\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)^3} \, dx\) [54]

Optimal result

Integrand size = 40, antiderivative size = 563 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\frac {B d^5 (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {5 b B d^4 (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 B d^2 (c+d x)}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B d (c+d x)^2}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 B (c+d x)^3}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}+\frac {5 b^2 B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^6 g^4 i^3}-\frac {d^5 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^6 g^4 i^3} \] Output:

1/4*B*d^5*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-5*b*B*d^4*(b*x+a)/(-a*d 
+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*B*d^2*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+ 
5/4*b^4*B*d*(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-1/9*b^5*B*(d*x+c)^3/( 
-a*d+b*c)^6/g^4/i^3/(b*x+a)^3+5*b^2*B*d^3*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c) 
^6/g^4/i^3-1/2*d^5*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^6/g^4/ 
i^3/(d*x+c)^2+5*b*d^4*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^6/g^4 
/i^3/(d*x+c)-10*b^3*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^6/g 
^4/i^3/(b*x+a)+5/2*b^4*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^ 
6/g^4/i^3/(b*x+a)^2-1/3*b^5*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b* 
c)^6/g^4/i^3/(b*x+a)^3-10*b^2*d^3*ln((b*x+a)/(d*x+c))*(A+B*ln(e*(b*x+a)/(d 
*x+c)))/(-a*d+b*c)^6/g^4/i^3
 

Mathematica [C] (verified)

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

Time = 1.51 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.13 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=-\frac {\frac {4 b^2 B (b c-a d)^3}{(a+b x)^3}-\frac {33 b^2 B d (b c-a d)^2}{(a+b x)^2}+\frac {216 b^3 B c d^2}{a+b x}-\frac {216 a b^2 B d^3}{a+b x}+\frac {66 b^2 B d^2 (b c-a d)}{a+b x}-\frac {9 B d^3 (b c-a d)^2}{(c+d x)^2}-\frac {144 b^2 B c d^3}{c+d x}+\frac {144 a b B d^4}{c+d x}-\frac {18 b B d^3 (b c-a d)}{c+d x}+120 b^2 B d^3 \log (a+b x)+\frac {12 b^2 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}-\frac {54 b^2 d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {216 b^2 d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+\frac {18 d^3 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}+\frac {144 b d^3 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+360 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-120 b^2 B d^3 \log (c+d x)-360 b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-180 b^2 B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+180 b^2 B d^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{36 (b c-a d)^6 g^4 i^3} \] Input:

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

Output:

-1/36*((4*b^2*B*(b*c - a*d)^3)/(a + b*x)^3 - (33*b^2*B*d*(b*c - a*d)^2)/(a 
 + b*x)^2 + (216*b^3*B*c*d^2)/(a + b*x) - (216*a*b^2*B*d^3)/(a + b*x) + (6 
6*b^2*B*d^2*(b*c - a*d))/(a + b*x) - (9*B*d^3*(b*c - a*d)^2)/(c + d*x)^2 - 
 (144*b^2*B*c*d^3)/(c + d*x) + (144*a*b*B*d^4)/(c + d*x) - (18*b*B*d^3*(b* 
c - a*d))/(c + d*x) + 120*b^2*B*d^3*Log[a + b*x] + (12*b^2*(b*c - a*d)^3*( 
A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^3 - (54*b^2*d*(b*c - a*d)^2 
*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (216*b^2*d^2*(b*c - a 
*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) + (18*d^3*(b*c - a*d)^ 
2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 + (144*b*d^3*(b*c - a* 
d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 360*b^2*d^3*Log[a + b 
*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 120*b^2*B*d^3*Log[c + d*x] - 36 
0*b^2*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 180*b^2*B*d^ 
3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Poly 
Log[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 180*b^2*B*d^3*((2*Log[(d*(a + b*x) 
)/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x) 
)/(b*c - a*d)]))/((b*c - a*d)^6*g^4*i^3)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.69, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2772, 27, 2010, 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.

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

\(\Big \downarrow \) 2962

\(\displaystyle \frac {\int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B \int -\frac {(c+d x)^4 \left (2 b^5-\frac {15 d (a+b x) b^4}{c+d x}+\frac {60 d^2 (a+b x)^2 b^3}{(c+d x)^2}+\frac {60 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right ) b^2}{(c+d x)^3}-\frac {30 d^4 (a+b x)^4 b}{(c+d x)^4}+\frac {3 d^5 (a+b x)^5}{(c+d x)^5}\right )}{6 (a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^5 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {10 b^3 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-10 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^5 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} B \int \frac {(c+d x)^4 \left (2 b^5-\frac {15 d (a+b x) b^4}{c+d x}+\frac {60 d^2 (a+b x)^2 b^3}{(c+d x)^2}+\frac {60 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right ) b^2}{(c+d x)^3}-\frac {30 d^4 (a+b x)^4 b}{(c+d x)^4}+\frac {3 d^5 (a+b x)^5}{(c+d x)^5}\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^5 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {10 b^3 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-10 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^5 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {\frac {1}{6} B \int \left (\frac {\left (2 b^5-\frac {15 d (a+b x) b^4}{c+d x}+\frac {60 d^2 (a+b x)^2 b^3}{(c+d x)^2}-\frac {30 d^4 (a+b x)^4 b}{(c+d x)^4}+\frac {3 d^5 (a+b x)^5}{(c+d x)^5}\right ) (c+d x)^4}{(a+b x)^4}+\frac {60 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) (c+d x)}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^5 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {10 b^3 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-10 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^5 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^5 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {10 b^3 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-10 b^2 d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {d^5 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+\frac {1}{6} B \left (-\frac {2 b^5 (c+d x)^3}{3 (a+b x)^3}+\frac {15 b^4 d (c+d x)^2}{2 (a+b x)^2}-\frac {60 b^3 d^2 (c+d x)}{a+b x}+30 b^2 d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )+\frac {3 d^5 (a+b x)^2}{2 (c+d x)^2}-\frac {30 b d^4 (a+b x)}{c+d x}\right )}{g^4 i^3 (b c-a d)^6}\)

Input:

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

Output:

((B*((3*d^5*(a + b*x)^2)/(2*(c + d*x)^2) - (30*b*d^4*(a + b*x))/(c + d*x) 
- (60*b^3*d^2*(c + d*x))/(a + b*x) + (15*b^4*d*(c + d*x)^2)/(2*(a + b*x)^2 
) - (2*b^5*(c + d*x)^3)/(3*(a + b*x)^3) + 30*b^2*d^3*Log[(a + b*x)/(c + d* 
x)]^2))/6 - (d^5*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(c + 
 d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d 
*x) - (10*b^3*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x 
) + (5*b^4*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x 
)^2) - (b^5*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(a + b*x) 
^3) - 10*b^2*d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d* 
x)]))/((b*c - a*d)^6*g^4*i^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

rule 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 = 8.76 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.40

method	result	size
parts	\(\frac {A \left (-\frac {d^{3}}{2 \left (d a -b c \right )^{4} \left (d x +c \right )^{2}}+\frac {10 d^{3} b^{2} \ln \left (d x +c \right )}{\left (d a -b c \right )^{6}}+\frac {4 d^{3} b}{\left (d a -b c \right )^{5} \left (d x +c \right )}+\frac {b^{2}}{3 \left (d a -b c \right )^{3} \left (b x +a \right )^{3}}-\frac {10 d^{3} b^{2} \ln \left (b x +a \right )}{\left (d a -b c \right )^{6}}+\frac {6 b^{2} d^{2}}{\left (d a -b c \right )^{5} \left (b x +a \right )}+\frac {3 b^{2} d}{2 \left (d a -b c \right )^{4} \left (b x +a \right )^{2}}\right )}{g^{4} i^{3}}-\frac {B d \left (\frac {d^{4} \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 )}{\left (d a -b c \right )^{4}}-\frac {5 d^{3} 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 )}{\left (d a -b c \right )^{4}}+\frac {5 d^{2} 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}}{\left (d a -b c \right )^{4}}-\frac {10 d \,b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{\left (d a -b c \right )^{4}}+\frac {5 b^{4} e^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{4}}-\frac {b^{5} e^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{d \left (d a -b c \right )^{4}}\right )}{g^{4} i^{3} \left (d a -b c \right )^{2} e^{2}}\)	\(787\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e^{2} A \,b^{5}}{3 i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {5 d^{3} e A \,b^{4}}{2 i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {10 d^{4} A \,b^{3}}{i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {10 d^{5} A \,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 )^{7} g^{4}}-\frac {5 d^{6} A 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 )^{7} g^{4}}+\frac {d^{7} A \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 )^{7} g^{4}}-\frac {d^{2} e^{2} B \,b^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}+\frac {5 d^{3} e B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}-\frac {10 d^{4} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}+\frac {5 d^{5} 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 )^{7} g^{4}}-\frac {5 d^{6} 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 )^{7} g^{4}}+\frac {d^{7} 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 )^{7} g^{4}}\right )}{d^{2}}\)	\(981\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {d^{2} e^{2} A \,b^{5}}{3 i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {5 d^{3} e A \,b^{4}}{2 i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {10 d^{4} A \,b^{3}}{i^{3} \left (d a -b c \right )^{7} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {10 d^{5} A \,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 )^{7} g^{4}}-\frac {5 d^{6} A 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 )^{7} g^{4}}+\frac {d^{7} A \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 )^{7} g^{4}}-\frac {d^{2} e^{2} B \,b^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}+\frac {5 d^{3} e B \,b^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}-\frac {10 d^{4} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right )}{i^{3} \left (d a -b c \right )^{7} g^{4}}+\frac {5 d^{5} 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 )^{7} g^{4}}-\frac {5 d^{6} 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 )^{7} g^{4}}+\frac {d^{7} 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 )^{7} g^{4}}\right )}{d^{2}}\)	\(981\)
risch	\(\text {Expression too large to display}\)	\(1254\)
parallelrisch	\(\text {Expression too large to display}\)	\(1922\)
norman	\(\text {Expression too large to display}\)	\(2527\)

Input:

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x,method=_RETU 
RNVERBOSE)
 

Output:

A/g^4/i^3*(-1/2*d^3/(a*d-b*c)^4/(d*x+c)^2+10*d^3/(a*d-b*c)^6*b^2*ln(d*x+c) 
+4*d^3/(a*d-b*c)^5*b/(d*x+c)+1/3*b^2/(a*d-b*c)^3/(b*x+a)^3-10*d^3/(a*d-b*c 
)^6*b^2*ln(b*x+a)+6*b^2/(a*d-b*c)^5*d^2/(b*x+a)+3/2*b^2/(a*d-b*c)^4*d/(b*x 
+a)^2)-B/g^4/i^3*d/(a*d-b*c)^2/e^2*(d^4/(a*d-b*c)^4*(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)-5*d^3/(a*d-b*c)^4*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)+5*d^2/(a*d-b*c)^4*b^2*e^ 
2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-10*d/(a*d-b*c)^4*b^3*e^3*(-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)))+5/(a*d-b*c)^4*b^4*e^4*(-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)-1/d/(a* 
d-b*c)^4*b^5*e^5*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)* 
e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1509 vs. \(2 (551) = 1102\).

Time = 0.13 (sec) , antiderivative size = 1509, normalized size of antiderivative = 2.68 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algo 
rithm="fricas")
 

Output:

-1/36*(4*(3*A + B)*b^5*c^5 - 45*(2*A + B)*a*b^4*c^4*d + 360*(A + B)*a^2*b^ 
3*c^3*d^2 - 10*(12*A + 49*B)*a^3*b^2*c^2*d^3 - 180*(A - B)*a^4*b*c*d^4 + 9 
*(2*A - B)*a^5*d^5 + 120*((3*A + B)*b^5*c*d^4 - (3*A + B)*a*b^4*d^5)*x^4 + 
 60*(3*(3*A + 2*B)*b^5*c^2*d^3 + 2*(3*A - 2*B)*a*b^4*c*d^4 - (15*A + 2*B)* 
a^2*b^3*d^5)*x^3 + 20*((6*A + 11*B)*b^5*c^3*d^2 + 21*(3*A + B)*a*b^4*c^2*d 
^3 - 3*(12*A + 13*B)*a^2*b^3*c*d^4 - (33*A - 7*B)*a^3*b^2*d^5)*x^2 + 180*( 
B*b^5*d^5*x^5 + B*a^3*b^2*c^2*d^3 + (2*B*b^5*c*d^4 + 3*B*a*b^4*d^5)*x^4 + 
(B*b^5*c^2*d^3 + 6*B*a*b^4*c*d^4 + 3*B*a^2*b^3*d^5)*x^3 + (3*B*a*b^4*c^2*d 
^3 + 6*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*x^2 + (3*B*a^2*b^3*c^2*d^3 + 2*B*a 
^3*b^2*c*d^4)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 5*((6*A + 5*B)*b^5*c^4*d 
 - 36*(2*A + 3*B)*a*b^4*c^3*d^2 - 6*(24*A - 13*B)*a^2*b^3*c^2*d^3 + 4*(48* 
A + 13*B)*a^3*b^2*c*d^4 + 9*(2*A - 3*B)*a^4*b*d^5)*x + 6*(20*(3*A + B)*b^5 
*d^5*x^5 + 2*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 60*B*a^2*b^3*c^3*d^2 + 60*A*a^ 
3*b^2*c^2*d^3 - 30*B*a^4*b*c*d^4 + 3*B*a^5*d^5 + 20*((6*A + 5*B)*b^5*c*d^4 
 + 9*A*a*b^4*d^5)*x^4 + 10*((6*A + 11*B)*b^5*c^2*d^3 + 18*(2*A + B)*a*b^4* 
c*d^4 + 9*(2*A - B)*a^2*b^3*d^5)*x^3 + 10*(2*B*b^5*c^3*d^2 + 9*(2*A + 3*B) 
*a*b^4*c^2*d^3 + 36*A*a^2*b^3*c*d^4 + 3*(2*A - 3*B)*a^3*b^2*d^5)*x^2 - 5*( 
B*b^5*c^4*d - 12*B*a*b^4*c^3*d^2 - 36*(A + B)*a^2*b^3*c^2*d^3 - 24*(A - B) 
*a^3*b^2*c*d^4 + 3*B*a^4*b*d^5)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^6 
*d^2 - 6*a*b^8*c^5*d^3 + 15*a^2*b^7*c^4*d^4 - 20*a^3*b^6*c^3*d^5 + 15*a...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4/(d*i*x+c*i)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3816 vs. \(2 (551) = 1102\).

Time = 0.38 (sec) , antiderivative size = 3816, normalized size of antiderivative = 6.78 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algo 
rithm="maxima")
 

Output:

-1/6*B*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 
+ 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2* 
b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a*b 
^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^ 
4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^ 
3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^3 + 
10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7 
)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c 
^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7*b* 
d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a 
^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a 
^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b^4*c^5*d 
^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^ 
6)*g^4*i^3*x + (a^3*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^ 
6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2*d^3*log(b 
*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^ 
3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*lo 
g(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3 
*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(b*e*x/( 
d*x + c) + a*e/(d*x + c)) - 1/6*A*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b...
                                                                                    
                                                                                    
 

Giac [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\right )}^{4} {\left (d i x + c i\right )}^{3}} \,d x } \] Input:

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algo 
rithm="giac")
 

Output:

integrate((B*log((b*x + a)*e/(d*x + c)) + A)/((b*g*x + a*g)^4*(d*i*x + c*i 
)^3), x)
 

Mupad [B] (verification not implemented)

Time = 38.95 (sec) , antiderivative size = 2291, normalized size of antiderivative = 4.07 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)^4*(c*i + d*i*x)^3) 
,x)
 

Output:

((12*A*b^4*c^4 - 18*A*a^4*d^4 + 9*B*a^4*d^4 + 4*B*b^4*c^4 + 282*A*a^2*b^2* 
c^2*d^2 + 319*B*a^2*b^2*c^2*d^2 - 78*A*a*b^3*c^3*d + 162*A*a^3*b*c*d^3 - 4 
1*B*a*b^3*c^3*d - 171*B*a^3*b*c*d^3)/(6*(a*d - b*c)) + (10*x^2*(33*A*a^2*b 
^2*d^4 - 7*B*a^2*b^2*d^4 + 6*A*b^4*c^2*d^2 + 11*B*b^4*c^2*d^2 + 69*A*a*b^3 
*c*d^3 + 32*B*a*b^3*c*d^3))/(3*(a*d - b*c)) + (5*x*(18*A*a^3*b*d^4 - 27*B* 
a^3*b*d^4 - 6*A*b^4*c^3*d - 5*B*b^4*c^3*d + 66*A*a*b^3*c^2*d^2 + 210*A*a^2 
*b^2*c*d^3 + 103*B*a*b^3*c^2*d^2 + 25*B*a^2*b^2*c*d^3))/(6*(a*d - b*c)) + 
(10*x^3*(15*A*a*b^3*d^4 + 2*B*a*b^3*d^4 + 9*A*b^4*c*d^3 + 6*B*b^4*c*d^3))/ 
(a*d - b*c) + (20*x^4*(3*A*b^4*d^4 + B*b^4*d^4))/(a*d - b*c))/(x^5*(6*a^4* 
b^3*d^6*g^4*i^3 + 6*b^7*c^4*d^2*g^4*i^3 - 24*a*b^6*c^3*d^3*g^4*i^3 - 24*a^ 
3*b^4*c*d^5*g^4*i^3 + 36*a^2*b^5*c^2*d^4*g^4*i^3) + x*(18*a^2*b^5*c^6*g^4* 
i^3 + 12*a^7*c*d^5*g^4*i^3 - 60*a^3*b^4*c^5*d*g^4*i^3 - 30*a^6*b*c^2*d^4*g 
^4*i^3 + 60*a^4*b^3*c^4*d^2*g^4*i^3) + x^2*(6*a^7*d^6*g^4*i^3 + 18*a*b^6*c 
^6*g^4*i^3 + 12*a^6*b*c*d^5*g^4*i^3 - 36*a^2*b^5*c^5*d*g^4*i^3 - 30*a^3*b^ 
4*c^4*d^2*g^4*i^3 + 120*a^4*b^3*c^3*d^3*g^4*i^3 - 90*a^5*b^2*c^2*d^4*g^4*i 
^3) + x^3*(6*b^7*c^6*g^4*i^3 + 18*a^6*b*d^6*g^4*i^3 + 12*a*b^6*c^5*d*g^4*i 
^3 - 36*a^5*b^2*c*d^5*g^4*i^3 - 90*a^2*b^5*c^4*d^2*g^4*i^3 + 120*a^3*b^4*c 
^3*d^3*g^4*i^3 - 30*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(18*a^5*b^2*d^6*g^4*i^3 
 + 12*b^7*c^5*d*g^4*i^3 - 30*a*b^6*c^4*d^2*g^4*i^3 - 60*a^4*b^3*c*d^5*g^4* 
i^3 + 60*a^3*b^4*c^2*d^4*g^4*i^3) + 6*a^3*b^4*c^6*g^4*i^3 + 6*a^7*c^2*d...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 3982, normalized size of antiderivative = 7.07 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x)
 

Output:

(i*( - 1080*log(a + b*x)*a**5*b**2*c**2*d**4 - 2160*log(a + b*x)*a**5*b**2 
*c*d**5*x - 1080*log(a + b*x)*a**5*b**2*d**6*x**2 - 720*log(a + b*x)*a**4* 
b**3*c**3*d**3 - 4680*log(a + b*x)*a**4*b**3*c**2*d**4*x - 7200*log(a + b* 
x)*a**4*b**3*c*d**5*x**2 - 3240*log(a + b*x)*a**4*b**3*d**6*x**3 - 2160*lo 
g(a + b*x)*a**3*b**4*c**3*d**3*x - 600*log(a + b*x)*a**3*b**4*c**3*d**3 - 
7560*log(a + b*x)*a**3*b**4*c**2*d**4*x**2 - 1200*log(a + b*x)*a**3*b**4*c 
**2*d**4*x - 8640*log(a + b*x)*a**3*b**4*c*d**5*x**3 - 600*log(a + b*x)*a* 
*3*b**4*c*d**5*x**2 - 3240*log(a + b*x)*a**3*b**4*d**6*x**4 - 2160*log(a + 
 b*x)*a**2*b**5*c**3*d**3*x**2 - 1800*log(a + b*x)*a**2*b**5*c**3*d**3*x - 
 5400*log(a + b*x)*a**2*b**5*c**2*d**4*x**3 - 3600*log(a + b*x)*a**2*b**5* 
c**2*d**4*x**2 - 4320*log(a + b*x)*a**2*b**5*c*d**5*x**4 - 1800*log(a + b* 
x)*a**2*b**5*c*d**5*x**3 - 1080*log(a + b*x)*a**2*b**5*d**6*x**5 - 720*log 
(a + b*x)*a*b**6*c**3*d**3*x**3 - 1800*log(a + b*x)*a*b**6*c**3*d**3*x**2 
- 1440*log(a + b*x)*a*b**6*c**2*d**4*x**4 - 3600*log(a + b*x)*a*b**6*c**2* 
d**4*x**3 - 720*log(a + b*x)*a*b**6*c*d**5*x**5 - 1800*log(a + b*x)*a*b**6 
*c*d**5*x**4 - 600*log(a + b*x)*b**7*c**3*d**3*x**3 - 1200*log(a + b*x)*b* 
*7*c**2*d**4*x**4 - 600*log(a + b*x)*b**7*c*d**5*x**5 + 1080*log(c + d*x)* 
a**5*b**2*c**2*d**4 + 2160*log(c + d*x)*a**5*b**2*c*d**5*x + 1080*log(c + 
d*x)*a**5*b**2*d**6*x**2 + 720*log(c + d*x)*a**4*b**3*c**3*d**3 + 4680*log 
(c + d*x)*a**4*b**3*c**2*d**4*x + 7200*log(c + d*x)*a**4*b**3*c*d**5*x*...