\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^5} \, dx\) [125]

Optimal result

Integrand size = 43, antiderivative size = 189 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\frac {B d i^2 n (c+d x)^3}{9 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B i^2 n (c+d x)^4}{16 (b c-a d)^2 g^5 (a+b x)^4}+\frac {d i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^2 g^5 (a+b x)^4} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(189)=378\).

Time = 0.46 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.51 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^3 g^5 (a+b x)^4}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^3}-\frac {d^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^3 g^5 (a+b x)^2}-\frac {B d^2 i^2 n \left (\frac {1}{(a+b x)^2}-\frac {2 d}{(b c-a d) (a+b x)}-\frac {2 d^2 \log (a+b x)}{(b c-a d)^2}+\frac {2 d^2 \log (c+d x)}{(b c-a d)^2}\right )}{4 b^3 g^5}-\frac {B d i^2 n \left (\frac {2 (b c-a d)}{(a+b x)^3}-\frac {3 d}{(a+b x)^2}+\frac {6 d^2}{(b c-a d) (a+b x)}+\frac {6 d^3 \log (a+b x)}{(b c-a d)^2}-\frac {6 d^3 \log (c+d x)}{(b c-a d)^2}\right )}{9 b^3 g^5}-\frac {B i^2 n \left (\frac {3 (b c-a d)^2}{(a+b x)^4}-\frac {4 d (b c-a d)}{(a+b x)^3}+\frac {6 d^2}{(a+b x)^2}-\frac {12 d^3}{(b c-a d) (a+b x)}-\frac {12 d^4 \log (a+b x)}{(b c-a d)^2}+\frac {12 d^4 \log (c+d x)}{(b c-a d)^2}\right )}{48 b^3 g^5} \] Input:

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + 
b*g*x)^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77, 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, 53, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x) 
^5,x]
 

Output:

(i^2*((B*n*((4*d*(c + d*x)^3)/(3*(a + b*x)^3) - (3*b*(c + d*x)^4)/(4*(a + 
b*x)^4)))/12 + (d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*( 
a + b*x)^3) - (b*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(a 
 + b*x)^4)))/((b*c - a*d)^2*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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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. \(854\) vs. \(2(181)=362\).

Time = 29.83 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.52

method	result	size
parallelrisch	\(\frac {48 A \,x^{3} a^{7} c \,d^{4} i^{2} n +144 A \,x^{3} a^{3} b^{4} c^{5} i^{2} n +48 B \,x^{2} a^{7} c^{2} d^{3} i^{2} n^{2}+54 B \,x^{2} a^{4} b^{3} c^{5} i^{2} n^{2}+144 A \,x^{2} a^{7} c^{2} d^{3} i^{2} n +216 A \,x^{2} a^{4} b^{3} c^{5} i^{2} n +48 B x \,a^{7} c^{3} d^{2} i^{2} n^{2}+36 B x \,a^{5} b^{2} c^{5} i^{2} n^{2}+144 A x \,a^{7} c^{3} d^{2} i^{2} n +144 A x \,a^{5} b^{2} c^{5} i^{2} n +48 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} c^{4} d \,i^{2} n -36 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b \,c^{5} i^{2} n +12 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b c \,d^{4} i^{2} n -72 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b \,c^{3} d^{2} i^{2} n -96 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} b \,c^{4} d \,i^{2} n +9 B \,x^{4} a^{2} b^{5} c^{5} i^{2} n^{2}+36 A \,x^{4} a^{2} b^{5} c^{5} i^{2} n +16 B \,x^{3} a^{7} c \,d^{4} i^{2} n^{2}+36 B \,x^{3} a^{3} b^{4} c^{5} i^{2} n^{2}+7 B \,x^{4} a^{6} b c \,d^{4} i^{2} n^{2}-16 B \,x^{4} a^{3} b^{4} c^{4} d \,i^{2} n^{2}+12 A \,x^{4} a^{6} b c \,d^{4} i^{2} n -48 A \,x^{4} a^{3} b^{4} c^{4} d \,i^{2} n +48 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} c \,d^{4} i^{2} n +12 B \,x^{3} a^{6} b \,c^{2} d^{3} i^{2} n^{2}-64 B \,x^{3} a^{4} b^{3} c^{4} d \,i^{2} n^{2}-192 A \,x^{3} a^{4} b^{3} c^{4} d \,i^{2} n +144 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} c^{2} d^{3} i^{2} n -6 B \,x^{2} a^{6} b \,c^{3} d^{2} i^{2} n^{2}-96 B \,x^{2} a^{5} b^{2} c^{4} d \,i^{2} n^{2}-72 A \,x^{2} a^{6} b \,c^{3} d^{2} i^{2} n -288 A \,x^{2} a^{5} b^{2} c^{4} d \,i^{2} n +144 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{7} c^{3} d^{2} i^{2} n -84 B x \,a^{6} b \,c^{4} d \,i^{2} n^{2}-288 A x \,a^{6} b \,c^{4} d \,i^{2} n}{144 g^{5} \left (b x +a \right )^{4} n \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) a^{6} c}\)	\(855\)

Input:

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x,method=_ 
RETURNVERBOSE)
 

Output:

1/144*(48*A*x^3*a^7*c*d^4*i^2*n+144*A*x^3*a^3*b^4*c^5*i^2*n+48*B*x^2*a^7*c 
^2*d^3*i^2*n^2+54*B*x^2*a^4*b^3*c^5*i^2*n^2+144*A*x^2*a^7*c^2*d^3*i^2*n+21 
6*A*x^2*a^4*b^3*c^5*i^2*n+48*B*x*a^7*c^3*d^2*i^2*n^2+36*B*x*a^5*b^2*c^5*i^ 
2*n^2+144*A*x*a^7*c^3*d^2*i^2*n+144*A*x*a^5*b^2*c^5*i^2*n+48*B*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^7*c^4*d*i^2*n-36*B*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c^5*i^ 
2*n+12*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c*d^4*i^2*n-72*B*x^2*ln(e*((b 
*x+a)/(d*x+c))^n)*a^6*b*c^3*d^2*i^2*n-96*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6 
*b*c^4*d*i^2*n+9*B*x^4*a^2*b^5*c^5*i^2*n^2+36*A*x^4*a^2*b^5*c^5*i^2*n+16*B 
*x^3*a^7*c*d^4*i^2*n^2+36*B*x^3*a^3*b^4*c^5*i^2*n^2+7*B*x^4*a^6*b*c*d^4*i^ 
2*n^2-16*B*x^4*a^3*b^4*c^4*d*i^2*n^2+12*A*x^4*a^6*b*c*d^4*i^2*n-48*A*x^4*a 
^3*b^4*c^4*d*i^2*n+48*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*c*d^4*i^2*n+12*B 
*x^3*a^6*b*c^2*d^3*i^2*n^2-64*B*x^3*a^4*b^3*c^4*d*i^2*n^2-192*A*x^3*a^4*b^ 
3*c^4*d*i^2*n+144*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*c^2*d^3*i^2*n-6*B*x^ 
2*a^6*b*c^3*d^2*i^2*n^2-96*B*x^2*a^5*b^2*c^4*d*i^2*n^2-72*A*x^2*a^6*b*c^3* 
d^2*i^2*n-288*A*x^2*a^5*b^2*c^4*d*i^2*n+144*B*x*ln(e*((b*x+a)/(d*x+c))^n)* 
a^7*c^3*d^2*i^2*n-84*B*x*a^6*b*c^4*d*i^2*n^2-288*A*x*a^6*b*c^4*d*i^2*n)/g^ 
5/(b*x+a)^4/n/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^6/c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (181) = 362\).

Time = 0.11 (sec) , antiderivative size = 710, normalized size of antiderivative = 3.76 \[ \int \frac {(c i+d i x)^2 \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^{2} n x^{3} - {\left (9 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 7 \, B a^{4} d^{4}\right )} i^{2} n - 12 \, {\left (3 \, A b^{4} c^{4} - 4 \, A a b^{3} c^{3} d + A a^{4} d^{4}\right )} i^{2} - 6 \, {\left ({\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^{2} n + 12 \, {\left (A b^{4} c^{2} d^{2} - 2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} i^{2}\right )} x^{2} - 4 \, {\left ({\left (5 \, B b^{4} c^{3} d - 12 \, B a b^{3} c^{2} d^{2} + 7 \, B a^{3} b d^{4}\right )} i^{2} n + 12 \, {\left (2 \, A b^{4} c^{3} d - 3 \, A a b^{3} c^{2} d^{2} + A a^{3} b d^{4}\right )} i^{2}\right )} x - 12 \, {\left (6 \, {\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} i^{2} x^{2} + 4 \, {\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + B a^{3} b d^{4}\right )} i^{2} x + {\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + B a^{4} d^{4}\right )} i^{2}\right )} \log \left (e\right ) + 12 \, {\left (B b^{4} d^{4} i^{2} n x^{4} + 4 \, B a b^{3} d^{4} i^{2} n x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 2 \, B a b^{3} c d^{3}\right )} i^{2} n x^{2} - 4 \, {\left (2 \, B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2}\right )} i^{2} n x - {\left (3 \, B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{144 \, {\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, 
algorithm="fricas")
 

Output:

1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i^2*n*x^3 - (9*B*b^4*c^4 - 16*B*a*b^ 
3*c^3*d + 7*B*a^4*d^4)*i^2*n - 12*(3*A*b^4*c^4 - 4*A*a*b^3*c^3*d + A*a^4*d 
^4)*i^2 - 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^2*n + 1 
2*(A*b^4*c^2*d^2 - 2*A*a*b^3*c*d^3 + A*a^2*b^2*d^4)*i^2)*x^2 - 4*((5*B*b^4 
*c^3*d - 12*B*a*b^3*c^2*d^2 + 7*B*a^3*b*d^4)*i^2*n + 12*(2*A*b^4*c^3*d - 3 
*A*a*b^3*c^2*d^2 + A*a^3*b*d^4)*i^2)*x - 12*(6*(B*b^4*c^2*d^2 - 2*B*a*b^3* 
c*d^3 + B*a^2*b^2*d^4)*i^2*x^2 + 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2 + B* 
a^3*b*d^4)*i^2*x + (3*B*b^4*c^4 - 4*B*a*b^3*c^3*d + B*a^4*d^4)*i^2)*log(e) 
 + 12*(B*b^4*d^4*i^2*n*x^4 + 4*B*a*b^3*d^4*i^2*n*x^3 - 6*(B*b^4*c^2*d^2 - 
2*B*a*b^3*c*d^3)*i^2*n*x^2 - 4*(2*B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2)*i^2*n*x 
 - (3*B*b^4*c^4 - 4*B*a*b^3*c^3*d)*i^2*n)*log((b*x + a)/(d*x + c)))/((b^9* 
c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + 
a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x 
^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 
2*a^5*b^4*c*d + a^6*b^3*d^2)*g^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)**2*(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. 2247 vs. \(2 (181) = 362\).

Time = 0.14 (sec) , antiderivative size = 2247, normalized size of antiderivative = 11.89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, 
algorithm="maxima")
 

Output:

1/48*B*c^2*i^2*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^2*i^2*n*((13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c 
*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6* 
(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(1 
0*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10 
*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c 
^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8 
*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b 
^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6 
*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c 
^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d ...
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.32 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {1}{144} \, {\left (\frac {12 \, {\left (3 \, B b i^{2} n - \frac {4 \, {\left (b x + a\right )} B d i^{2} n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {9 \, B b i^{2} n - \frac {16 \, {\left (b x + a\right )} B d i^{2} n}{d x + c} + 36 \, B b i^{2} \log \left (e\right ) - \frac {48 \, {\left (b x + a\right )} B d i^{2} \log \left (e\right )}{d x + c} + 36 \, A b i^{2} - \frac {48 \, {\left (b x + a\right )} A d i^{2}}{d x + c}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, 
algorithm="giac")
 

Output:

-1/144*(12*(3*B*b*i^2*n - 4*(b*x + a)*B*d*i^2*n/(d*x + c))*log((b*x + a)/( 
d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c) 
^4) + (9*B*b*i^2*n - 16*(b*x + a)*B*d*i^2*n/(d*x + c) + 36*B*b*i^2*log(e) 
- 48*(b*x + a)*B*d*i^2*log(e)/(d*x + c) + 36*A*b*i^2 - 48*(b*x + a)*A*d*i^ 
2/(d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*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 = 27.08 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.45 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3\,i^2-36\,A\,b^3\,c^3\,i^2+7\,B\,a^3\,d^3\,i^2\,n-9\,B\,b^3\,c^3\,i^2\,n+12\,A\,a\,b^2\,c^2\,d\,i^2+12\,A\,a^2\,b\,c\,d^2\,i^2+7\,B\,a\,b^2\,c^2\,d\,i^2\,n+7\,B\,a^2\,b\,c\,d^2\,i^2\,n}{12\,\left (a\,d-b\,c\right )}+\frac {x\,\left (12\,A\,a^2\,b\,d^3\,i^2-24\,A\,b^3\,c^2\,d\,i^2+12\,A\,a\,b^2\,c\,d^2\,i^2+7\,B\,a^2\,b\,d^3\,i^2\,n-5\,B\,b^3\,c^2\,d\,i^2\,n+7\,B\,a\,b^2\,c\,d^2\,i^2\,n\right )}{3\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (12\,A\,a\,b^2\,d^3\,i^2-12\,A\,b^3\,c\,d^2\,i^2+7\,B\,a\,b^2\,d^3\,i^2\,n-B\,b^3\,c\,d^2\,i^2\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^3\,d^3\,i^2\,n\,x^3}{a\,d-b\,c}}{12\,a^4\,b^3\,g^5+48\,a^3\,b^4\,g^5\,x+72\,a^2\,b^5\,g^5\,x^2+48\,a\,b^6\,g^5\,x^3+12\,b^7\,g^5\,x^4}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^3}+\frac {B\,c\,d\,i^2}{6\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{12\,b^3}+\frac {B\,c\,d\,i^2}{6\,b^2}\right )+\frac {B\,a\,d^2\,i^2}{4\,b^2}+\frac {B\,c\,d\,i^2}{2\,b}\right )+\frac {B\,c^2\,i^2}{4\,b}+\frac {B\,d^2\,i^2\,x^2}{2\,b}\right )}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {B\,d^4\,i^2\,n\,\mathrm {atanh}\left (\frac {12\,b^5\,c^2\,g^5-12\,a^2\,b^3\,d^2\,g^5}{12\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{6\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2} \] Input:

int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x) 
^5,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1333, normalized size of antiderivative = 7.05 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x)
 

Output:

( - 24*log(a + b*x)*a**5*b**2*c*d**3*n + 12*log(a + b*x)*a**4*b**3*c**2*d* 
*2*n - 96*log(a + b*x)*a**4*b**3*c*d**3*n*x + 48*log(a + b*x)*a**3*b**4*c* 
*2*d**2*n*x - 144*log(a + b*x)*a**3*b**4*c*d**3*n*x**2 + 72*log(a + b*x)*a 
**2*b**5*c**2*d**2*n*x**2 - 96*log(a + b*x)*a**2*b**5*c*d**3*n*x**3 + 48*l 
og(a + b*x)*a*b**6*c**2*d**2*n*x**3 - 24*log(a + b*x)*a*b**6*c*d**3*n*x**4 
 + 12*log(a + b*x)*b**7*c**2*d**2*n*x**4 + 24*log(c + d*x)*a**5*b**2*c*d** 
3*n - 12*log(c + d*x)*a**4*b**3*c**2*d**2*n + 96*log(c + d*x)*a**4*b**3*c* 
d**3*n*x - 48*log(c + d*x)*a**3*b**4*c**2*d**2*n*x + 144*log(c + d*x)*a**3 
*b**4*c*d**3*n*x**2 - 72*log(c + d*x)*a**2*b**5*c**2*d**2*n*x**2 + 96*log( 
c + d*x)*a**2*b**5*c*d**3*n*x**3 - 48*log(c + d*x)*a*b**6*c**2*d**2*n*x**3 
 + 24*log(c + d*x)*a*b**6*c*d**3*n*x**4 - 12*log(c + d*x)*b**7*c**2*d**2*n 
*x**4 + 24*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*b**2*c*d**3 - 12*log((( 
a + b*x)**n*e)/(c + d*x)**n)*a**4*b**3*c**2*d**2 + 96*log(((a + b*x)**n*e) 
/(c + d*x)**n)*a**4*b**3*c*d**3*x - 48*log(((a + b*x)**n*e)/(c + d*x)**n)* 
a**3*b**4*c**3*d - 192*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**4*c**2*d 
**2*x - 48*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**4*d**4*x**3 + 36*log 
(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**5*c**4 + 96*log(((a + b*x)**n*e)/( 
c + d*x)**n)*a**2*b**5*c**3*d*x + 96*log(((a + b*x)**n*e)/(c + d*x)**n)*a* 
*2*b**5*c*d**3*x**3 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**5*d**4 
*x**4 - 48*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**6*c**2*d**2*x**3 + 2...