\(\int \frac {(A+B \log (\frac {e (c+d x)}{a+b x}))^2}{(a g+b g x)^4} \, dx\) [189]

Optimal result

Integrand size = 32, antiderivative size = 399 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}+\frac {B^2 d^3 \log ^2\left (\frac {c+d x}{a+b x}\right )}{3 b (b c-a d)^3 g^4}+\frac {2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}-\frac {b B d (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}+\frac {2 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^3 \log \left (\frac {c+d x}{a+b x}\right ) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3} \] 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+1/3*B 
^2*d^3*ln((d*x+c)/(b*x+a))^2/b/(-a*d+b*c)^3/g^4+2*B*d^2*(d*x+c)*(A+B*ln(e* 
(d*x+c)/(b*x+a)))/(-a*d+b*c)^3/g^4/(b*x+a)-b*B*d*(d*x+c)^2*(A+B*ln(e*(d*x+ 
c)/(b*x+a)))/(-a*d+b*c)^3/g^4/(b*x+a)^2+2/9*b^2*B*(d*x+c)^3*(A+B*ln(e*(d*x 
+c)/(b*x+a)))/(-a*d+b*c)^3/g^4/(b*x+a)^3-2/3*B*d^3*ln((d*x+c)/(b*x+a))*(A+ 
B*ln(e*(d*x+c)/(b*x+a)))/b/(-a*d+b*c)^3/g^4-1/3*(A+B*ln(e*(d*x+c)/(b*x+a)) 
)^2/b/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 = 0.61 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.46 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {-18 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B \left (12 A (b c-a d)^3-4 B (b c-a d)^3-18 A d (b c-a d)^2 (a+b x)+15 B d (b c-a d)^2 (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (-b c+a d) (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)-66 B d^3 (a+b x)^3 \log (a+b x)+18 B d^3 (a+b x)^3 \log ^2(a+b x)-36 A d^3 (a+b x)^3 \log (c+d x)+66 B d^3 (a+b x)^3 \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+18 B d^3 (a+b x)^3 \log ^2(c+d x)-36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+12 B (b c-a d)^3 \log \left (\frac {e (c+d x)}{a+b x}\right )-18 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (c+d x)}{a+b x}\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )-36 B d^3 (a+b x)^3 \log (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )-36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{54 b g^4 (a+b x)^3} \] Input:

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

Output:

(-18*(A + B*Log[(e*(c + d*x))/(a + b*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 - 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] + 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)] + 12*B* 
(b*c - a*d)^3*Log[(e*(c + d*x))/(a + b*x)] - 18*B*d*(b*c - a*d)^2*(a + b*x 
)*Log[(e*(c + d*x))/(a + b*x)] + 36*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*( 
c + d*x))/(a + b*x)] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(e*(c + d*x)) 
/(a + b*x)] - 36*B*d^3*(a + b*x)^3*Log[c + d*x]*Log[(e*(c + d*x))/(a + b*x 
)] - 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)/(54 
*b*g^4*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.79, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2952, 2756, 2772, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((-1/3*((d - (b*(c + d*x))/(a + b*x))^3*(A + B*Log[(e*(c + d*x))/(a + b*x 
)])^2)/b + (2*B*(-(B*((-3*b*d^2*(c + d*x))/(a + b*x) + (3*b^2*d*(c + d*x)^ 
2)/(4*(a + b*x)^2) - (b^3*(c + d*x)^3)/(9*(a + b*x)^3) + (d^3*Log[(c + d*x 
)/(a + b*x)]^2)/2)) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x 
)]))/(a + b*x) + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]) 
)/(2*(a + b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)])) 
/(3*(a + b*x)^3) + d^3*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x))/( 
a + b*x)])))/(3*b))/((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 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

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 2952

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/d)^m   Subst[Int[(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] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(835\) vs. \(2(387)=774\).

Time = 1.48 (sec) , antiderivative size = 836, normalized size of antiderivative = 2.10

method	result	size
parts	\(-\frac {A^{2}}{3 g^{4} \left (b x +a \right )^{3} b}+\frac {B^{2} b^{2} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{3}-\frac {2 \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{9}+\frac {2 \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{3}}{27}-\frac {2 d e \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{2}+\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{b}+\frac {d^{2} e^{2} \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}-2 \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {2 e \left (d a -b c \right )}{b \left (b x +a \right )}+\frac {2 d e}{b}\right )}{b^{2}}\right )}{g^{4} e^{3} \left (d a -b c \right )^{3}}+\frac {2 B A \,b^{2} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{3}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{3}}{9}-\frac {2 d e \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{b}+\frac {d^{2} e^{2} \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{b^{2}}\right )}{g^{4} e^{3} \left (d a -b c \right )^{3}}\)	\(836\)
norman	\(\frac {\frac {B^{2} a^{2} d^{3} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) g}+\frac {B^{2} a b \,d^{3} x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\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 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\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 (d x +c \right )}{b x +a}\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 (d x +c \right )}{b x +a}\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 (d x +c \right )}{b x +a}\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 (d x +c \right )}{b x +a}\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 (d x +c \right )}{b x +a}\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 (d x +c \right )}{b x +a}\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 )}}{g^{3} \left (b x +a \right )^{3}}\)	\(928\)
parallelrisch	\(-\frac {147 B^{2} x \,a^{2} b^{5} d^{4}+15 B^{2} x \,b^{7} c^{2} d^{2}+66 B^{2} x^{2} a \,b^{6} d^{4}-66 B^{2} x^{2} b^{7} c \,d^{3}-54 A^{2} a^{2} b^{5} c \,d^{3}+54 A^{2} a \,b^{6} c^{2} d^{2}+108 B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{6} c \,d^{3}-108 A B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{5} c \,d^{3}+108 A B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) 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}-18 B^{2} x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} b^{7} d^{4}+66 B^{2} x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} d^{4}-18 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} b^{7} c^{3} d +12 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} c^{3} d +108 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{5} c \,d^{3}-54 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{6} c^{2} d^{2}-36 A B \,x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} d^{4}-54 B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} a \,b^{6} d^{4}+162 B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{6} d^{4}+36 B^{2} x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} c \,d^{3}-54 B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} a^{2} b^{5} d^{4}+108 B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{5} d^{4}-18 B^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} c^{2} d^{2}-54 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{2} a^{2} b^{5} c \,d^{3}+54 B^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )^{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 \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{7} c^{3} d -36 A B \,x^{2} a \,b^{6} d^{4}+36 A B \,x^{2} b^{7} c \,d^{3}-90 A B x \,a^{2} b^{5} d^{4}-18 A B x \,b^{7} c^{2} d^{2}-162 B^{2} x a \,b^{6} c \,d^{3}+108 A B x a \,b^{6} c \,d^{3}-108 A B \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{6} d^{4}-108 A B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{2} b^{5} d^{4}}{54 g^{4} \left (b x +a \right )^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{5} d}\)	\(988\)
orering	\(\text {Expression too large to display}\)	\(1049\)
derivativedivides	\(\text {Expression too large to display}\)	\(1061\)
default	\(\text {Expression too large to display}\)	\(1061\)
risch	\(\text {Expression too large to display}\)	\(1819\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.70 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b 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 {d e x + c e}{b x + a}\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 {d e x + c e}{b x + a}\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*(d*x+c)/(b*x+a)))^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((d*e*x + c*e)/(b*x + a))^ 
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((d*e*x + c*e)/(b*x + a) 
))/((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. \(2 (362) = 724\).

Time = 12.43 (sec) , antiderivative size = 1544, normalized size of antiderivative = 3.87 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*ln(e*(d*x+c)/(b*x+a)))**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 - 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)/(1 
2*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*(c + d*x)/(a + b*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. 1420 vs. \(2 (387) = 774\).

Time = 0.13 (sec) , antiderivative size = 1420, normalized size of antiderivative = 3.56 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(d*x+c)/(b*x+a)))^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(d*e*x/(b*x + a) + c*e/(b*x + a)) - (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*a...
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.79 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(18*((d*e*x + c*e)^3*B^2*b^2/((b^2*c^2*e^2*g^4 - 2*a*b*c*d*e^2*g^4 + 
 a^2*d^2*e^2*g^4)*(b*x + a)^3) - 3*(d*e*x + c*e)^2*B^2*b*d/((b^2*c^2*e*g^4 
 - 2*a*b*c*d*e*g^4 + a^2*d^2*e*g^4)*(b*x + a)^2) + 3*(d*e*x + c*e)*B^2*d^2 
/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(b*x + a)))*log((d*e*x + c*e 
)/(b*x + a))^2 + 6*(2*(3*A*B*b^2 - B^2*b^2)*(d*e*x + c*e)^3/((b^2*c^2*e^2* 
g^4 - 2*a*b*c*d*e^2*g^4 + a^2*d^2*e^2*g^4)*(b*x + a)^3) - 9*(2*A*B*b*d - B 
^2*b*d)*(d*e*x + c*e)^2/((b^2*c^2*e*g^4 - 2*a*b*c*d*e*g^4 + a^2*d^2*e*g^4) 
*(b*x + a)^2) + 18*(A*B*d^2 - B^2*d^2)*(d*e*x + c*e)/((b^2*c^2*g^4 - 2*a*b 
*c*d*g^4 + a^2*d^2*g^4)*(b*x + a)))*log((d*e*x + c*e)/(b*x + a)) + 2*(9*A^ 
2*b^2 - 6*A*B*b^2 + 2*B^2*b^2)*(d*e*x + c*e)^3/((b^2*c^2*e^2*g^4 - 2*a*b*c 
*d*e^2*g^4 + a^2*d^2*e^2*g^4)*(b*x + a)^3) - 27*(2*A^2*b*d - 2*A*B*b*d + B 
^2*b*d)*(d*e*x + c*e)^2/((b^2*c^2*e*g^4 - 2*a*b*c*d*e*g^4 + a^2*d^2*e*g^4) 
*(b*x + a)^2) + 54*(A^2*d^2 - 2*A*B*d^2 + 2*B^2*d^2)*(d*e*x + c*e)/((b^2*c 
^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(b*x + a)))*(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 = 29.56 (sec) , antiderivative size = 1064, normalized size of antiderivative = 2.67 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((A + B*log((e*(c + d*x))/(a + b*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*(c + d*x))/(a + b*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*(c + d*x))/(a + b*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 = 0.17 (sec) , antiderivative size = 1459, normalized size of antiderivative = 3.66 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(d*x+c)/(b*x+a)))^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*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 + 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((c*e + d*e*x)/(a 
+ b*x))**2*a**3*b**3*c*d**2 + 54*log((c*e + d*e*x)/(a + b*x))**2*a**3*b**3 
*d**3*x - 54*log((c*e + d*e*x)/(a + b*x))**2*a**2*b**4*c**2*d + 54*log((c* 
e + d*e*x)/(a + b*x))**2*a**2*b**4*d**3*x**2 + 18*log((c*e + d*e*x)/(a + b 
*x))**2*a*b**5*c**3 + 18*log((c*e + d*e*x)/(a + b*x))**2*a*b**5*d**3*x**3 
- 36*log((c*e + d*e*x)/(a + b*x))*a**5*b*d**3 + 108*log((c*e + d*e*x)/(a + 
 b*x))*a**4*b**2*c*d**2 + 54*log((c*e + d*e*x)/(a + b*x))*a**4*b**2*d**3 - 
 108*log((c*e + d*e*x)/(a + b*x))*a**3*b**3*c**2*d - 96*log((c*e + d*e*x)/ 
(a + b*x))*a**3*b**3*c*d**2 + 54*log((c*e + d*e*x)/(a + b*x))*a**3*b**3...