\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)^3} \, dx\) [158]

Optimal result

Integrand size = 43, antiderivative size = 587 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\frac {B d^5 n (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {5 b B d^4 n (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 B d^2 n (c+d x)}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B d n (c+d x)^2}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 B n (c+d x)^3}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {d^5 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^6 g^4 i^3}+\frac {5 b^2 B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^6 g^4 i^3} \] Output:

1/4*B*d^5*n*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-5*b*B*d^4*n*(b*x+a)/( 
-a*d+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*B*d^2*n*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b 
*x+a)+5/4*b^4*B*d*n*(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-1/9*b^5*B*n*( 
d*x+c)^3/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-1/2*d^5*(b*x+a)^2*(A+B*ln(e*((b*x+ 
a)/(d*x+c))^n))/(-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))^n))/(-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))^n))/(-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))^n))/(-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))^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-10 
*b^2*d^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+b*c)^6/ 
g^4/i^3+5*b^2*B*d^3*n*ln((b*x+a)/(d*x+c))^2/(-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 = 1.63 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.14 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=-\frac {\frac {4 b^2 B (b c-a d)^3 n}{(a+b x)^3}-\frac {33 b^2 B d (b c-a d)^2 n}{(a+b x)^2}+\frac {216 b^3 B c d^2 n}{a+b x}-\frac {216 a b^2 B d^3 n}{a+b x}+\frac {66 b^2 B d^2 (b c-a d) n}{a+b x}-\frac {9 B d^3 (b c-a d)^2 n}{(c+d x)^2}-\frac {144 b^2 B c d^3 n}{c+d x}+\frac {144 a b B d^4 n}{c+d x}-\frac {18 b B d^3 (b c-a d) n}{c+d x}+120 b^2 B d^3 n \log (a+b x)+\frac {12 b^2 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3}-\frac {54 b^2 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {216 b^2 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+\frac {18 d^3 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}+\frac {144 b d^3 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+360 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-120 b^2 B d^3 n \log (c+d x)-360 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-180 b^2 B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+180 b^2 B d^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{36 (b c-a d)^6 g^4 i^3} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((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*n)/(a + b*x)^3 - (33*b^2*B*d*(b*c - a*d)^2*n 
)/(a + b*x)^2 + (216*b^3*B*c*d^2*n)/(a + b*x) - (216*a*b^2*B*d^3*n)/(a + b 
*x) + (66*b^2*B*d^2*(b*c - a*d)*n)/(a + b*x) - (9*B*d^3*(b*c - a*d)^2*n)/( 
c + d*x)^2 - (144*b^2*B*c*d^3*n)/(c + d*x) + (144*a*b*B*d^4*n)/(c + d*x) - 
 (18*b*B*d^3*(b*c - a*d)*n)/(c + d*x) + 120*b^2*B*d^3*n*Log[a + b*x] + (12 
*b^2*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^3 - ( 
54*b^2*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 
 + (216*b^2*d^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b 
*x) + (18*d^3*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d 
*x)^2 + (144*b*d^3*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c 
+ d*x) + 360*b^2*d^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 
 120*b^2*B*d^3*n*Log[c + d*x] - 360*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d 
*x))^n])*Log[c + d*x] - 180*b^2*B*d^3*n*(Log[a + b*x]*(Log[a + b*x] - 2*Lo 
g[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] 
) + 180*b^2*B*d^3*n*((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 = 0.63 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.69, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-1/2*(d^5*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 
 + (5*b*d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - 
(10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + 
(5*b^4*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^ 
2) - (b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x) 
^3) - 10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + 
 d*x)] + (B*n*((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)/((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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2148\) vs. \(2(575)=1150\).

Time = 211.34 (sec) , antiderivative size = 2149, normalized size of antiderivative = 3.66

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2149\)

Input:

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

Output:

-1/36*(-720*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^7*c*d^8*n-540*B*x^2*ln(e*( 
(b*x+a)/(d*x+c))^n)*a^3*b^7*d^9*n+120*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^10 
*c^3*d^6*n-780*B*x^2*a^2*b^8*c*d^8*n^2+420*B*x^2*a*b^9*c^2*d^7*n^2+2160*A* 
x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^8*c*d^8+1080*A*x^2*ln(e*((b*x+a)/(d*x+ 
c))^n)*a*b^9*c^2*d^7-720*A*x^2*a^2*b^8*c*d^8*n+1260*A*x^2*a*b^9*c^2*d^7*n+ 
360*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b^7*c*d^8+540*B*x*ln(e*((b*x+a)/(d 
*x+c))^n)^2*a^2*b^8*c^2*d^7-90*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^6*d^9*n 
-30*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^10*c^4*d^5*n-260*B*x*a^3*b^7*c*d^8*n^2 
-390*B*x*a^2*b^8*c^2*d^7*n^2+540*B*x*a*b^9*c^3*d^6*n^2+720*A*x*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^3*b^7*c*d^8+1080*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^8*c^ 
2*d^7-960*A*x*a^3*b^7*c*d^8*n+720*A*x*a^2*b^8*c^2*d^7*n+360*A*x*a*b^9*c^3* 
d^6*n-180*B*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^6*c*d^8*n+360*B*ln(e*((b*x+a)/ 
(d*x+c))^n)*a^2*b^8*c^3*d^6*n-90*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^9*c^4*d^5 
*n+120*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^10*d^9*n+540*B*x^4*ln(e*((b*x+a)/ 
(d*x+c))^n)^2*a*b^9*d^9+360*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*b^10*c*d^8-1 
20*B*x^4*a*b^9*d^9*n^2+120*B*x^4*b^10*c*d^8*n^2+1080*A*x^4*ln(e*((b*x+a)/( 
d*x+c))^n)*a*b^9*d^9+720*A*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^10*c*d^8-360*A* 
x^4*a*b^9*d^9*n+360*A*x^4*b^10*c*d^8*n+540*B*x^3*ln(e*((b*x+a)/(d*x+c))^n) 
^2*a^2*b^8*d^9+180*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^10*c^2*d^7-120*B*x^ 
3*a^2*b^8*d^9*n^2+360*B*x^3*b^10*c^2*d^7*n^2+1080*A*x^3*ln(e*((b*x+a)/(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2181 vs. \(2 (575) = 1150\).

Time = 0.18 (sec) , antiderivative size = 2181, normalized size of antiderivative = 3.72 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/36*(12*A*b^5*c^5 - 90*A*a*b^4*c^4*d + 360*A*a^2*b^3*c^3*d^2 - 120*A*a^3 
*b^2*c^2*d^3 - 180*A*a^4*b*c*d^4 + 18*A*a^5*d^5 + 120*(3*A*b^5*c*d^4 - 3*A 
*a*b^4*d^5 + (B*b^5*c*d^4 - B*a*b^4*d^5)*n)*x^4 + 60*(9*A*b^5*c^2*d^3 + 6* 
A*a*b^4*c*d^4 - 15*A*a^2*b^3*d^5 + 2*(3*B*b^5*c^2*d^3 - 2*B*a*b^4*c*d^4 - 
B*a^2*b^3*d^5)*n)*x^3 + 20*(6*A*b^5*c^3*d^2 + 63*A*a*b^4*c^2*d^3 - 36*A*a^ 
2*b^3*c*d^4 - 33*A*a^3*b^2*d^5 + (11*B*b^5*c^3*d^2 + 21*B*a*b^4*c^2*d^3 - 
39*B*a^2*b^3*c*d^4 + 7*B*a^3*b^2*d^5)*n)*x^2 + 180*(B*b^5*d^5*n*x^5 + B*a^ 
3*b^2*c^2*d^3*n + (2*B*b^5*c*d^4 + 3*B*a*b^4*d^5)*n*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)*n*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)*n*x^2 + (3*B*a^2*b^3*c^2*d^3 + 2*B*a^3*b^2*c*d^4 
)*n*x)*log((b*x + a)/(d*x + c))^2 + (4*B*b^5*c^5 - 45*B*a*b^4*c^4*d + 360* 
B*a^2*b^3*c^3*d^2 - 490*B*a^3*b^2*c^2*d^3 + 180*B*a^4*b*c*d^4 - 9*B*a^5*d^ 
5)*n - 5*(6*A*b^5*c^4*d - 72*A*a*b^4*c^3*d^2 - 144*A*a^2*b^3*c^2*d^3 + 192 
*A*a^3*b^2*c*d^4 + 18*A*a^4*b*d^5 + (5*B*b^5*c^4*d - 108*B*a*b^4*c^3*d^2 + 
 78*B*a^2*b^3*c^2*d^3 + 52*B*a^3*b^2*c*d^4 - 27*B*a^4*b*d^5)*n)*x + 6*(2*B 
*b^5*c^5 - 15*B*a*b^4*c^4*d + 60*B*a^2*b^3*c^3*d^2 - 20*B*a^3*b^2*c^2*d^3 
- 30*B*a^4*b*c*d^4 + 3*B*a^5*d^5 + 60*(B*b^5*c*d^4 - B*a*b^4*d^5)*x^4 + 30 
*(3*B*b^5*c^2*d^3 + 2*B*a*b^4*c*d^4 - 5*B*a^2*b^3*d^5)*x^3 + 10*(2*B*b^5*c 
^3*d^2 + 21*B*a*b^4*c^2*d^3 - 12*B*a^2*b^3*c*d^4 - 11*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 - 24*B*a^2*b^3*c^2*d^3 + 32*B*a^3*...
 

Sympy [F(-1)]

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

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(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. 3819 vs. \(2 (575) = 1150\).

Time = 0.33 (sec) , antiderivative size = 3819, normalized size of antiderivative = 6.51 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, 
algorithm="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(e*(b*x/ 
(d*x + c) + a/(d*x + c))^n) - 1/36*(4*b^5*c^5 - 45*a*b^4*c^4*d + 360*a^...
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 31.38 (sec) , antiderivative size = 2400, normalized size of antiderivative = 4.09 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log(e*((a + b*x)/(c + d*x))^n)*((x*((5*B*(2*a*b*d^2 + b^2*c*d)*(a*d + b*c) 
)/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (5*B*b*d)/(6*(a^2*d^2 + b^2*c^2 
- 2*a*b*c*d)) + (5*B*a*b^2*c*d^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + x^2 
*((5*B*b*d*(2*a*b*d^2 + b^2*c*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + 
(5*B*b^2*d^2*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (B*(3*a*d + 
 2*b*c))/(6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (5*B*a*c*(2*a*b*d^2 + b^2*c 
*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (5*B*b^3*d^3*x^3)/(a^2*d^2 + 
b^2*c^2 - 2*a*b*c*d)^2)/(x*(2*a^3*c*d*g^4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2 
*(a^3*d^2*g^4*i^3 + 3*a*b^2*c^2*g^4*i^3 + 6*a^2*b*c*d*g^4*i^3) + x^3*(b^3* 
c^2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6*a*b^2*c*d*g^4*i^3) + x^4*(2*b^3*c*d* 
g^4*i^3 + 3*a*b^2*d^2*g^4*i^3) + a^3*c^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5) + 
(10*B*b^2*d^3*(x^2*((g^4*i^3*n*(a*d + b*c)^2*(a*d - b*c))/d + 2*a*b*c*g^4* 
i^3*n*(a*d - b*c)) + b^2*d*g^4*i^3*n*x^4*(a*d - b*c) + (a^2*c^2*g^4*i^3*n* 
(a*d - b*c))/d + 2*b*g^4*i^3*n*x^3*(a*d + b*c)*(a*d - b*c) + (2*a*c*g^4*i^ 
3*n*x*(a*d + b*c)*(a*d - b*c))/d))/(g^4*i^3*n*(a*d - b*c)^6*(x*(2*a^3*c*d* 
g^4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2*(a^3*d^2*g^4*i^3 + 3*a*b^2*c^2*g^4*i^ 
3 + 6*a^2*b*c*d*g^4*i^3) + x^3*(b^3*c^2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6* 
a*b^2*c*d*g^4*i^3) + x^4*(2*b^3*c*d*g^4*i^3 + 3*a*b^2*d^2*g^4*i^3) + a^3*c 
^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5))) + ((12*A*b^4*c^4 - 18*A*a^4*d^4 + 9*B* 
a^4*d^4*n + 4*B*b^4*c^4*n + 282*A*a^2*b^2*c^2*d^2 - 78*A*a*b^3*c^3*d + ...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 4291, normalized size of antiderivative = 7.31 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(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*n - 2160*log(a + b*x)*a**5*b* 
*2*c*d**5*n*x - 1080*log(a + b*x)*a**5*b**2*d**6*n*x**2 - 720*log(a + b*x) 
*a**4*b**3*c**3*d**3*n - 4680*log(a + b*x)*a**4*b**3*c**2*d**4*n*x - 7200* 
log(a + b*x)*a**4*b**3*c*d**5*n*x**2 - 3240*log(a + b*x)*a**4*b**3*d**6*n* 
x**3 - 600*log(a + b*x)*a**3*b**4*c**3*d**3*n**2 - 2160*log(a + b*x)*a**3* 
b**4*c**3*d**3*n*x - 1200*log(a + b*x)*a**3*b**4*c**2*d**4*n**2*x - 7560*l 
og(a + b*x)*a**3*b**4*c**2*d**4*n*x**2 - 600*log(a + b*x)*a**3*b**4*c*d**5 
*n**2*x**2 - 8640*log(a + b*x)*a**3*b**4*c*d**5*n*x**3 - 3240*log(a + b*x) 
*a**3*b**4*d**6*n*x**4 - 1800*log(a + b*x)*a**2*b**5*c**3*d**3*n**2*x - 21 
60*log(a + b*x)*a**2*b**5*c**3*d**3*n*x**2 - 3600*log(a + b*x)*a**2*b**5*c 
**2*d**4*n**2*x**2 - 5400*log(a + b*x)*a**2*b**5*c**2*d**4*n*x**3 - 1800*l 
og(a + b*x)*a**2*b**5*c*d**5*n**2*x**3 - 4320*log(a + b*x)*a**2*b**5*c*d** 
5*n*x**4 - 1080*log(a + b*x)*a**2*b**5*d**6*n*x**5 - 1800*log(a + b*x)*a*b 
**6*c**3*d**3*n**2*x**2 - 720*log(a + b*x)*a*b**6*c**3*d**3*n*x**3 - 3600* 
log(a + b*x)*a*b**6*c**2*d**4*n**2*x**3 - 1440*log(a + b*x)*a*b**6*c**2*d* 
*4*n*x**4 - 1800*log(a + b*x)*a*b**6*c*d**5*n**2*x**4 - 720*log(a + b*x)*a 
*b**6*c*d**5*n*x**5 - 600*log(a + b*x)*b**7*c**3*d**3*n**2*x**3 - 1200*log 
(a + b*x)*b**7*c**2*d**4*n**2*x**4 - 600*log(a + b*x)*b**7*c*d**5*n**2*x** 
5 + 1080*log(c + d*x)*a**5*b**2*c**2*d**4*n + 2160*log(c + d*x)*a**5*b**2* 
c*d**5*n*x + 1080*log(c + d*x)*a**5*b**2*d**6*n*x**2 + 720*log(c + d*x)...