\(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^5} \, dx\) [116]

Optimal result

Integrand size = 41, antiderivative size = 281 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {B d^2 i n (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b B d i n (c+d x)^3}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i n (c+d x)^4}{16 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^5 (a+b x)^4} \] Output:

-1/4*B*d^2*i*n*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+2/9*b*B*d*i*n*(d*x+c)^ 
3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/16*b^2*B*i*n*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b* 
x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^ 
5/(b*x+a)^2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c) 
^3/g^5/(b*x+a)^3-1/4*b^2*i*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d 
+b*c)^3/g^5/(b*x+a)^4
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {i \left (\frac {36 A b c}{(a+b x)^4}-\frac {36 a A d}{(a+b x)^4}+\frac {9 b B c n}{(a+b x)^4}-\frac {9 a B d n}{(a+b x)^4}+\frac {48 A d}{(a+b x)^3}+\frac {4 B d n}{(a+b x)^3}-\frac {6 B d^2 n}{(b c-a d) (a+b x)^2}+\frac {12 B d^3 n}{(b c-a d)^2 (a+b x)}+\frac {12 B d^4 n \log (a+b x)}{(b c-a d)^3}+\frac {12 B (3 b c+a d+4 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4}-\frac {12 B d^4 n \log (c+d x)}{(b c-a d)^3}\right )}{144 b^2 g^5} \] Input:

Integrate[((c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b* 
g*x)^5,x]
 

Output:

-1/144*(i*((36*A*b*c)/(a + b*x)^4 - (36*a*A*d)/(a + b*x)^4 + (9*b*B*c*n)/( 
a + b*x)^4 - (9*a*B*d*n)/(a + b*x)^4 + (48*A*d)/(a + b*x)^3 + (4*B*d*n)/(a 
 + b*x)^3 - (6*B*d^2*n)/((b*c - a*d)*(a + b*x)^2) + (12*B*d^3*n)/((b*c - a 
*d)^2*(a + b*x)) + (12*B*d^4*n*Log[a + b*x])/(b*c - a*d)^3 + (12*B*(3*b*c 
+ a*d + 4*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^4 - (12*B*d^4*n 
*Log[c + d*x])/(b*c - a*d)^3))/(b^2*g^5)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2961, 2772, 27, 1140, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

\(\displaystyle \frac {i \left (-B n \int -\frac {(c+d x)^5 \left (3 b^2-\frac {8 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{12 (a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i \left (\frac {1}{12} B n \int \frac {(c+d x)^5 \left (3 b^2-\frac {8 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^5}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\)

\(\Big \downarrow \) 1140

\(\displaystyle \frac {i \left (\frac {1}{12} B n \int \left (\frac {3 b^2 (c+d x)^5}{(a+b x)^5}-\frac {8 b d (c+d x)^4}{(a+b x)^4}+\frac {6 d^2 (c+d x)^3}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \left (-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {1}{12} B n \left (-\frac {3 b^2 (c+d x)^4}{4 (a+b x)^4}-\frac {3 d^2 (c+d x)^2}{(a+b x)^2}+\frac {8 b d (c+d x)^3}{3 (a+b x)^3}\right )\right )}{g^5 (b c-a d)^3}\)

Input:

Int[((c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^5 
,x]
 

Output:

(i*((B*n*((-3*d^2*(c + d*x)^2)/(a + b*x)^2 + (8*b*d*(c + d*x)^3)/(3*(a + b 
*x)^3) - (3*b^2*(c + d*x)^4)/(4*(a + b*x)^4)))/12 - (d^2*(c + d*x)^2*(A + 
B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + (2*b*d*(c + d*x)^3*(A 
 + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) - (b^2*(c + d*x)^4*( 
A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(a + b*x)^4)))/((b*c - a*d)^3*g^ 
5)
 

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 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

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. \(985\) vs. \(2(269)=538\).

Time = 29.72 (sec) , antiderivative size = 986, normalized size of antiderivative = 3.51

method	result	size
parallelrisch	\(\frac {48 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b c \,d^{4} i n -144 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b \,c^{3} d^{2} i n +48 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c^{4} d i n -9 B \,x^{4} a^{2} b^{6} c^{5} i \,n^{2}-36 A \,x^{4} a^{2} b^{6} c^{5} i n -36 B \,x^{3} a^{3} b^{5} c^{5} i \,n^{2}-144 A \,x^{3} a^{3} b^{5} c^{5} i n +36 B \,x^{2} a^{8} c \,d^{4} i \,n^{2}-54 B \,x^{2} a^{4} b^{4} c^{5} i \,n^{2}+72 A \,x^{2} a^{8} c \,d^{4} i n -216 A \,x^{2} a^{4} b^{4} c^{5} i n +72 B x \,a^{8} c^{2} d^{3} i \,n^{2}-36 B x \,a^{5} b^{3} c^{5} i \,n^{2}+144 A x \,a^{8} c^{2} d^{3} i n -144 A x \,a^{5} b^{3} c^{5} i n +72 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c^{3} d^{2} i n +36 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c^{5} i n +128 B \,x^{3} a^{4} b^{4} c^{4} d i \,n^{2}+48 A \,x^{3} a^{7} b c \,d^{4} i n -288 A \,x^{3} a^{5} b^{3} c^{3} d^{2} i n +384 A \,x^{3} a^{4} b^{4} c^{4} d i n +72 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c \,d^{4} i n +48 B \,x^{2} a^{7} b \,c^{2} d^{3} i \,n^{2}-222 B \,x^{2} a^{6} b^{2} c^{3} d^{2} i \,n^{2}+192 B \,x^{2} a^{5} b^{3} c^{4} d i \,n^{2}-432 A \,x^{2} a^{6} b^{2} c^{3} d^{2} i n +576 A \,x^{2} a^{5} b^{3} c^{4} d i n +144 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{8} c^{2} d^{3} i n -168 B x \,a^{7} b \,c^{3} d^{2} i \,n^{2}+132 B x \,a^{6} b^{2} c^{4} d i \,n^{2}-432 A x \,a^{7} b \,c^{3} d^{2} i n +432 A x \,a^{6} b^{2} c^{4} d i n -96 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} b \,c^{4} d i n +12 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b^{2} c \,d^{4} i n +13 B \,x^{4} a^{6} b^{2} c \,d^{4} i \,n^{2}-36 B \,x^{4} a^{4} b^{4} c^{3} d^{2} i \,n^{2}+32 B \,x^{4} a^{3} b^{5} c^{4} d i \,n^{2}+12 A \,x^{4} a^{6} b^{2} c \,d^{4} i n -72 A \,x^{4} a^{4} b^{4} c^{3} d^{2} i n +96 A \,x^{4} a^{3} b^{5} c^{4} d i n +40 B \,x^{3} a^{7} b c \,d^{4} i \,n^{2}+12 B \,x^{3} a^{6} b^{2} c^{2} d^{3} i \,n^{2}-144 B \,x^{3} a^{5} b^{3} c^{3} d^{2} i \,n^{2}}{144 g^{5} \left (b x +a \right )^{4} n \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{6} c}\)	\(986\)

Input:

int((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x,method=_RE 
TURNVERBOSE)
 

Output:

1/144*(48*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c*d^4*i*n-144*B*x*ln(e*((b 
*x+a)/(d*x+c))^n)*a^7*b*c^3*d^2*i*n+48*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b 
^2*c^4*d*i*n-9*B*x^4*a^2*b^6*c^5*i*n^2-36*A*x^4*a^2*b^6*c^5*i*n-36*B*x^3*a 
^3*b^5*c^5*i*n^2-144*A*x^3*a^3*b^5*c^5*i*n+36*B*x^2*a^8*c*d^4*i*n^2-54*B*x 
^2*a^4*b^4*c^5*i*n^2+72*A*x^2*a^8*c*d^4*i*n-216*A*x^2*a^4*b^4*c^5*i*n+72*B 
*x*a^8*c^2*d^3*i*n^2-36*B*x*a^5*b^3*c^5*i*n^2+144*A*x*a^8*c^2*d^3*i*n-144* 
A*x*a^5*b^3*c^5*i*n+72*B*ln(e*((b*x+a)/(d*x+c))^n)*a^8*c^3*d^2*i*n+36*B*ln 
(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^5*i*n+128*B*x^3*a^4*b^4*c^4*d*i*n^2+48*A 
*x^3*a^7*b*c*d^4*i*n-288*A*x^3*a^5*b^3*c^3*d^2*i*n+384*A*x^3*a^4*b^4*c^4*d 
*i*n+72*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^8*c*d^4*i*n+48*B*x^2*a^7*b*c^2*d 
^3*i*n^2-222*B*x^2*a^6*b^2*c^3*d^2*i*n^2+192*B*x^2*a^5*b^3*c^4*d*i*n^2-432 
*A*x^2*a^6*b^2*c^3*d^2*i*n+576*A*x^2*a^5*b^3*c^4*d*i*n+144*B*x*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^8*c^2*d^3*i*n-168*B*x*a^7*b*c^3*d^2*i*n^2+132*B*x*a^6*b^2 
*c^4*d*i*n^2-432*A*x*a^7*b*c^3*d^2*i*n+432*A*x*a^6*b^2*c^4*d*i*n-96*B*ln(e 
*((b*x+a)/(d*x+c))^n)*a^7*b*c^4*d*i*n+12*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a 
^6*b^2*c*d^4*i*n+13*B*x^4*a^6*b^2*c*d^4*i*n^2-36*B*x^4*a^4*b^4*c^3*d^2*i*n 
^2+32*B*x^4*a^3*b^5*c^4*d*i*n^2+12*A*x^4*a^6*b^2*c*d^4*i*n-72*A*x^4*a^4*b^ 
4*c^3*d^2*i*n+96*A*x^4*a^3*b^5*c^4*d*i*n+40*B*x^3*a^7*b*c*d^4*i*n^2+12*B*x 
^3*a^6*b^2*c^2*d^3*i*n^2-144*B*x^3*a^5*b^3*c^3*d^2*i*n^2)/g^5/(b*x+a)^4/n/ 
(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^6/c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (269) = 538\).

Time = 0.11 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.75 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i n x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} i n x^{2} + {\left (9 \, B b^{4} c^{4} - 32 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 13 \, B a^{4} d^{4}\right )} i n + 12 \, {\left (3 \, A b^{4} c^{4} - 8 \, A a b^{3} c^{3} d + 6 \, A a^{2} b^{2} c^{2} d^{2} - A a^{4} d^{4}\right )} i + 4 \, {\left ({\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} i n + 12 \, {\left (A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + 3 \, A a^{2} b^{2} c d^{3} - A a^{3} b d^{4}\right )} i\right )} x + 12 \, {\left (4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} i x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - B a^{4} d^{4}\right )} i\right )} \log \left (e\right ) + 12 \, {\left (B b^{4} d^{4} i n x^{4} + 4 \, B a b^{3} d^{4} i n x^{3} + 6 \, B a^{2} b^{2} d^{4} i n x^{2} + 4 \, {\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i n x + {\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, al 
gorithm="fricas")
 

Output:

-1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i*n*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a* 
b^3*c*d^3 + 7*B*a^2*b^2*d^4)*i*n*x^2 + (9*B*b^4*c^4 - 32*B*a*b^3*c^3*d + 3 
6*B*a^2*b^2*c^2*d^2 - 13*B*a^4*d^4)*i*n + 12*(3*A*b^4*c^4 - 8*A*a*b^3*c^3* 
d + 6*A*a^2*b^2*c^2*d^2 - A*a^4*d^4)*i + 4*((B*b^4*c^3*d - 6*B*a*b^3*c^2*d 
^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*i*n + 12*(A*b^4*c^3*d - 3*A*a*b^ 
3*c^2*d^2 + 3*A*a^2*b^2*c*d^3 - A*a^3*b*d^4)*i)*x + 12*(4*(B*b^4*c^3*d - 3 
*B*a*b^3*c^2*d^2 + 3*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*i*x + (3*B*b^4*c^4 - 8 
*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2 - B*a^4*d^4)*i)*log(e) + 12*(B*b^4*d^ 
4*i*n*x^4 + 4*B*a*b^3*d^4*i*n*x^3 + 6*B*a^2*b^2*d^4*i*n*x^2 + 4*(B*b^4*c^3 
*d - 3*B*a*b^3*c^2*d^2 + 3*B*a^2*b^2*c*d^3)*i*n*x + (3*B*b^4*c^4 - 8*B*a*b 
^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*i*n)*log((b*x + a)/(d*x + c)))/((b^9*c^3 - 
 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3 
*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 
 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 
 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 
 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (269) = 538\).

Time = 0.10 (sec) , antiderivative size = 1398, normalized size of antiderivative = 4.98 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, al 
gorithm="maxima")
 

Output:

1/48*B*c*i*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^ 
2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2* 
c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3 
*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4 
*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a 
^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - 
 a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a 
^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3 
*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c 
^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) 
) - 1/144*B*d*i*n*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a 
^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^ 
2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 
- 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d 
^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d 
^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4 
*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b 
^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2 
*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d 
 + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*...
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.40 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b^{2} i n - \frac {8 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {9 \, B b^{2} i n - \frac {32 \, {\left (b x + a\right )} B b d i n}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B d^{2} i n}{{\left (d x + c\right )}^{2}} + 36 \, B b^{2} i \log \left (e\right ) - \frac {96 \, {\left (b x + a\right )} B b d i \log \left (e\right )}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} B d^{2} i \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 36 \, A b^{2} i - \frac {96 \, {\left (b x + a\right )} A b d i}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A d^{2} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b x + a\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b x + a\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, al 
gorithm="giac")
 

Output:

-1/144*(12*(3*B*b^2*i*n - 8*(b*x + a)*B*b*d*i*n/(d*x + c) + 6*(b*x + a)^2* 
B*d^2*i*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/((b*x + a)^4*b^2*c^2*g^5/( 
d*x + c)^4 - 2*(b*x + a)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*x + a)^4*a^2*d^2*g 
^5/(d*x + c)^4) + (9*B*b^2*i*n - 32*(b*x + a)*B*b*d*i*n/(d*x + c) + 36*(b* 
x + a)^2*B*d^2*i*n/(d*x + c)^2 + 36*B*b^2*i*log(e) - 96*(b*x + a)*B*b*d*i* 
log(e)/(d*x + c) + 72*(b*x + a)^2*B*d^2*i*log(e)/(d*x + c)^2 + 36*A*b^2*i 
- 96*(b*x + a)*A*b*d*i/(d*x + c) + 72*(b*x + a)^2*A*d^2*i/(d*x + c)^2)/((b 
*x + a)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*x + a)^4*a*b*c*d*g^5/(d*x + c)^4 
+ (b*x + a)^4*a^2*d^2*g^5/(d*x + c)^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a* 
d)^2)
 

Mupad [B] (verification not implemented)

Time = 26.91 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.17 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\frac {B\,d^4\,i\,n\,\mathrm {atanh}\left (\frac {12\,a^3\,b^2\,d^3\,g^5-12\,a^2\,b^3\,c\,d^2\,g^5-12\,a\,b^4\,c^2\,d\,g^5+12\,b^5\,c^3\,g^5}{12\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{6\,b^2\,g^5\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{4\,b}+\frac {B\,a\,d\,i}{12\,b^2}+\frac {B\,d\,i\,x}{3\,b}\right )}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {\frac {12\,A\,a^3\,d^3\,i+36\,A\,b^3\,c^3\,i+13\,B\,a^3\,d^3\,i\,n+9\,B\,b^3\,c^3\,i\,n-60\,A\,a\,b^2\,c^2\,d\,i+12\,A\,a^2\,b\,c\,d^2\,i-23\,B\,a\,b^2\,c^2\,d\,i\,n+13\,B\,a^2\,b\,c\,d^2\,i\,n}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i+12\,A\,b^3\,c^2\,d\,i-24\,A\,a\,b^2\,c\,d^2\,i+13\,B\,a^2\,b\,d^3\,i\,n+B\,b^3\,c^2\,d\,i\,n-5\,B\,a\,b^2\,c\,d^2\,i\,n\right )}{3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {d\,x^2\,\left (B\,b^3\,c\,d\,i\,n-7\,B\,a\,b^2\,d^2\,i\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^3\,d^3\,i\,n\,x^3}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{12\,a^4\,b^2\,g^5+48\,a^3\,b^3\,g^5\,x+72\,a^2\,b^4\,g^5\,x^2+48\,a\,b^5\,g^5\,x^3+12\,b^6\,g^5\,x^4} \] Input:

int(((c*i + d*i*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x)^5 
,x)
 

Output:

(B*d^4*i*n*atanh((12*b^5*c^3*g^5 + 12*a^3*b^2*d^3*g^5 - 12*a*b^4*c^2*d*g^5 
 - 12*a^2*b^3*c*d^2*g^5)/(12*b^2*g^5*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + 
b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(6*b^2*g^5*(a*d - b*c)^3) - (log(e*( 
(a + b*x)/(c + d*x))^n)*((B*c*i)/(4*b) + (B*a*d*i)/(12*b^2) + (B*d*i*x)/(3 
*b)))/(a^4*g^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2*b^2*g^5*x^2 + 4*a^3 
*b*g^5*x) - ((12*A*a^3*d^3*i + 36*A*b^3*c^3*i + 13*B*a^3*d^3*i*n + 9*B*b^3 
*c^3*i*n - 60*A*a*b^2*c^2*d*i + 12*A*a^2*b*c*d^2*i - 23*B*a*b^2*c^2*d*i*n 
+ 13*B*a^2*b*c*d^2*i*n)/(12*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(12*A*a^ 
2*b*d^3*i + 12*A*b^3*c^2*d*i - 24*A*a*b^2*c*d^2*i + 13*B*a^2*b*d^3*i*n + B 
*b^3*c^2*d*i*n - 5*B*a*b^2*c*d^2*i*n))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) 
 - (d*x^2*(B*b^3*c*d*i*n - 7*B*a*b^2*d^2*i*n))/(2*(a^2*d^2 + b^2*c^2 - 2*a 
*b*c*d)) + (B*b^3*d^3*i*n*x^3)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(12*a^4*b^ 
2*g^5 + 12*b^6*g^5*x^4 + 48*a^3*b^3*g^5*x + 48*a*b^5*g^5*x^3 + 72*a^2*b^4* 
g^5*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 994, normalized size of antiderivative = 3.54 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x)
 

Output:

(i*(12*log(a + b*x)*a**5*b*d**4*n + 48*log(a + b*x)*a**4*b**2*d**4*n*x + 7 
2*log(a + b*x)*a**3*b**3*d**4*n*x**2 + 48*log(a + b*x)*a**2*b**4*d**4*n*x* 
*3 + 12*log(a + b*x)*a*b**5*d**4*n*x**4 - 12*log(c + d*x)*a**5*b*d**4*n - 
48*log(c + d*x)*a**4*b**2*d**4*n*x - 72*log(c + d*x)*a**3*b**3*d**4*n*x**2 
 - 48*log(c + d*x)*a**2*b**4*d**4*n*x**3 - 12*log(c + d*x)*a*b**5*d**4*n*x 
**4 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*b*d**4 - 48*log(((a + b*x 
)**n*e)/(c + d*x)**n)*a**4*b**2*d**4*x + 72*log(((a + b*x)**n*e)/(c + d*x) 
**n)*a**3*b**3*c**2*d**2 + 144*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b** 
3*c*d**3*x - 96*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c**3*d - 144* 
log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c**2*d**2*x + 36*log(((a + b* 
x)**n*e)/(c + d*x)**n)*a*b**5*c**4 + 48*log(((a + b*x)**n*e)/(c + d*x)**n) 
*a*b**5*c**3*d*x - 12*a**6*d**4 - 10*a**5*b*d**4*n - 48*a**5*b*d**4*x + 72 
*a**4*b**2*c**2*d**2 - 3*a**4*b**2*c*d**3*n + 144*a**4*b**2*c*d**3*x - 40* 
a**4*b**2*d**4*n*x - 96*a**3*b**3*c**3*d + 36*a**3*b**3*c**2*d**2*n - 144* 
a**3*b**3*c**2*d**2*x + 60*a**3*b**3*c*d**3*n*x - 24*a**3*b**3*d**4*n*x**2 
 + 36*a**2*b**4*c**4 - 32*a**2*b**4*c**3*d*n + 48*a**2*b**4*c**3*d*x - 24* 
a**2*b**4*c**2*d**2*n*x + 30*a**2*b**4*c*d**3*n*x**2 + 9*a*b**5*c**4*n + 4 
*a*b**5*c**3*d*n*x - 6*a*b**5*c**2*d**2*n*x**2 + 3*a*b**5*d**4*n*x**4 - 3* 
b**6*c*d**3*n*x**4))/(144*a*b**2*g**5*(a**7*d**3 - 3*a**6*b*c*d**2 + 4*a** 
6*b*d**3*x + 3*a**5*b**2*c**2*d - 12*a**5*b**2*c*d**2*x + 6*a**5*b**2*d...