))^ 
2 - 3*((6*A*B + 5*B^2)*b^3*c^2*d - 18*(2*A*B + 3*B^2)*a*b^2*c*d^2 + (30*A* 
B + 49*B^2)*a^2*b*d^3)*x + 6*((6*A*B + 11*B^2)*b^3*d^3*x^3 + 2*(3*A*B + B^ 
2)*b^3*c^3 - 9*(2*A*B + B^2)*a*b^2*c^2*d + 18*(A*B + B^2)*a^2*b*c*d^2 + 3* 
(2*B^2*b^3*c*d^2 + 3*(2*A*B + 3*B^2)*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2*d - 6 
*B^2*a*b^2*c*d^2 - 6*(A*B + B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c) 
))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3* 
(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3* 
(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a 
^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (384) = 768\).

Time = 12.47 (sec) , antiderivative size = 1544, normalized size of antiderivative = 3.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d**4 + 6*A*B*b*c*d**3 + 11*B**2*a*d* 
*4 + 11*B**2*b*c*d**3 - B*a**4*d**7*(6*A + 11*B)/(a*d - b*c)**3 + 4*B*a**3 
*b*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5*(6*A + 11* 
B)/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**4*(6*A + 11*B)/(a*d - b*c)**3 - B*b 
**4*c**4*d**3*(6*A + 11*B)/(a*d - b*c)**3)/(12*A*B*b*d**4 + 22*B**2*b*d**4 
))/(9*b*g**4*(a*d - b*c)**3) + B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d**4 + 
 6*A*B*b*c*d**3 + 11*B**2*a*d**4 + 11*B**2*b*c*d**3 + B*a**4*d**7*(6*A + 1 
1*B)/(a*d - b*c)**3 - 4*B*a**3*b*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 + 6*B* 
a**2*b**2*c**2*d**5*(6*A + 11*B)/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**4*(6* 
A + 11*B)/(a*d - b*c)**3 + B*b**4*c**4*d**3*(6*A + 11*B)/(a*d - b*c)**3)/( 
12*A*B*b*d**4 + 22*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + (3*B**2*a**2* 
c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**2*d + 3*B**2*a*b*d**3*x**2 + B 
**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(a + b*x)/(c + d*x))**2/(3*a**6 
*d**3*g**4 - 9*a**5*b*c*d**2*g**4 + 9*a**5*b*d**3*g**4*x + 9*a**4*b**2*c** 
2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*x + 9*a**4*b**2*d**3*g**4*x**2 - 3*a** 
3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x - 27*a**3*b**3*c*d**2*g**4*x 
**2 + 3*a**3*b**3*d**3*g**4*x**3 - 9*a**2*b**4*c**3*g**4*x + 27*a**2*b**4* 
c**2*d*g**4*x**2 - 9*a**2*b**4*c*d**2*g**4*x**3 - 9*a*b**5*c**3*g**4*x**2 
+ 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4*x**3) + (-6*A*B*a**2*d**2 + 
 12*A*B*a*b*c*d - 6*A*B*b**2*c**2 - 11*B**2*a**2*d**2 + 7*B**2*a*b*c*d ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1419 vs. \(2 (410) = 820\).

Time = 0.13 (sec) , antiderivative size = 1419, normalized size of antiderivative = 3.39 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="maxim 
a")
 

Output:

-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))*log(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^5*b^2*d^3*g^4)*x))*B^2 - 1/9*A*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^2 - 2*...
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.73 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(18*(B^2*b^2*e^4 - 3*(b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 3*(b*e*x 
+ a*e)^2*B^2*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + 
 a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^ 
3 + (b*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3) + 6*(6*A*B*b^2*e^4 + 2*B^2*b^ 
2*e^4 - 18*(b*e*x + a*e)*A*B*b*d*e^3/(d*x + c) - 9*(b*e*x + a*e)*B^2*b*d*e 
^3/(d*x + c) + 18*(b*e*x + a*e)^2*A*B*d^2*e^2/(d*x + c)^2 + 18*(b*e*x + a* 
e)^2*B^2*d^2*e^2/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^ 
3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b 
*e*x + a*e)^3*a^2*d^2*g^4/(d*x + c)^3) + (18*A^2*b^2*e^4 + 12*A*B*b^2*e^4 
+ 4*B^2*b^2*e^4 - 54*(b*e*x + a*e)*A^2*b*d*e^3/(d*x + c) - 54*(b*e*x + a*e 
)*A*B*b*d*e^3/(d*x + c) - 27*(b*e*x + a*e)*B^2*b*d*e^3/(d*x + c) + 54*(b*e 
*x + a*e)^2*A^2*d^2*e^2/(d*x + c)^2 + 108*(b*e*x + a*e)^2*A*B*d^2*e^2/(d*x 
 + c)^2 + 108*(b*e*x + a*e)^2*B^2*d^2*e^2/(d*x + c)^2)/((b*e*x + a*e)^3*b^ 
2*c^2*g^4/(d*x + c)^3 - 2*(b*e*x + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*e*x 
 + a*e)^3*a^2*d^2*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 = 28.29 (sec) , antiderivative size = 1064, normalized size of antiderivative = 2.55 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 85*B^2*a^2*d^2 + 4*B^2*b^2*c^2 + 66*A* 
B*a^2*d^2 + 12*A*B*b^2*c^2 - 36*A^2*a*b*c*d - 23*B^2*a*b*c*d - 42*A*B*a*b* 
c*d)/(6*(a*d - b*c)) + (x*(49*B^2*a*b*d^2 - 5*B^2*b^2*c*d + 30*A*B*a*b*d^2 
 - 6*A*B*b^2*c*d))/(2*(a*d - b*c)) + (d*x^2*(11*B^2*b^2*d + 6*A*B*b^2*d))/ 
(a*d - b*c))/(x*(27*a^2*b^3*c*g^4 - 27*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d* 
g^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*c*g^ 
4 - 9*a^4*b*d*g^4) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(3*b^2*g^4*(3*a^2 
*x + a^3/b + b^2*x^3 + 3*a*b*x^2)) - (B^2*d^3)/(3*b*g^4*(a^3*d^3 - b^3*c^3 
 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*((2*A* 
B)/(3*b^2*d*g^4) + (2*B^2*d^3*(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*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3 
*a^2*b*c*d^2)) - (2*B^2*d^3*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c 
))/(3*d^2)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) 
 + (2*B^2*d^3*x*(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*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c 
*d^2))))/((3*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/d) - (B*d^3* 
atan((B*d^3*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - a*b^3*c^2*d*g^4 - a^2*b^2*c*d^ 
2*g^4)/(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*d*g^4) + 2*b*d*x)*(6*A ...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1459, normalized size of antiderivative = 3.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(36*log(a + b*x)*a**5*b*d**3 + 108*log(a + b*x)*a**4*b**2*d**3*x + 54*log( 
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 + 162*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 + 162*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 + 54*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 - 54*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 - 162*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 - 162*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 - 54*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**3*b**3*c*d**2 + 54*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**3*d* 
*3*x - 54*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**4*c**2*d + 54*log((a*e + 
 b*e*x)/(c + d*x))**2*a**2*b**4*d**3*x**2 + 18*log((a*e + b*e*x)/(c + d*x) 
)**2*a*b**5*c**3 + 18*log((a*e + b*e*x)/(c + d*x))**2*a*b**5*d**3*x**3 - 3 
6*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*c*d**2 - 54*log((a*e + b*e*x)/(c + d*x))*a**4*b**2*d**3 - 10 
8*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*c**2*d + 96*log((a*e + b*e*x)/(c 
+ d*x))*a**3*b**3*c*d**2 - 54*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*d*...