\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x) (c i+d i x)^2} \, dx\) [96]

Optimal result

Integrand size = 42, antiderivative size = 214 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {2 A B d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}-\frac {2 B^2 d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}+\frac {2 B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^2 g i^2 (c+d x)}-\frac {d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^2 g i^2 (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^2 g i^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {3 b \left (A^2-2 A B+2 B^2\right ) (c+d x) \log (a+b x)+6 (A-B) B (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (-B d (a+b x)+A b (c+d x)) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+b B^2 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-3 \left (A^2-2 A B+2 B^2\right ) (-b c+a d+b (c+d x) \log (c+d x))}{3 (b c-a d)^2 g i^2 (c+d x)} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.53 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2962, 2788, 2733, 2009, 2739, 15}

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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2739

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

\(\Big \downarrow \) 15

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2009

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

rule 2733

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b 
*Log[c*x^n])^p, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; 
 FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 2739

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( 
b*n)   Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} 
, x]
 

rule 2788

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

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 [A] (verified)

Time = 1.78 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.86

method	result	size
norman	\(\frac {\frac {\left (A^{2} b c -2 A B a d +2 B^{2} a d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B \left (A b c -B a d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {d \left (A^{2} b -2 A B b +2 B^{2} b \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (A^{2}-2 A B +2 B^{2}\right ) d x}{g i c \left (d a -b c \right )}+\frac {b d B \left (A -B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {B^{2} b c \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {b \,B^{2} d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3}}{3 g i \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}}{i \left (d x +c \right )}\)	\(398\)
parallelrisch	\(\frac {-3 A^{2} a \,b^{2} d^{4}+3 A^{2} b^{3} c \,d^{3}-6 B^{2} a \,b^{2} d^{4}+6 B^{2} b^{3} c \,d^{3}-3 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4}+B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{3} c \,d^{3}+3 A^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}+6 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}-3 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a \,b^{2} d^{4}+3 A^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{3}+6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4}+B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{3} b^{3} d^{4}+6 A B a \,b^{2} d^{4}-6 A B \,b^{3} c \,d^{3}+3 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4}-6 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}+3 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} c \,d^{3}-6 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4}}{3 i^{2} g \left (d x +c \right ) b^{2} d^{3} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}\)	\(446\)
parts	\(\frac {A^{2} \left (-\frac {1}{\left (d a -b c \right ) \left (d x +c \right )}-\frac {b \ln \left (d x +c \right )}{\left (d a -b c \right )^{2}}+\frac {b \ln \left (b x +a \right )}{\left (d a -b c \right )^{2}}\right )}{g \,i^{2}}-\frac {B^{2} \left (\frac {d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{d a -b c}-\frac {b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 \left (d a -b c \right )}\right )}{g \,i^{2} \left (d a -b c \right ) e}-\frac {2 A B \left (\frac {d \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{d a -b c}-\frac {b e \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (d a -b c \right )}\right )}{g \,i^{2} \left (d a -b c \right ) e}\)	\(459\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}-\frac {d^{2} A B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {2 d^{3} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}-\frac {d^{2} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {d^{3} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}\right )}{d^{2}}\)	\(521\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {d^{2} A^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {d^{3} A^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}-\frac {d^{2} A B b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {2 d^{3} A B \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}-\frac {d^{2} B^{2} b \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 e \,i^{2} \left (d a -b c \right )^{3} g}+\frac {d^{3} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (d a -b c \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{e^{2} i^{2} \left (d a -b c \right )^{3} g}\right )}{d^{2}}\)	\(521\)
risch	\(-\frac {A^{2}}{g \,i^{2} \left (d a -b c \right ) \left (d x +c \right )}-\frac {A^{2} b \ln \left (d x +c \right )}{g \,i^{2} \left (d a -b c \right )^{2}}+\frac {A^{2} b \ln \left (b x +a \right )}{g \,i^{2} \left (d a -b c \right )^{2}}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b}{g \,i^{2} \left (d a -b c \right )^{2}}-\frac {B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} a}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b c}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b}{g \,i^{2} \left (d a -b c \right )^{2}}+\frac {2 B^{2} d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b c}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} d a}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 B^{2} b c}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 B^{2} b}{g \,i^{2} \left (d a -b c \right )^{2}}+\frac {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 g \,i^{2} \left (d a -b c \right )^{2}}-\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b}{g \,i^{2} \left (d a -b c \right )^{2}}-\frac {2 A B d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) a}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) b c}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 A B d a}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {2 A B b c}{g \,i^{2} \left (d a -b c \right )^{2} \left (d x +c \right )}+\frac {2 A B b}{g \,i^{2} \left (d a -b c \right )^{2}}+\frac {A B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} b}{g \,i^{2} \left (d a -b c \right )^{2}}\)	\(848\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.10 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - 3 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d + 3 \, {\left (A B b c - B^{2} a d + {\left (A B - B^{2}\right )} b d x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left (A^{2} b c + {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b d x - 2 \, {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{3 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (185) = 370\).

Time = 0.72 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.52 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {B^{2} b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g i^{2} - 6 a b c d g i^{2} + 3 b^{2} c^{2} g i^{2}} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A^{2} - 2 A B + 2 B^{2}\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) + \frac {\left (A B b c + A B b d x - B^{2} a d - B^{2} b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{2} c d^{2} g i^{2} + a^{2} d^{3} g i^{2} x - 2 a b c^{2} d g i^{2} - 2 a b c d^{2} g i^{2} x + b^{2} c^{3} g i^{2} + b^{2} c^{2} d g i^{2} x} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1004 vs. \(2 (212) = 424\).

Time = 0.11 (sec) , antiderivative size = 1004, normalized size of antiderivative = 4.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {1}{3} \, {\left (\frac {B^{2} b e \log \left (\frac {b e x + a e}{d x + c}\right )^{3}}{b c g i^{2} - a d g i^{2}} + \frac {3 \, A^{2} b e \log \left (\frac {b e x + a e}{d x + c}\right )}{b c g i^{2} - a d g i^{2}} + 3 \, {\left (\frac {A B b e}{b c g i^{2} - a d g i^{2}} - \frac {{\left (b e x + a e\right )} B^{2} d}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - \frac {6 \, {\left (A B d - B^{2} d\right )} {\left (b e x + a e\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (A^{2} d - 2 \, A B d + 2 \, B^{2} d\right )} {\left (b e x + a e\right )}}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 28.06 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx={\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B\,b\,\left (A-B\right )}{g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}+\frac {B^2\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {2\,B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (A-B\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {b\,\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (b\,A^2-2\,b\,A\,B+2\,b\,B^2\right )}\right )\,\left (A^2-2\,A\,B+2\,B^2\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {-3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{2} c d x +6 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,d^{2} x -6 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c d -6 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} d^{2} x -6 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,d^{2} x +6 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} d^{2} x -\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{3} b^{3} c^{2}-3 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,c^{2}+3 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,c^{2}-3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{2} c^{2}-6 a \,b^{2} d^{2} x +6 b^{3} c d x +6 \,\mathrm {log}\left (b x +a \right ) a^{2} b c d +3 a^{2} b c d x -\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{3} b^{3} c d x +6 a^{2} b \,d^{2} x +6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} c d x -6 \,\mathrm {log}\left (d x +c \right ) a^{2} b c d +6 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c d -6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b \,d^{2} x +6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{2} d^{2} x -6 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{3} c d x -6 a \,b^{2} c d x -3 \,\mathrm {log}\left (b x +a \right ) a^{2} b c d x +3 \,\mathrm {log}\left (d x +c \right ) a^{2} b c d x +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} b^{3} c d x -3 a^{3} d^{2} x +3 \mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} a \,b^{2} c d}{3 c g \left (a^{2} d^{3} x -2 a b c \,d^{2} x +b^{2} c^{2} d x +a^{2} c \,d^{2}-2 a b \,c^{2} d +b^{2} c^{3}\right )} \] Input:

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

Output:

( - 3*log(a + b*x)*a**2*b*c**2 - 3*log(a + b*x)*a**2*b*c*d*x + 6*log(a + b 
*x)*a**2*b*c*d + 6*log(a + b*x)*a**2*b*d**2*x - 6*log(a + b*x)*a*b**2*c*d 
- 6*log(a + b*x)*a*b**2*d**2*x + 3*log(c + d*x)*a**2*b*c**2 + 3*log(c + d* 
x)*a**2*b*c*d*x - 6*log(c + d*x)*a**2*b*c*d - 6*log(c + d*x)*a**2*b*d**2*x 
 + 6*log(c + d*x)*a*b**2*c*d + 6*log(c + d*x)*a*b**2*d**2*x - log((a*e + b 
*e*x)/(c + d*x))**3*b**3*c**2 - log((a*e + b*e*x)/(c + d*x))**3*b**3*c*d*x 
 - 3*log((a*e + b*e*x)/(c + d*x))**2*a*b**2*c**2 - 3*log((a*e + b*e*x)/(c 
+ d*x))**2*a*b**2*c*d*x + 3*log((a*e + b*e*x)/(c + d*x))**2*a*b**2*c*d + 3 
*log((a*e + b*e*x)/(c + d*x))**2*b**3*c*d*x - 6*log((a*e + b*e*x)/(c + d*x 
))*a**2*b*d**2*x + 6*log((a*e + b*e*x)/(c + d*x))*a*b**2*c*d*x + 6*log((a* 
e + b*e*x)/(c + d*x))*a*b**2*d**2*x - 6*log((a*e + b*e*x)/(c + d*x))*b**3* 
c*d*x - 3*a**3*d**2*x + 3*a**2*b*c*d*x + 6*a**2*b*d**2*x - 6*a*b**2*c*d*x 
- 6*a*b**2*d**2*x + 6*b**3*c*d*x)/(3*c*g*(a**2*c*d**2 + a**2*d**3*x - 2*a* 
b*c**2*d - 2*a*b*c*d**2*x + b**2*c**3 + b**2*c**2*d*x))