\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) [156]

Optimal result

Integrand size = 43, antiderivative size = 381 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {B d^3 n (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 n (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B n (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^2 i^3}+\frac {3 b^2 B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^2 i^3} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.69 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {-\frac {4 b^3 B c n}{a+b x}+\frac {4 a b^2 B d n}{a+b x}+\frac {B d (b c-a d)^2 n}{(c+d x)^2}+\frac {8 b^2 B c d n}{c+d x}-\frac {8 a b B d^2 n}{c+d x}+\frac {2 b B d (b c-a d) n}{c+d x}+6 b^2 B d n \log (a+b x)-\frac {4 b^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}-\frac {2 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}-\frac {8 b d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-12 b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b^2 B d n \log (c+d x)+12 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+6 b^2 B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-6 b^2 B d n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^2 i^3} \] Input:

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

Output:

((-4*b^3*B*c*n)/(a + b*x) + (4*a*b^2*B*d*n)/(a + b*x) + (B*d*(b*c - a*d)^2 
*n)/(c + d*x)^2 + (8*b^2*B*c*d*n)/(c + d*x) - (8*a*b*B*d^2*n)/(c + d*x) + 
(2*b*B*d*(b*c - a*d)*n)/(c + d*x) + 6*b^2*B*d*n*Log[a + b*x] - (4*b^2*(b*c 
 - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (2*d*(b*c - a* 
d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 - (8*b*d*(b*c - a 
*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - 12*b^2*d*Log[a + b 
*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b^2*B*d*n*Log[c + d*x] + 12 
*b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 6*b^2*B*d*n*( 
Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog 
[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 6*b^2*B*d*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)]))/(4*(b*c - a*d)^4*g^2*i^3)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 2772, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

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. \(976\) vs. \(2(375)=750\).

Time = 30.64 (sec) , antiderivative size = 977, normalized size of antiderivative = 2.56

method	result	size
parallelrisch	\(-\frac {6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} d^{7}+12 A \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{7}-12 A x a \,b^{5} c \,d^{6} n -12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{4} c \,d^{6} n -B \,a^{3} b^{3} d^{7} n^{2}+4 B \,b^{6} c^{3} d^{4} n^{2}+2 A \,a^{3} b^{3} d^{7} n +4 A \,b^{6} c^{3} d^{4} n -18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{7} n +12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} c \,d^{6}-6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{4} d^{7} n +12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{5} n -6 B x a \,b^{5} c \,d^{6} n^{2}+24 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{6}+12 B \,a^{2} b^{4} c \,d^{6} n^{2}-15 B a \,b^{5} c^{2} d^{5} n^{2}-12 A \,a^{2} b^{4} c \,d^{6} n +6 A a \,b^{5} c^{2} d^{5} n -24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{6} n -6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{7} n +6 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} d^{7}+12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} c \,d^{6}+6 B \,x^{2} a \,b^{5} d^{7} n^{2}-6 B \,x^{2} b^{6} c \,d^{6} n^{2}+12 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{7}+24 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c \,d^{6}-12 A \,x^{2} a \,b^{5} d^{7} n +12 A \,x^{2} b^{6} c \,d^{6} n +6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} c^{2} d^{5}+9 B x \,a^{2} b^{4} d^{7} n^{2}-3 B x \,b^{6} c^{2} d^{5} n^{2}+12 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{5}-6 A x \,a^{2} b^{4} d^{7} n +18 A x \,b^{6} c^{2} d^{5} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} c^{2} d^{5}+2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{3} d^{7} n +4 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{3} d^{4} n +12 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c^{2} d^{5}}{4 i^{3} g^{2} \left (d x +c \right )^{2} \left (b x +a \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n \left (d a -b c \right ) b^{3} d^{4}}\)	\(977\)

Input:

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

Output:

-1/4*(6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*d^7+12*A*x^3*ln(e*((b*x+a)/( 
d*x+c))^n)*b^6*d^7-12*A*x*a*b^5*c*d^6*n-12*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2 
*b^4*c*d^6*n-B*a^3*b^3*d^7*n^2+4*B*b^6*c^3*d^4*n^2+2*A*a^3*b^3*d^7*n+4*A*b 
^6*c^3*d^4*n-18*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^7*n+12*B*x*ln(e*(( 
b*x+a)/(d*x+c))^n)^2*a*b^5*c*d^6-6*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d 
^7*n+12*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^5*n-6*B*x*a*b^5*c*d^6*n^2+ 
24*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^6+12*B*a^2*b^4*c*d^6*n^2-15*B*a 
*b^5*c^2*d^5*n^2-12*A*a^2*b^4*c*d^6*n+6*A*a*b^5*c^2*d^5*n-24*B*x*ln(e*((b* 
x+a)/(d*x+c))^n)*a*b^5*c*d^6*n-6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^7*n 
+6*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*d^7+12*B*x^2*ln(e*((b*x+a)/(d*x 
+c))^n)^2*b^6*c*d^6+6*B*x^2*a*b^5*d^7*n^2-6*B*x^2*b^6*c*d^6*n^2+12*A*x^2*l 
n(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^7+24*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^6* 
c*d^6-12*A*x^2*a*b^5*d^7*n+12*A*x^2*b^6*c*d^6*n+6*B*x*ln(e*((b*x+a)/(d*x+c 
))^n)^2*b^6*c^2*d^5+9*B*x*a^2*b^4*d^7*n^2-3*B*x*b^6*c^2*d^5*n^2+12*A*x*ln( 
e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^5-6*A*x*a^2*b^4*d^7*n+18*A*x*b^6*c^2*d^5* 
n+6*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c^2*d^5+2*B*ln(e*((b*x+a)/(d*x+c)) 
^n)*a^3*b^3*d^7*n+4*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d^4*n+12*A*ln(e*(( 
b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^5)/i^3/g^2/(d*x+c)^2/(b*x+a)/(a^3*d^3-3*a^2 
*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/n/(a*d-b*c)/b^3/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (375) = 750\).

Time = 0.12 (sec) , antiderivative size = 949, normalized size of antiderivative = 2.49 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**2/(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. 1724 vs. \(2 (375) = 750\).

Time = 0.14 (sec) , antiderivative size = 1724, normalized size of antiderivative = 4.52 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/2*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + 
a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5) 
*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b* 
c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 
+ 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d 
+ 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4*c^4 
 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 
 6*b^2*d*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^ 
3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/ 
4*(4*b^3*c^3 - 15*a*b^2*c^2*d + 12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - 
a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)* 
x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b^3*d^3*x^3 + a*b 
^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)* 
log(d*x + c)^2 - 3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^ 
3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2 
*c*d^2)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a* 
b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 + a*b^2*c^2* 
d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x 
 + a))*log(d*x + c))*B*n/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6* 
a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2*i^3 + a^5*c^2*d^4*g^2*i^3...
 

Giac [F]

\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3}} \,d x } \] Input:

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

Mupad [B] (verification not implemented)

Time = 28.64 (sec) , antiderivative size = 1018, normalized size of antiderivative = 2.67 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((4*A*b^2*c^2 - 2*A*a^2*d^2 + B*a^2*d^2*n + 4*B*b^2*c^2*n + 10*A*a*b*c*d - 
 11*B*a*b*c*d*n)/(2*(a*d - b*c)) + (3*x^2*(2*A*b^2*d^2 - B*b^2*d^2*n))/(a* 
d - b*c) + (3*x*(2*A*a*b*d^2 + 6*A*b^2*c*d - 3*B*a*b*d^2*n - B*b^2*c*d*n)) 
/(2*(a*d - b*c)))/(x*(2*b^3*c^4*g^2*i^3 + 4*a^3*c*d^3*g^2*i^3 - 6*a^2*b*c^ 
2*d^2*g^2*i^3) + x^2*(2*a^3*d^4*g^2*i^3 + 4*b^3*c^3*d*g^2*i^3 - 6*a*b^2*c^ 
2*d^2*g^2*i^3) + x^3*(2*b^3*c^2*d^2*g^2*i^3 + 2*a^2*b*d^4*g^2*i^3 - 4*a*b^ 
2*c*d^3*g^2*i^3) + 2*a^3*c^2*d^2*g^2*i^3 + 2*a*b^2*c^4*g^2*i^3 - 4*a^2*b*c 
^3*d*g^2*i^3) - log(e*((a + b*x)/(c + d*x))^n)*(((B*(a*d + 2*b*c))/(2*(a^2 
*d^2 + b^2*c^2 - 2*a*b*c*d)) + (3*B*b*d*x)/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c 
*d)))/(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i^3) + x^2*(a*d^2*g^2*i^3 + 2*b*c*d* 
g^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^3*x^3) - (3*B*b^2*d*(d*g^2*i^3*n*x^ 
2*(a*d - b*c) + (a*c*g^2*i^3*n*(a*d - b*c))/b + (g^2*i^3*n*x*(a*d + b*c)*( 
a*d - b*c))/b))/(g^2*i^3*n*(a*d - b*c)^4*(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i 
^3) + x^2*(a*d^2*g^2*i^3 + 2*b*c*d*g^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^ 
3*x^3))) + (b^2*d*atan((b^2*d*(2*A - B*n)*((a^4*d^4*g^2*i^3 - b^4*c^4*g^2* 
i^3 + 2*a*b^3*c^3*d*g^2*i^3 - 2*a^3*b*c*d^3*g^2*i^3)/(a^3*d^3*g^2*i^3 - b^ 
3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3 - 3*a^2*b*c*d^2*g^2*i^3) + 2*b*d*x)* 
(a^3*d^3*g^2*i^3 - b^3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3 - 3*a^2*b*c*d^2 
*g^2*i^3)*3i)/(g^2*i^3*(6*A*b^2*d - 3*B*b^2*d*n)*(a*d - b*c)^4))*(2*A - B* 
n)*3i)/(g^2*i^3*(a*d - b*c)^4) - (3*B*b^2*d*log(e*((a + b*x)/(c + d*x))...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1928, normalized size of antiderivative = 5.06 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 12*log(a + b*x)*a**3*b**2*c**2*d**2*n - 24*log(a + b*x)*a**3*b**2*c 
*d**3*n*x - 12*log(a + b*x)*a**3*b**2*d**4*n*x**2 - 24*log(a + b*x)*a**2*b 
**3*c**3*d*n + 18*log(a + b*x)*a**2*b**3*c**2*d**2*n**2 - 60*log(a + b*x)* 
a**2*b**3*c**2*d**2*n*x + 36*log(a + b*x)*a**2*b**3*c*d**3*n**2*x - 48*log 
(a + b*x)*a**2*b**3*c*d**3*n*x**2 + 18*log(a + b*x)*a**2*b**3*d**4*n**2*x* 
*2 - 12*log(a + b*x)*a**2*b**3*d**4*n*x**3 - 24*log(a + b*x)*a*b**4*c**3*d 
*n*x + 18*log(a + b*x)*a*b**4*c**2*d**2*n**2*x - 48*log(a + b*x)*a*b**4*c* 
*2*d**2*n*x**2 + 36*log(a + b*x)*a*b**4*c*d**3*n**2*x**2 - 24*log(a + b*x) 
*a*b**4*c*d**3*n*x**3 + 18*log(a + b*x)*a*b**4*d**4*n**2*x**3 + 12*log(c + 
 d*x)*a**3*b**2*c**2*d**2*n + 24*log(c + d*x)*a**3*b**2*c*d**3*n*x + 12*lo 
g(c + d*x)*a**3*b**2*d**4*n*x**2 + 24*log(c + d*x)*a**2*b**3*c**3*d*n - 18 
*log(c + d*x)*a**2*b**3*c**2*d**2*n**2 + 60*log(c + d*x)*a**2*b**3*c**2*d* 
*2*n*x - 36*log(c + d*x)*a**2*b**3*c*d**3*n**2*x + 48*log(c + d*x)*a**2*b* 
*3*c*d**3*n*x**2 - 18*log(c + d*x)*a**2*b**3*d**4*n**2*x**2 + 12*log(c + d 
*x)*a**2*b**3*d**4*n*x**3 + 24*log(c + d*x)*a*b**4*c**3*d*n*x - 18*log(c + 
 d*x)*a*b**4*c**2*d**2*n**2*x + 48*log(c + d*x)*a*b**4*c**2*d**2*n*x**2 - 
36*log(c + d*x)*a*b**4*c*d**3*n**2*x**2 + 24*log(c + d*x)*a*b**4*c*d**3*n* 
x**3 - 18*log(c + d*x)*a*b**4*d**4*n**2*x**3 - 6*log(((a + b*x)**n*e)/(c + 
 d*x)**n)**2*a**2*b**3*c**2*d**2 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)** 
2*a**2*b**3*c*d**3*x - 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**...