\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^3} \, dx\) [103]

Optimal result

Integrand size = 32, antiderivative size = 268 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {2 B^2 d (c+d x)}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B^2 (c+d x)^2}{4 (b c-a d)^2 g^3 (a+b x)^2}+\frac {2 B d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 g^3 (a+b x)}-\frac {b B (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^2 g^3 (a+b x)^2}+\frac {d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g^3 (a+b x)}-\frac {b (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^2 g^3 (a+b x)^2} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.48 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.65 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^2}}{4 b g^3 (a+b x)^2} \] Input:

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^3,x]
 

Output:

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

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 [A] (verified)

Time = 1.20 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.81

method	result	size
norman	\(\frac {\frac {B d \left (2 A d a +2 B a d +B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B^{2} a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c +4 A B a d -2 A B b c +4 B^{2} a d -B^{2} b c \right ) x}{2 a g \left (d a -b c \right )}+\frac {B c \left (4 A d a -2 A b c +4 B a d -B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B^{2} c \left (2 d a -b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (2 A^{2} a d -2 A^{2} b c +6 A B a d -2 A B b c +7 B^{2} a d -B^{2} b c \right ) b \,x^{2}}{4 g \,a^{2} \left (d a -b c \right )}+\frac {b \,d^{2} B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B b \,d^{2} \left (2 A +3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}}{\left (b x +a \right )^{2} g^{2}}\)	\(485\)
parallelrisch	\(-\frac {-2 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} d^{3}-6 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{3}+2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} c^{2} d +6 B^{2} x a \,b^{4} d^{3}-6 B^{2} x \,b^{5} c \,d^{2}+2 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d -4 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{3}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{4} d^{3}-8 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{3}-4 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{2}-4 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{4} c \,d^{2}+4 A B x a \,b^{4} d^{3}-4 A B x \,b^{5} c \,d^{2}+4 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d -8 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c \,d^{2}+2 A^{2} a^{2} b^{3} d^{3}+2 A^{2} b^{5} c^{2} d +7 B^{2} a^{2} b^{3} d^{3}+B^{2} b^{5} c^{2} d -4 A^{2} a \,b^{4} c \,d^{2}+6 A B \,a^{2} b^{3} d^{3}+2 A B \,b^{5} c^{2} d -8 B^{2} a \,b^{4} c \,d^{2}-8 A B a \,b^{4} c \,d^{2}-8 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{3}-8 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c \,d^{2}}{4 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{4} d}\)	\(574\)
parts	\(-\frac {A^{2}}{2 g^{3} \left (b x +a \right )^{2} b}-\frac {B^{2} \left (d a -b c \right ) e \left (\frac {d^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {2}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}-\frac {d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} d^{2}}-\frac {2 B A \left (d a -b c \right ) e \left (\frac {d^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{3}}-\frac {d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{3}}\right )}{g^{3} d^{2}}\)	\(609\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b e}{2 \left (d a -b c \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{3} A^{2}}{\left (d a -b c \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{2} A B 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 )^{3} g^{3}}+\frac {2 d^{3} 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 )}{\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 )^{3} g^{3}}-\frac {d^{2} B^{2} 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 )^{3} g^{3}}+\frac {d^{3} 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}}{\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 )^{3} g^{3}}\right )}{d^{2}}\)	\(689\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b e}{2 \left (d a -b c \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{3} A^{2}}{\left (d a -b c \right )^{3} g^{3} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{2} A B 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 )^{3} g^{3}}+\frac {2 d^{3} 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 )}{\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 )^{3} g^{3}}-\frac {d^{2} B^{2} 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 )^{3} g^{3}}+\frac {d^{3} 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}}{\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 )^{3} g^{3}}\right )}{d^{2}}\)	\(689\)
orering	\(-\frac {\left (b x +a \right ) \left (90 b^{2} d^{3} x^{3}+122 a b \,d^{3} x^{2}+148 b^{2} c \,d^{2} x^{2}+13 a^{2} d^{3} x +218 a b c \,d^{2} x +39 b^{2} c^{2} d x +13 a^{2} d^{2} c +96 a b \,c^{2} d -19 b^{2} c^{3}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{8 \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 g x +a g \right )^{3}}-\frac {\left (b x +a \right )^{2} \left (d x +c \right ) \left (54 b^{2} d^{2} x^{2}+70 a b \,d^{2} x +38 b^{2} c d x +7 a^{2} d^{2}+56 a c d b -9 c^{2} b^{2}\right ) \left (\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right )}{\left (b g x +a g \right )^{3} e \left (b x +a \right )}-\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2} b g}{\left (b g x +a g \right )^{4}}\right )}{8 b \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 b d x +7 d a -b c \right ) \left (d x +c \right )^{2} \left (b x +a \right )^{3} \left (\frac {2 B^{2} \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right )^{2} \left (d x +c \right )^{2}}{e^{2} \left (b x +a \right )^{2} \left (b g x +a g \right )^{3}}-\frac {12 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right ) b g}{\left (b g x +a g \right )^{4} e \left (b x +a \right )}+\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (-\frac {2 e b d}{\left (d x +c \right )^{2}}+\frac {2 e \left (b x +a \right ) d^{2}}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )}{\left (b g x +a g \right )^{3} e \left (b x +a \right )}-\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) \left (d x +c \right ) b}{\left (b g x +a g \right )^{3} e \left (b x +a \right )^{2}}+\frac {2 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right ) B \left (\frac {e b}{d x +c}-\frac {e \left (b x +a \right ) d}{\left (d x +c \right )^{2}}\right ) d}{\left (b g x +a g \right )^{3} e \left (b x +a \right )}+\frac {12 \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2} b^{2} g^{2}}{\left (b g x +a g \right )^{5}}\right )}{8 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(851\)
risch	\(\text {Expression too large to display}\)	\(1504\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)
 

Output:

(B/g*d*(2*A*a*d+2*B*a*d+B*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/ 
(d*x+c))+B^2*a*d^2/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*ln(e*(b*x+a)/(d*x+c))^2 
+1/2*(2*A^2*a*d-2*A^2*b*c+4*A*B*a*d-2*A*B*b*c+4*B^2*a*d-B^2*b*c)/a/g/(a*d- 
b*c)*x+1/2*B*c*(4*A*a*d-2*A*b*c+4*B*a*d-B*b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^ 
2)*ln(e*(b*x+a)/(d*x+c))+1/2*B^2*c*(2*a*d-b*c)/g/(a^2*d^2-2*a*b*c*d+b^2*c^ 
2)*ln(e*(b*x+a)/(d*x+c))^2+1/4*(2*A^2*a*d-2*A^2*b*c+6*A*B*a*d-2*A*B*b*c+7* 
B^2*a*d-B^2*b*c)/g/a^2*b/(a*d-b*c)*x^2+1/2*b*d^2*B^2/g/(a^2*d^2-2*a*b*c*d+ 
b^2*c^2)*x^2*ln(e*(b*x+a)/(d*x+c))^2+1/2*B*b/g*d^2*(2*A+3*B)/(a^2*d^2-2*a* 
b*c*d+b^2*c^2)*x^2*ln(e*(b*x+a)/(d*x+c)))/(b*x+a)^2/g^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.37 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 4 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b c d + {\left (2 \, A^{2} + 6 \, A B + 7 \, B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x - B^{2} b^{2} c^{2} + 2 \, B^{2} a b c d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B + 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A B + 3 \, B^{2}\right )} b^{2} d^{2} x^{2} - {\left (2 \, A B + B^{2}\right )} b^{2} c^{2} + 4 \, {\left (A B + B^{2}\right )} a b c d + 2 \, {\left (B^{2} b^{2} c d + 2 \, {\left (A B + B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algorithm="frica 
s")
 

Output:

-1/4*((2*A^2 + 2*A*B + B^2)*b^2*c^2 - 4*(A^2 + 2*A*B + 2*B^2)*a*b*c*d + (2 
*A^2 + 6*A*B + 7*B^2)*a^2*d^2 - 2*(B^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2*x - B^2 
*b^2*c^2 + 2*B^2*a*b*c*d)*log((b*e*x + a*e)/(d*x + c))^2 - 2*((2*A*B + 3*B 
^2)*b^2*c*d - (2*A*B + 3*B^2)*a*b*d^2)*x - 2*((2*A*B + 3*B^2)*b^2*d^2*x^2 
- (2*A*B + B^2)*b^2*c^2 + 4*(A*B + B^2)*a*b*c*d + 2*(B^2*b^2*c*d + 2*(A*B 
+ B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^2 - 2*a*b^4*c*d + 
 a^2*b^3*d^2)*g^3*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*g^3*x 
+ (a^2*b^3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*g^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 894 vs. \(2 (241) = 482\).

Time = 2.74 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.34 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx =\text {Too large to display} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**3,x)
 

Output:

-B*d**2*(2*A + 3*B)*log(x + (2*A*B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 
 + 3*B**2*b*c*d**2 - B*a**3*d**5*(2*A + 3*B)/(a*d - b*c)**2 + 3*B*a**2*b*c 
*d**4*(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**3*(2*A + 3*B)/(a*d - 
 b*c)**2 + B*b**3*c**3*d**2*(2*A + 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6* 
B**2*b*d**3))/(2*b*g**3*(a*d - b*c)**2) + B*d**2*(2*A + 3*B)*log(x + (2*A* 
B*a*d**3 + 2*A*B*b*c*d**2 + 3*B**2*a*d**3 + 3*B**2*b*c*d**2 + B*a**3*d**5* 
(2*A + 3*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**4*(2*A + 3*B)/(a*d - b*c)**2 
+ 3*B*a*b**2*c**2*d**3*(2*A + 3*B)/(a*d - b*c)**2 - B*b**3*c**3*d**2*(2*A 
+ 3*B)/(a*d - b*c)**2)/(4*A*B*b*d**3 + 6*B**2*b*d**3))/(2*b*g**3*(a*d - b* 
c)**2) + (2*B**2*a*c*d + 2*B**2*a*d**2*x - B**2*b*c**2 + B**2*b*d**2*x**2) 
*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**2*g**3 - 4*a**3*b*c*d*g**3 + 4*a 
**3*b*d**2*g**3*x + 2*a**2*b**2*c**2*g**3 - 8*a**2*b**2*c*d*g**3*x + 2*a** 
2*b**2*d**2*g**3*x**2 + 4*a*b**3*c**2*g**3*x - 4*a*b**3*c*d*g**3*x**2 + 2* 
b**4*c**2*g**3*x**2) + (-2*A*B*a*d + 2*A*B*b*c - 3*B**2*a*d + B**2*b*c - 2 
*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**3*b*d*g**3 - 2*a**2*b**2*c*g 
**3 + 4*a**2*b**2*d*g**3*x - 4*a*b**3*c*g**3*x + 2*a*b**3*d*g**3*x**2 - 2* 
b**4*c*g**3*x**2) + (-2*A**2*a*d + 2*A**2*b*c - 6*A*B*a*d + 2*A*B*b*c - 7* 
B**2*a*d + B**2*b*c + x*(-4*A*B*b*d - 6*B**2*b*d))/(4*a**3*b*d*g**3 - 4*a* 
*2*b**2*c*g**3 + x**2*(4*a*b**3*d*g**3 - 4*b**4*c*g**3) + x*(8*a**2*b**2*d 
*g**3 - 8*a*b**3*c*g**3))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (262) = 524\).

Time = 0.08 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algorithm="maxim 
a")
 

Output:

1/4*(2*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - 
a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) + 2*d^2*log(b*x + a)/((b^3*c 
^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^ 
2*c*d + a^2*b*d^2)*g^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (b^2*c^2 - 
 8*a*b*c*d + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + 
 a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x + c)^2 - 6*(b^2*c* 
d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2* 
(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + 
a^2*d^2)*log(b*x + a))*log(d*x + c))/(a^2*b^3*c^2*g^3 - 2*a^3*b^2*c*d*g^3 
+ a^4*b*d^2*g^3 + (b^5*c^2*g^3 - 2*a*b^4*c*d*g^3 + a^2*b^3*d^2*g^3)*x^2 + 
2*(a*b^4*c^2*g^3 - 2*a^2*b^3*c*d*g^3 + a^3*b^2*d^2*g^3)*x))*B^2 + 1/2*A*B* 
((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2 
*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x 
+ c))/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a)/((b^3 
*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a* 
b^2*c*d + a^2*b*d^2)*g^3)) - 1/2*B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^ 
2/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) - 1/2*A^2/(b^3*g^3*x^2 + 2*a*b 
^2*g^3*x + a^2*b*g^3)
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (B^{2} b e^{3} - \frac {2 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, {\left (2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {4 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {2 \, A^{2} b e^{3} + 2 \, A B b e^{3} + B^{2} b e^{3} - \frac {4 \, {\left (b e x + a e\right )} A^{2} d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} A B d e^{2}}{d x + c} - \frac {8 \, {\left (b e x + a e\right )} B^{2} d e^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b e x + a e\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x, algorithm="giac" 
)
 

Output:

-1/4*(2*(B^2*b*e^3 - 2*(b*e*x + a*e)*B^2*d*e^2/(d*x + c))*log((b*e*x + a*e 
)/(d*x + c))^2/((b*e*x + a*e)^2*b*c*g^3/(d*x + c)^2 - (b*e*x + a*e)^2*a*d* 
g^3/(d*x + c)^2) + 2*(2*A*B*b*e^3 + B^2*b*e^3 - 4*(b*e*x + a*e)*A*B*d*e^2/ 
(d*x + c) - 4*(b*e*x + a*e)*B^2*d*e^2/(d*x + c))*log((b*e*x + a*e)/(d*x + 
c))/((b*e*x + a*e)^2*b*c*g^3/(d*x + c)^2 - (b*e*x + a*e)^2*a*d*g^3/(d*x + 
c)^2) + (2*A^2*b*e^3 + 2*A*B*b*e^3 + B^2*b*e^3 - 4*(b*e*x + a*e)*A^2*d*e^2 
/(d*x + c) - 8*(b*e*x + a*e)*A*B*d*e^2/(d*x + c) - 8*(b*e*x + a*e)*B^2*d*e 
^2/(d*x + c))/((b*e*x + a*e)^2*b*c*g^3/(d*x + c)^2 - (b*e*x + a*e)^2*a*d*g 
^3/(d*x + c)^2))*(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 = 27.06 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+7\,B^2\,a\,d-B^2\,b\,c+6\,A\,B\,a\,d-2\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (3\,b\,d\,B^2+2\,A\,b\,d\,B\right )}{a\,d-b\,c}}{2\,a^2\,b\,g^3+4\,a\,b^2\,g^3\,x+2\,b^3\,g^3\,x^2}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {B^2\,d^2}{2\,b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {A\,B}{b^2\,d\,g^3}+\frac {B^2\,x\,\left (a\,d-b\,c\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,d^2\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{b\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{\frac {b\,x^2}{d}+\frac {a^2}{b\,d}+\frac {2\,a\,x}{d}}-\frac {B\,d^2\,\mathrm {atan}\left (\frac {B\,d^2\,\left (2\,b\,d\,x-\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,\left (a\,d-b\,c\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,B^2\,d^2+2\,A\,B\,d^2\right )}\right )\,\left (2\,A+3\,B\right )\,1{}\mathrm {i}}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))^2/(a*g + b*g*x)^3,x)
 

Output:

- ((2*A^2*a*d - 2*A^2*b*c + 7*B^2*a*d - B^2*b*c + 6*A*B*a*d - 2*A*B*b*c)/( 
2*(a*d - b*c)) + (x*(3*B^2*b*d + 2*A*B*b*d))/(a*d - b*c))/(2*a^2*b*g^3 + 2 
*b^3*g^3*x^2 + 4*a*b^2*g^3*x) - log((e*(a + b*x))/(c + d*x))^2*(B^2/(2*b^2 
*g^3*(2*a*x + b*x^2 + a^2/b)) - (B^2*d^2)/(2*b*g^3*(a^2*d^2 + b^2*c^2 - 2* 
a*b*c*d))) - (log((e*(a + b*x))/(c + d*x))*((A*B)/(b^2*d*g^3) + (B^2*x*(a* 
d - b*c))/(b*g^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*d^2*((2*a^2*d^2 + 
 b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(b*g^3*(a^2* 
d^2 + b^2*c^2 - 2*a*b*c*d))))/((b*x^2)/d + a^2/(b*d) + (2*a*x)/d) - (B*d^2 
*atan((B*d^2*(2*b*d*x - (b^3*c^2*g^3 - a^2*b*d^2*g^3)/(b*g^3*(a*d - b*c))) 
*(2*A + 3*B)*1i)/((a*d - b*c)*(3*B^2*d^2 + 2*A*B*d^2)))*(2*A + 3*B)*1i)/(b 
*g^3*(a*d - b*c)^2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 882, normalized size of antiderivative = 3.29 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^3,x)
 

Output:

(4*log(a + b*x)*a**4*b*d**2 + 8*log(a + b*x)*a**3*b**2*d**2*x + 4*log(a + 
b*x)*a**3*b**2*d**2 + 2*log(a + b*x)*a**2*b**3*c*d + 4*log(a + b*x)*a**2*b 
**3*d**2*x**2 + 8*log(a + b*x)*a**2*b**3*d**2*x + 4*log(a + b*x)*a*b**4*c* 
d*x + 4*log(a + b*x)*a*b**4*d**2*x**2 + 2*log(a + b*x)*b**5*c*d*x**2 - 4*l 
og(c + d*x)*a**4*b*d**2 - 8*log(c + d*x)*a**3*b**2*d**2*x - 4*log(c + d*x) 
*a**3*b**2*d**2 - 2*log(c + d*x)*a**2*b**3*c*d - 4*log(c + d*x)*a**2*b**3* 
d**2*x**2 - 8*log(c + d*x)*a**2*b**3*d**2*x - 4*log(c + d*x)*a*b**4*c*d*x 
- 4*log(c + d*x)*a*b**4*d**2*x**2 - 2*log(c + d*x)*b**5*c*d*x**2 + 4*log(( 
a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c*d + 4*log((a*e + b*e*x)/(c + d*x))* 
*2*a**2*b**3*d**2*x - 2*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*c**2 + 2*lo 
g((a*e + b*e*x)/(c + d*x))**2*a*b**4*d**2*x**2 - 4*log((a*e + b*e*x)/(c + 
d*x))*a**4*b*d**2 + 8*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d - 4*log(( 
a*e + b*e*x)/(c + d*x))*a**3*b**2*d**2 - 4*log((a*e + b*e*x)/(c + d*x))*a* 
*2*b**3*c**2 + 6*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c*d - 2*log((a*e + 
 b*e*x)/(c + d*x))*a*b**4*c**2 + 2*log((a*e + b*e*x)/(c + d*x))*a*b**4*d** 
2*x**2 - 2*log((a*e + b*e*x)/(c + d*x))*b**5*c*d*x**2 - 2*a**5*d**2 + 4*a* 
*4*b*c*d - 4*a**4*b*d**2 - 2*a**3*b**2*c**2 + 6*a**3*b**2*c*d - 4*a**3*b** 
2*d**2 - 2*a**2*b**3*c**2 + 5*a**2*b**3*c*d + 2*a**2*b**3*d**2*x**2 - a*b* 
*4*c**2 - 2*a*b**4*c*d*x**2 + 3*a*b**4*d**2*x**2 - 3*b**5*c*d*x**2)/(4*a*b 
*g**3*(a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x + a**2*b**2*c**2 - 4*...