\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) [97]

Optimal result

Integrand size = 42, antiderivative size = 365 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {2 A B d^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}+\frac {2 B^2 d^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B^2 (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 B^2 d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g^2 i^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.84 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {-3 \left (A^2-2 A B+2 B^2\right ) d (-b c+a d) (a+b x)+3 b \left (A^2+2 A B+2 B^2\right ) (b c-a d) (c+d x)+6 b \left (A^2+2 B^2\right ) d (a+b x) (c+d x) \log (a+b x)+6 B (b c-a d) (A b c+b B c+a A d-a B d+2 A b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B \left (-a^2 B d^2+2 a b d (-B d x+A (c+d x))+b^2 (2 A d x (c+d x)+B c (c+2 d x))\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+2 b B^2 d (a+b x) (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b \left (A^2+2 B^2\right ) d (a+b x) (c+d x) \log (c+d x)}{3 (b c-a d)^3 g^2 i^2 (a+b x) (c+d x)} \] Input:

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

Output:

-1/3*(-3*(A^2 - 2*A*B + 2*B^2)*d*(-(b*c) + a*d)*(a + b*x) + 3*b*(A^2 + 2*A 
*B + 2*B^2)*(b*c - a*d)*(c + d*x) + 6*b*(A^2 + 2*B^2)*d*(a + b*x)*(c + d*x 
)*Log[a + b*x] + 6*B*(b*c - a*d)*(A*b*c + b*B*c + a*A*d - a*B*d + 2*A*b*d* 
x)*Log[(e*(a + b*x))/(c + d*x)] + 3*B*(-(a^2*B*d^2) + 2*a*b*d*(-(B*d*x) + 
A*(c + d*x)) + b^2*(2*A*d*x*(c + d*x) + B*c*(c + 2*d*x)))*Log[(e*(a + b*x) 
)/(c + d*x)]^2 + 2*b*B^2*d*(a + b*x)*(c + d*x)*Log[(e*(a + b*x))/(c + d*x) 
]^3 - 6*b*(A^2 + 2*B^2)*d*(a + b*x)*(c + d*x)*Log[c + d*x])/((b*c - a*d)^3 
*g^2*i^2*(a + b*x)*(c + d*x))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 2795, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2795

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

rule 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(363)=726\).

Time = 2.00 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.10

method	result	size
parts	\(\frac {A^{2} \left (-\frac {d}{\left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 d b \ln \left (d x +c \right )}{\left (d a -b c \right )^{3}}-\frac {b}{\left (d a -b c \right )^{2} \left (b x +a \right )}+\frac {2 d b \ln \left (b x +a \right )}{\left (d a -b c \right )^{3}}\right )}{g^{2} i^{2}}-\frac {B^{2} \left (\frac {d^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \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 ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (d a -b c \right )^{2}}-\frac {2 b d e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (d a -b c \right )^{2}}+\frac {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}}{\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 )^{2}}\right )}{g^{2} i^{2} \left (d a -b c \right ) e}-\frac {2 A B \left (\frac {d^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (d a -b c \right )^{2}}-\frac {b d e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{\left (d a -b c \right )^{2}}+\frac {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 )}{\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 )^{2}}\right )}{g^{2} i^{2} \left (d a -b c \right ) e}\)	\(766\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} b^{2}}{i^{2} \left (d a -b c \right )^{4} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{3} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {2 d^{2} A B \,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 )}{\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 )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {2 d^{3} A B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {2 d^{4} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{2} B^{2} 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 )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {2 d^{3} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \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 ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}\right )}{d^{2}}\)	\(872\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} b^{2}}{i^{2} \left (d a -b c \right )^{4} g^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}-\frac {2 d^{3} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {2 d^{2} A B \,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 )}{\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 )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {2 d^{3} A B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {2 d^{4} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{2} B^{2} 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 )}{i^{2} \left (d a -b c \right )^{4} g^{2}}-\frac {2 d^{3} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (d a -b c \right )^{4} g^{2}}+\frac {d^{4} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \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 ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{4} g^{2}}\right )}{d^{2}}\)	\(872\)
norman	\(\frac {\frac {\left (2 A^{2} a b c d -2 A B \,a^{2} d^{2}+2 A B \,b^{2} c^{2}+2 B^{2} a^{2} d^{2}+2 B^{2} b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \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 \left (2 A a b c d -B \,a^{2} d^{2}+B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \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 (2 A^{2} a b \,d^{2}+2 A^{2} b^{2} c d -4 A B a b \,d^{2}+4 A B \,b^{2} c d +4 B^{2} a b \,d^{2}+4 B^{2} b^{2} c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \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 (2 A^{2} b d +4 B^{2} b d \right ) b d \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (d a -b c \right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {A^{2} a b \,d^{2}+A^{2} b^{2} c d -2 A B a b \,d^{2}+2 A B \,b^{2} c d +2 B^{2} a b \,d^{2}+2 B^{2} b^{2} c d}{i g \left (d a -b c \right )^{2} d b}-\frac {\left (2 A^{2} b^{2} d^{2}+4 B^{2} b^{2} d^{2}\right ) x}{b d g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {2 b^{2} B^{2} d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 i 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 {2 A B \,b^{2} d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \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 {2 B^{2} a b c d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \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 {2 b d B \left (A d a +A b c -B a d +B b c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \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 {2 \left (d a +b c \right ) B^{2} b d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \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 \left (b x +a \right ) i \left (d x +c \right )}\)	\(940\)
parallelrisch	\(\text {Expression too large to display}\)	\(1161\)
risch	\(\text {Expression too large to display}\)	\(1282\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.41 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {12 \, A B a b c d - 3 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} + 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + B^{2} a b c d + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} - 3 \, {\left (2 \, A B b^{2} d^{2} x^{2} + B^{2} b^{2} c^{2} + 2 \, A B a b c d - B^{2} a^{2} d^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} b^{2} c d + {\left (A B - B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 6 \, {\left ({\left (A^{2} + 2 \, B^{2}\right )} b^{2} c d - {\left (A^{2} + 2 \, B^{2}\right )} a b d^{2}\right )} x - 6 \, {\left ({\left (A^{2} + 2 \, B^{2}\right )} b^{2} d^{2} x^{2} + A^{2} a b c d + {\left (A B + B^{2}\right )} b^{2} c^{2} - {\left (A B - B^{2}\right )} a^{2} d^{2} + {\left ({\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{2} c d + {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}\right )}} \] Input:

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

Output:

1/3*(12*A*B*a*b*c*d - 3*(A^2 + 2*A*B + 2*B^2)*b^2*c^2 + 3*(A^2 - 2*A*B + 2 
*B^2)*a^2*d^2 - 2*(B^2*b^2*d^2*x^2 + B^2*a*b*c*d + (B^2*b^2*c*d + B^2*a*b* 
d^2)*x)*log((b*e*x + a*e)/(d*x + c))^3 - 3*(2*A*B*b^2*d^2*x^2 + B^2*b^2*c^ 
2 + 2*A*B*a*b*c*d - B^2*a^2*d^2 + 2*((A*B + B^2)*b^2*c*d + (A*B - B^2)*a*b 
*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 6*((A^2 + 2*B^2)*b^2*c*d - (A^2 
+ 2*B^2)*a*b*d^2)*x - 6*((A^2 + 2*B^2)*b^2*d^2*x^2 + A^2*a*b*c*d + (A*B + 
B^2)*b^2*c^2 - (A*B - B^2)*a^2*d^2 + ((A^2 + 2*A*B + 2*B^2)*b^2*c*d + (A^2 
 - 2*A*B + 2*B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^4*c^3*d - 
3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*g^2*i^2*x^2 + (b^4*c^4 - 2* 
a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*g^2*i^2*x + (a*b^3*c^4 - 3*a^2*b^2* 
c^3*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*g^2*i^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1404 vs. \(2 (335) = 670\).

Time = 3.06 (sec) , antiderivative size = 1404, normalized size of antiderivative = 3.85 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*B**2*b*d*log(e*(a + b*x)/(c + d*x))**3/(3*a**3*d**3*g**2*i**2 - 9*a**2*b 
*c*d**2*g**2*i**2 + 9*a*b**2*c**2*d*g**2*i**2 - 3*b**3*c**3*g**2*i**2) - 2 
*b*d*(A**2 + 2*B**2)*log(x + (2*A**2*a*b*d**2 + 2*A**2*b**2*c*d + 4*B**2*a 
*b*d**2 + 4*B**2*b**2*c*d - 2*a**4*b*d**5*(A**2 + 2*B**2)/(a*d - b*c)**3 + 
 8*a**3*b**2*c*d**4*(A**2 + 2*B**2)/(a*d - b*c)**3 - 12*a**2*b**3*c**2*d** 
3*(A**2 + 2*B**2)/(a*d - b*c)**3 + 8*a*b**4*c**3*d**2*(A**2 + 2*B**2)/(a*d 
 - b*c)**3 - 2*b**5*c**4*d*(A**2 + 2*B**2)/(a*d - b*c)**3)/(4*A**2*b**2*d* 
*2 + 8*B**2*b**2*d**2))/(g**2*i**2*(a*d - b*c)**3) + 2*b*d*(A**2 + 2*B**2) 
*log(x + (2*A**2*a*b*d**2 + 2*A**2*b**2*c*d + 4*B**2*a*b*d**2 + 4*B**2*b** 
2*c*d + 2*a**4*b*d**5*(A**2 + 2*B**2)/(a*d - b*c)**3 - 8*a**3*b**2*c*d**4* 
(A**2 + 2*B**2)/(a*d - b*c)**3 + 12*a**2*b**3*c**2*d**3*(A**2 + 2*B**2)/(a 
*d - b*c)**3 - 8*a*b**4*c**3*d**2*(A**2 + 2*B**2)/(a*d - b*c)**3 + 2*b**5* 
c**4*d*(A**2 + 2*B**2)/(a*d - b*c)**3)/(4*A**2*b**2*d**2 + 8*B**2*b**2*d** 
2))/(g**2*i**2*(a*d - b*c)**3) + (-2*A*B*a*d - 2*A*B*b*c - 4*A*B*b*d*x + 2 
*B**2*a*d - 2*B**2*b*c)*log(e*(a + b*x)/(c + d*x))/(a**3*c*d**2*g**2*i**2 
+ a**3*d**3*g**2*i**2*x - 2*a**2*b*c**2*d*g**2*i**2 - a**2*b*c*d**2*g**2*i 
**2*x + a**2*b*d**3*g**2*i**2*x**2 + a*b**2*c**3*g**2*i**2 - a*b**2*c**2*d 
*g**2*i**2*x - 2*a*b**2*c*d**2*g**2*i**2*x**2 + b**3*c**3*g**2*i**2*x + b* 
*3*c**2*d*g**2*i**2*x**2) + (2*A*B*a*b*c*d + 2*A*B*a*b*d**2*x + 2*A*B*b**2 
*c*d*x + 2*A*B*b**2*d**2*x**2 - B**2*a**2*d**2 - 2*B**2*a*b*d**2*x + B*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (363) = 726\).

Time = 0.17 (sec) , antiderivative size = 1995, normalized size of antiderivative = 5.47 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-B^2*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*g^2*i 
^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*g^2*i^2*x + (a*b^ 
2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^2) + 2*b*d*log(b*x + a)/((b^3*c^3 
 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2) - 2*b*d*log(d*x + c)/ 
((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2*i^2))*log(b*e*x/( 
d*x + c) + a*e/(d*x + c))^2 - 2*A*B*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2 
*a*b^2*c*d^2 + a^2*b*d^3)*g^2*i^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d 
^2 + a^3*d^3)*g^2*i^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2*i^2) 
 + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) 
*g^2*i^2) - 2*b*d*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - 
 a^3*d^3)*g^2*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 2/3*B^2*(3*(b^2 
*c^2 - 2*a*b*c*d + a^2*d^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)* 
x)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log( 
b*x + a)*log(d*x + c) - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*lo 
g(d*x + c)^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a*b^3*c^4*g^2*i^2 - 3* 
a^2*b^2*c^3*d*g^2*i^2 + 3*a^3*b*c^2*d^2*g^2*i^2 - a^4*c*d^3*g^2*i^2 + (b^4 
*c^3*d*g^2*i^2 - 3*a*b^3*c^2*d^2*g^2*i^2 + 3*a^2*b^2*c*d^3*g^2*i^2 - a^3*b 
*d^4*g^2*i^2)*x^2 + (b^4*c^4*g^2*i^2 - 2*a*b^3*c^3*d*g^2*i^2 + 2*a^3*b*c*d 
^3*g^2*i^2 - a^4*d^4*g^2*i^2)*x) + (3*b^2*c^2 - 3*a^2*d^2 + (b^2*d^2*x^2 + 
 a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^3 + 3*(b^2*d^2*x^2 + a*b...
 

Giac [A] (verification not implemented)

Time = 57.73 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.56 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-{\left (\frac {{\left (d x + c\right )} B^{2} e^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )} g^{2} i^{2}} + \frac {2 \, {\left (A B e^{2} + B^{2} e^{2}\right )} {\left (d x + c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )} g^{2} i^{2}} + \frac {{\left (A^{2} e^{2} + 2 \, A B e^{2} + 2 \, B^{2} e^{2}\right )} {\left (d x + c\right )}}{{\left (b e x + a e\right )} g^{2} i^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 28.81 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.00 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {2\,B^2\,b\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g^2\,i^2\,{\left (a\,d-b\,c\right )}^3}-\frac {\frac {A^2\,a\,d+A^2\,b\,c+2\,B^2\,a\,d+2\,B^2\,b\,c-2\,A\,B\,a\,d+2\,A\,B\,b\,c}{a\,d-b\,c}+\frac {2\,x\,\left (b\,d\,A^2+2\,b\,d\,B^2\right )}{a\,d-b\,c}}{x\,\left (a^2\,d^2\,g^2\,i^2-b^2\,c^2\,g^2\,i^2\right )+x^2\,\left (a\,b\,d^2\,g^2\,i^2-b^2\,c\,d\,g^2\,i^2\right )-a\,b\,c^2\,g^2\,i^2+a^2\,c\,d\,g^2\,i^2}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,\left (B^2\,b\,c-B^2\,a\,d+A\,B\,a\,d+A\,B\,b\,c\right )}{g^2\,i^2\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}+\frac {4\,A\,B\,x}{g^2\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )}{x^2+\frac {x\,\left (a\,d+b\,c\right )}{b\,d}+\frac {a\,c}{b\,d}}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,\left (a\,d+b\,c\right )}{g^2\,i^2\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}+\frac {2\,B^2\,x}{g^2\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x^2+\frac {x\,\left (a\,d+b\,c\right )}{b\,d}+\frac {a\,c}{b\,d}}-\frac {2\,A\,B\,b\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3}\right )-\frac {b\,d\,\mathrm {atan}\left (\frac {b\,d\,\left (A^2+2\,B^2\right )\,\left (\frac {a^3\,d^3\,g^2\,i^2-a^2\,b\,c\,d^2\,g^2\,i^2-a\,b^2\,c^2\,d\,g^2\,i^2+b^3\,c^3\,g^2\,i^2}{a^2\,d^2\,g^2\,i^2-2\,a\,b\,c\,d\,g^2\,i^2+b^2\,c^2\,g^2\,i^2}+2\,b\,d\,x\right )\,\left (a^2\,d^2\,g^2\,i^2-2\,a\,b\,c\,d\,g^2\,i^2+b^2\,c^2\,g^2\,i^2\right )\,2{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3\,\left (2\,b\,d\,A^2+4\,b\,d\,B^2\right )}\right )\,\left (A^2+2\,B^2\right )\,4{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1957, normalized size of antiderivative = 5.36 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*log(a + b*x)*a**4*b*c*d**2 - 6*log(a + b*x)*a**4*b*d**3*x - 6*log(a 
+ b*x)*a**3*b**2*c**2*d - 12*log(a + b*x)*a**3*b**2*c*d**2*x + 12*log(a + 
b*x)*a**3*b**2*c*d**2 - 6*log(a + b*x)*a**3*b**2*d**3*x**2 + 12*log(a + b* 
x)*a**3*b**2*d**3*x - 6*log(a + b*x)*a**2*b**3*c**2*d*x - 12*log(a + b*x)* 
a**2*b**3*c**2*d - 6*log(a + b*x)*a**2*b**3*c*d**2*x**2 - 12*log(a + b*x)* 
a**2*b**3*c*d**2 + 12*log(a + b*x)*a**2*b**3*d**3*x**2 - 12*log(a + b*x)*a 
**2*b**3*d**3*x - 12*log(a + b*x)*a*b**4*c**2*d*x - 12*log(a + b*x)*a*b**4 
*c**2*d - 12*log(a + b*x)*a*b**4*c*d**2*x**2 - 24*log(a + b*x)*a*b**4*c*d* 
*2*x - 12*log(a + b*x)*a*b**4*d**3*x**2 - 12*log(a + b*x)*b**5*c**2*d*x - 
12*log(a + b*x)*b**5*c*d**2*x**2 + 6*log(c + d*x)*a**4*b*c*d**2 + 6*log(c 
+ d*x)*a**4*b*d**3*x + 6*log(c + d*x)*a**3*b**2*c**2*d + 12*log(c + d*x)*a 
**3*b**2*c*d**2*x - 12*log(c + d*x)*a**3*b**2*c*d**2 + 6*log(c + d*x)*a**3 
*b**2*d**3*x**2 - 12*log(c + d*x)*a**3*b**2*d**3*x + 6*log(c + d*x)*a**2*b 
**3*c**2*d*x + 12*log(c + d*x)*a**2*b**3*c**2*d + 6*log(c + d*x)*a**2*b**3 
*c*d**2*x**2 + 12*log(c + d*x)*a**2*b**3*c*d**2 - 12*log(c + d*x)*a**2*b** 
3*d**3*x**2 + 12*log(c + d*x)*a**2*b**3*d**3*x + 12*log(c + d*x)*a*b**4*c* 
*2*d*x + 12*log(c + d*x)*a*b**4*c**2*d + 12*log(c + d*x)*a*b**4*c*d**2*x** 
2 + 24*log(c + d*x)*a*b**4*c*d**2*x + 12*log(c + d*x)*a*b**4*d**3*x**2 + 1 
2*log(c + d*x)*b**5*c**2*d*x + 12*log(c + d*x)*b**5*c*d**2*x**2 - 2*log((a 
*e + b*e*x)/(c + d*x))**3*a**2*b**3*c*d**2 - 2*log((a*e + b*e*x)/(c + d...