\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x) (c i+d i x)^3} \, dx\) [104]

Optimal result

Integrand size = 42, antiderivative size = 375 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {B^2 d^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{3 B (b c-a d)^3 g i^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.77 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {\frac {3 \left (2 A^2-2 A B+B^2\right ) (b c-a d)^2}{(c+d x)^2}+\frac {6 b \left (2 A^2-6 A B+7 B^2\right ) (b c-a d)}{c+d x}+6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (a+b x)+\frac {6 B (b c-a d) (B (-7 b c+a d-6 b d x)+A (6 b c-2 a d+4 b d x)) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+\frac {6 B \left (2 A b^2 (c+d x)^2+B d (a+b x) (-4 b c+a d-3 b d x)\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+4 b^2 B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (c+d x)}{12 (b c-a d)^3 g i^3} \] Input:

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

Output:

((3*(2*A^2 - 2*A*B + B^2)*(b*c - a*d)^2)/(c + d*x)^2 + (6*b*(2*A^2 - 6*A*B 
 + 7*B^2)*(b*c - a*d))/(c + d*x) + 6*b^2*(2*A^2 - 6*A*B + 7*B^2)*Log[a + b 
*x] + (6*B*(b*c - a*d)*(B*(-7*b*c + a*d - 6*b*d*x) + A*(6*b*c - 2*a*d + 4* 
b*d*x))*Log[(e*(a + b*x))/(c + d*x)])/(c + d*x)^2 + (6*B*(2*A*b^2*(c + d*x 
)^2 + B*d*(a + b*x)*(-4*b*c + a*d - 3*b*d*x))*Log[(e*(a + b*x))/(c + d*x)] 
^2)/(c + d*x)^2 + 4*b^2*B^2*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*b^2*(2*A^2 
- 6*A*B + 7*B^2)*Log[c + d*x])/(12*(b*c - a*d)^3*g*i^3)
 

Rubi [A] (warning: unable to verify)

Time = 0.86 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2962, 2788, 2767, 2009, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2767

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2739

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

\(\Big \downarrow \) 15

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

Input:

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

Output:

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

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 2767

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

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(768\) vs. \(2(367)=734\).

Time = 2.09 (sec) , antiderivative size = 769, normalized size of antiderivative = 2.05

method	result	size
parts	\(\frac {A^{2} \left (-\frac {1}{2 \left (d a -b c \right ) \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{\left (d a -b c \right )^{3}}+\frac {b}{\left (d a -b c \right )^{2} \left (d x +c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (d a -b c \right )^{3}}\right )}{g \,i^{3}}-\frac {B^{2} d \left (\frac {d \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{d a -b c}-\frac {2 b e \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^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}{3 d \left (d a -b c \right )}\right )}{g \,i^{3} \left (d a -b c \right )^{2} e^{2}}-\frac {2 A B d \left (\frac {d \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{d a -b c}-\frac {2 b e \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^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 d \left (d a -b c \right )}\right )}{g \,i^{3} \left (d a -b c \right )^{2} e^{2}}\)	\(769\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (d a -b c \right )^{4} g}-\frac {2 d^{3} A^{2} b \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (d a -b c \right )^{4} g}+\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 )^{2}}{2 e^{3} i^{3} \left (d a -b c \right )^{4} g}+\frac {d^{2} A B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{3} \left (d a -b c \right )^{4} g}-\frac {4 d^{3} A B 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^{3} \left (d a -b c \right )^{4} g}+\frac {2 d^{4} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (d a -b c \right )^{4} g}+\frac {d^{2} B^{2} b^{2} \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^{3} \left (d a -b c \right )^{4} g}-\frac {2 d^{3} B^{2} 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 )^{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^{3} \left (d a -b c \right )^{4} g}+\frac {d^{4} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (d a -b c \right )^{4} g}\right )}{d^{2}}\)	\(881\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} A^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (d a -b c \right )^{4} g}-\frac {2 d^{3} A^{2} b \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (d a -b c \right )^{4} g}+\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 )^{2}}{2 e^{3} i^{3} \left (d a -b c \right )^{4} g}+\frac {d^{2} A B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{e \,i^{3} \left (d a -b c \right )^{4} g}-\frac {4 d^{3} A B 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^{3} \left (d a -b c \right )^{4} g}+\frac {2 d^{4} A B \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (d a -b c \right )^{4} g}+\frac {d^{2} B^{2} b^{2} \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^{3} \left (d a -b c \right )^{4} g}-\frac {2 d^{3} B^{2} 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 )^{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^{3} \left (d a -b c \right )^{4} g}+\frac {d^{4} B^{2} \left (\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2}+\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (d a -b c \right )^{4} g}\right )}{d^{2}}\)	\(881\)
norman	\(\frac {-\frac {2 A^{2} a \,d^{3}-6 A^{2} b c \,d^{2}-2 A B a \,d^{3}+14 A B b c \,d^{2}+B^{2} a \,d^{3}-15 B^{2} b c \,d^{2}}{4 i g \,d^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {\left (2 A^{2} b^{2} c^{2}+2 A B \,a^{2} d^{2}-8 A B a b c d -B^{2} a^{2} d^{2}+8 B^{2} a b c d \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 {B \left (2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-4 B a b c d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{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} b \,d^{2}-6 A B b \,d^{2}+7 B^{2} b \,d^{2}\right ) x}{2 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g i}-\frac {B^{2} b^{2} c^{2} \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 {d \left (2 A^{2} b^{2} c -2 A B a b d -4 A B \,b^{2} c +3 B^{2} a b d +4 B^{2} b^{2} c \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 {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 {\left (2 A^{2} b^{2}-6 A B \,b^{2}+7 B^{2} b^{2}\right ) d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (d a -b c \right ) \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {b d B \left (2 A b c -B a d -2 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 d \,B^{2} b^{2} c 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 )}-\frac {d^{2} B \,b^{2} \left (2 A -3 B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{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 )}}{i^{2} \left (d x +c \right )^{2}}\)	\(937\)
parallelrisch	\(\text {Expression too large to display}\)	\(1122\)
risch	\(\text {Expression too large to display}\)	\(1661\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.45 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {3 \, {\left (6 \, A^{2} - 14 \, A B + 15 \, B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2} + 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + B^{2} b^{2} c^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 6 \, {\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A B b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2} - 2 \, {\left (B^{2} a b d^{2} - 2 \, {\left (A B - B^{2}\right )} b^{2} c d\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} b^{2} c^{2} - 8 \, {\left (A B - B^{2}\right )} a b c d + {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} + 2 \, {\left (2 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B - 3 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i)^3,x, algo 
rithm="fricas")
 

Output:

1/12*(3*(6*A^2 - 14*A*B + 15*B^2)*b^2*c^2 - 24*(A^2 - 2*A*B + 2*B^2)*a*b*c 
*d + 3*(2*A^2 - 2*A*B + B^2)*a^2*d^2 + 4*(B^2*b^2*d^2*x^2 + 2*B^2*b^2*c*d* 
x + B^2*b^2*c^2)*log((b*e*x + a*e)/(d*x + c))^3 + 6*((2*A*B - 3*B^2)*b^2*d 
^2*x^2 + 2*A*B*b^2*c^2 - 4*B^2*a*b*c*d + B^2*a^2*d^2 - 2*(B^2*a*b*d^2 - 2* 
(A*B - B^2)*b^2*c*d)*x)*log((b*e*x + a*e)/(d*x + c))^2 + 6*((2*A^2 - 6*A*B 
 + 7*B^2)*b^2*c*d - (2*A^2 - 6*A*B + 7*B^2)*a*b*d^2)*x + 6*((2*A^2 - 6*A*B 
 + 7*B^2)*b^2*d^2*x^2 + 2*A^2*b^2*c^2 - 8*(A*B - B^2)*a*b*c*d + (2*A*B - B 
^2)*a^2*d^2 + 2*(2*(A^2 - 2*A*B + 2*B^2)*b^2*c*d - (2*A*B - 3*B^2)*a*b*d^2 
)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2 
*b*c*d^4 - a^3*d^5)*g*i^3*x^2 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c 
^2*d^3 - a^3*c*d^4)*g*i^3*x + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - 
 a^3*c^2*d^3)*g*i^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1488 vs. \(2 (333) = 666\).

Time = 4.17 (sec) , antiderivative size = 1488, normalized size of antiderivative = 3.97 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-B**2*b**2*log(e*(a + b*x)/(c + d*x))**3/(3*a**3*d**3*g*i**3 - 9*a**2*b*c* 
d**2*g*i**3 + 9*a*b**2*c**2*d*g*i**3 - 3*b**3*c**3*g*i**3) + b**2*(2*A**2 
- 6*A*B + 7*B**2)*log(x + (2*A**2*a*b**2*d + 2*A**2*b**3*c - 6*A*B*a*b**2* 
d - 6*A*B*b**3*c + 7*B**2*a*b**2*d + 7*B**2*b**3*c - a**4*b**2*d**4*(2*A** 
2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 + 4*a**3*b**3*c*d**3*(2*A**2 - 6*A*B + 
7*B**2)/(a*d - b*c)**3 - 6*a**2*b**4*c**2*d**2*(2*A**2 - 6*A*B + 7*B**2)/( 
a*d - b*c)**3 + 4*a*b**5*c**3*d*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 - 
 b**6*c**4*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b**3*d - 12*A 
*B*b**3*d + 14*B**2*b**3*d))/(2*g*i**3*(a*d - b*c)**3) - b**2*(2*A**2 - 6* 
A*B + 7*B**2)*log(x + (2*A**2*a*b**2*d + 2*A**2*b**3*c - 6*A*B*a*b**2*d - 
6*A*B*b**3*c + 7*B**2*a*b**2*d + 7*B**2*b**3*c + a**4*b**2*d**4*(2*A**2 - 
6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a**3*b**3*c*d**3*(2*A**2 - 6*A*B + 7*B* 
*2)/(a*d - b*c)**3 + 6*a**2*b**4*c**2*d**2*(2*A**2 - 6*A*B + 7*B**2)/(a*d 
- b*c)**3 - 4*a*b**5*c**3*d*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 + b** 
6*c**4*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b**3*d - 12*A*B*b 
**3*d + 14*B**2*b**3*d))/(2*g*i**3*(a*d - b*c)**3) + (-2*A*B*a*d + 6*A*B*b 
*c + 4*A*B*b*d*x + B**2*a*d - 7*B**2*b*c - 6*B**2*b*d*x)*log(e*(a + b*x)/( 
c + d*x))/(2*a**2*c**2*d**2*g*i**3 + 4*a**2*c*d**3*g*i**3*x + 2*a**2*d**4* 
g*i**3*x**2 - 4*a*b*c**3*d*g*i**3 - 8*a*b*c**2*d**2*g*i**3*x - 4*a*b*c*d** 
3*g*i**3*x**2 + 2*b**2*c**4*g*i**3 + 4*b**2*c**3*d*g*i**3*x + 2*b**2*c*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2116 vs. \(2 (367) = 734\).

Time = 0.29 (sec) , antiderivative size = 2116, normalized size of antiderivative = 5.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)/(d*i*x+c*i)^3,x, algo 
rithm="maxima")
 

Output:

1/2*B^2*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g* 
i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + (b^2*c^4 - 2 
*a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*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*i^3) - 2*b^2*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*i^3))*log(b*e*x/(d*x + c) + a*e/ 
(d*x + c))^2 + A*B*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + 
a^2*d^4)*g*i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + ( 
b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*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*i^3) - 2*b^2*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*i^3))*log(b*e*x/(d*x 
+ c) + a*e/(d*x + c)) - 1/12*B^2*(6*(7*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*( 
b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*b 
^2*c*d*x + b^2*c^2)*log(d*x + c)^2 + 6*(b^2*c*d - a*b*d^2)*x + 6*(b^2*d^2* 
x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a) - 2*(3*b^2*d^2*x^2 + 6*b^2*c*d*x 
 + 3*b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a))*log(d 
*x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^3*c^5*g*i^3 - 3*a*b^2*c^4 
*d*g*i^3 + 3*a^2*b*c^3*d^2*g*i^3 - a^3*c^2*d^3*g*i^3 + (b^3*c^3*d^2*g*i^3 
- 3*a*b^2*c^2*d^3*g*i^3 + 3*a^2*b*c*d^4*g*i^3 - a^3*d^5*g*i^3)*x^2 + 2*(b^ 
3*c^4*d*g*i^3 - 3*a*b^2*c^3*d^2*g*i^3 + 3*a^2*b*c^2*d^3*g*i^3 - a^3*c*d^4* 
g*i^3)*x) - (45*b^2*c^2 - 48*a*b*c*d + 3*a^2*d^2 + 4*(b^2*d^2*x^2 + 2*b...
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.79 \[ \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)^3} \, dx=\frac {1}{12} \, {\left (\frac {4 \, B^{2} b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )^{3}}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + \frac {12 \, A^{2} b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + 6 \, {\left (\frac {2 \, A B b^{2} e}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} - \frac {4 \, {\left (b e x + a e\right )} B^{2} b d}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}} + \frac {{\left (b e x + a e\right )}^{2} B^{2} d^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left (\frac {{\left (2 \, A B d^{2} - B^{2} d^{2}\right )} {\left (b e x + a e\right )}^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (A B b d - B^{2} b d\right )} {\left (b e x + a e\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) + \frac {3 \, {\left (2 \, A^{2} d^{2} - 2 \, A B d^{2} + B^{2} d^{2}\right )} {\left (b e x + a e\right )}^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {24 \, {\left (A^{2} b d - 2 \, A B b d + 2 \, B^{2} b d\right )} {\left (b e x + a e\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\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)^3,x, algo 
rithm="giac")
 

Output:

1/12*(4*B^2*b^2*e*log((b*e*x + a*e)/(d*x + c))^3/(b^2*c^2*g*i^3 - 2*a*b*c* 
d*g*i^3 + a^2*d^2*g*i^3) + 12*A^2*b^2*e*log((b*e*x + a*e)/(d*x + c))/(b^2* 
c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) + 6*(2*A*B*b^2*e/(b^2*c^2*g*i 
^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) - 4*(b*e*x + a*e)*B^2*b*d/((b^2*c^2* 
g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)) + (b*e*x + a*e)^2*B^2* 
d^2/((b^2*c^2*e*g*i^3 - 2*a*b*c*d*e*g*i^3 + a^2*d^2*e*g*i^3)*(d*x + c)^2)) 
*log((b*e*x + a*e)/(d*x + c))^2 + 6*((2*A*B*d^2 - B^2*d^2)*(b*e*x + a*e)^2 
/((b^2*c^2*e*g*i^3 - 2*a*b*c*d*e*g*i^3 + a^2*d^2*e*g*i^3)*(d*x + c)^2) - 8 
*(A*B*b*d - B^2*b*d)*(b*e*x + a*e)/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2 
*d^2*g*i^3)*(d*x + c)))*log((b*e*x + a*e)/(d*x + c)) + 3*(2*A^2*d^2 - 2*A* 
B*d^2 + B^2*d^2)*(b*e*x + a*e)^2/((b^2*c^2*e*g*i^3 - 2*a*b*c*d*e*g*i^3 + a 
^2*d^2*e*g*i^3)*(d*x + c)^2) - 24*(A^2*b*d - 2*A*B*b*d + 2*B^2*b*d)*(b*e*x 
 + a*e)/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(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 = 30.06 (sec) , antiderivative size = 984, normalized size of antiderivative = 2.62 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(b^2*atan((b^2*(A^2 + (7*B^2)/2 - 3*A*B)*(2*a^3*d^3*g*i^3 + 2*b^3*c^3*g*i^ 
3 - 2*a*b^2*c^2*d*g*i^3 - 2*a^2*b*c*d^2*g*i^3)*1i)/(g*i^3*(a*d - b*c)^3*(2 
*A^2*b^2 + 7*B^2*b^2 - 6*A*B*b^2)) + (b^3*d*x*(a^2*d^2*g*i^3 + b^2*c^2*g*i 
^3 - 2*a*b*c*d*g*i^3)*(A^2 + (7*B^2)/2 - 3*A*B)*4i)/(g*i^3*(a*d - b*c)^3*( 
2*A^2*b^2 + 7*B^2*b^2 - 6*A*B*b^2)))*(A^2 + (7*B^2)/2 - 3*A*B)*2i)/(g*i^3* 
(a*d - b*c)^3) - ((2*A^2*a*d - 6*A^2*b*c + B^2*a*d - 15*B^2*b*c - 2*A*B*a* 
d + 14*A*B*b*c)/(2*(a*d - b*c)) - (x*(2*A^2*b*d + 7*B^2*b*d - 6*A*B*b*d))/ 
(a*d - b*c))/(x^2*(2*a*d^3*g*i^3 - 2*b*c*d^2*g*i^3) + x*(4*a*c*d^2*g*i^3 - 
 4*b*c^2*d*g*i^3) - 2*b*c^3*g*i^3 + 2*a*c^2*d*g*i^3) - (log((e*(a + b*x))/ 
(c + d*x))*(B^2/(b*d*g*i^3*(a*d - b*c)) + (B*b^2*((a^2*d^2 + 2*b^2*c^2 - 3 
*a*b*c*d)/(2*b^3*d) - (c*(a*d - b*c))/(2*b^2*d))*(2*A - 3*B))/(g*i^3*(a^3* 
d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B*x*(2*A - 3*B)*(a*d - 
b*c))/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((d*x^ 
2)/b + c^2/(b*d) + (2*c*x)/b) - log((e*(a + b*x))/(c + d*x))^2*(((B^2*b^2* 
((a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d)/(2*b^3*d) - (c*(a*d - b*c))/(2*b^2*d))) 
/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*x*(a*d 
 - b*c))/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/((d* 
x^2)/b + c^2/(b*d) + (2*c*x)/b) + (B*b^2*(2*A - 3*B))/(2*g*i^3*(a^3*d^3 - 
b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (B^2*b^2*log((e*(a + b*x))/(c 
 + d*x))^3)/(3*g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1457, normalized size of antiderivative = 3.89 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 12*log(a + b*x)*a**2*b**2*c**3 - 24*log(a + b*x)*a**2*b**2*c**2*d*x 
 + 12*log(a + b*x)*a**2*b**2*c**2*d - 12*log(a + b*x)*a**2*b**2*c*d**2*x** 
2 + 24*log(a + b*x)*a**2*b**2*c*d**2*x + 12*log(a + b*x)*a**2*b**2*d**3*x* 
*2 + 24*log(a + b*x)*a*b**3*c**3 + 48*log(a + b*x)*a*b**3*c**2*d*x - 18*lo 
g(a + b*x)*a*b**3*c**2*d + 24*log(a + b*x)*a*b**3*c*d**2*x**2 - 36*log(a + 
 b*x)*a*b**3*c*d**2*x - 18*log(a + b*x)*a*b**3*d**3*x**2 - 24*log(a + b*x) 
*b**4*c**3 - 48*log(a + b*x)*b**4*c**2*d*x - 24*log(a + b*x)*b**4*c*d**2*x 
**2 + 12*log(c + d*x)*a**2*b**2*c**3 + 24*log(c + d*x)*a**2*b**2*c**2*d*x 
- 12*log(c + d*x)*a**2*b**2*c**2*d + 12*log(c + d*x)*a**2*b**2*c*d**2*x**2 
 - 24*log(c + d*x)*a**2*b**2*c*d**2*x - 12*log(c + d*x)*a**2*b**2*d**3*x** 
2 - 24*log(c + d*x)*a*b**3*c**3 - 48*log(c + d*x)*a*b**3*c**2*d*x + 18*log 
(c + d*x)*a*b**3*c**2*d - 24*log(c + d*x)*a*b**3*c*d**2*x**2 + 36*log(c + 
d*x)*a*b**3*c*d**2*x + 18*log(c + d*x)*a*b**3*d**3*x**2 + 24*log(c + d*x)* 
b**4*c**3 + 48*log(c + d*x)*b**4*c**2*d*x + 24*log(c + d*x)*b**4*c*d**2*x* 
*2 - 4*log((a*e + b*e*x)/(c + d*x))**3*b**4*c**3 - 8*log((a*e + b*e*x)/(c 
+ d*x))**3*b**4*c**2*d*x - 4*log((a*e + b*e*x)/(c + d*x))**3*b**4*c*d**2*x 
**2 - 6*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*c*d**2 - 12*log((a*e + b 
*e*x)/(c + d*x))**2*a*b**3*c**3 - 24*log((a*e + b*e*x)/(c + d*x))**2*a*b** 
3*c**2*d*x + 24*log((a*e + b*e*x)/(c + d*x))**2*a*b**3*c**2*d - 12*log((a* 
e + b*e*x)/(c + d*x))**2*a*b**3*c*d**2*x**2 + 12*log((a*e + b*e*x)/(c +...