\(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^5} \, dx\) [167]

Optimal result

Integrand size = 43, antiderivative size = 475 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {B^2 d^2 i n^2 (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B^2 d i n^2 (c+d x)^3}{27 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B^2 i n^2 (c+d x)^4}{32 (b c-a d)^3 g^5 (a+b x)^4}-\frac {B d^2 i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B d i n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d)^3 g^5 (a+b x)^4} \] Output:

-1/4*B^2*d^2*i*n^2*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+4/27*b*B^2*d*i*n^2 
*(d*x+c)^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/32*b^2*B^2*i*n^2*(d*x+c)^4/(-a*d+b 
*c)^3/g^5/(b*x+a)^4-1/2*B*d^2*i*n*(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+4/9*b*B*d*i*n*(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)^3-1/8*b^2*B*i*n*(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)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln 
(e*((b*x+a)/(d*x+c))^n))^2/(-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))^2/(-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))^2/(-a*d+b*c)^3/g^5/(b*x+a)^4
 

Mathematica [C] (verified)

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

Time = 1.31 (sec) , antiderivative size = 1319, normalized size of antiderivative = 2.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/864*(i*(216*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 28 
8*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 
16*B*d*n*(a + b*x)*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3*n - 18*A*d*(b*c 
 - a*d)^2*(a + b*x) - 15*B*d*(b*c - a*d)^2*n*(a + b*x) + 36*A*d^2*(b*c - a 
*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*n*(a + b*x)^2 + 36*A*d^3*(a + b*x)^ 
3*Log[a + b*x] + 66*B*d^3*n*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*n*(a + b*x 
)^3*Log[a + b*x]^2 + 12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n] - 1 
8*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^2*(b 
*c - a*d)*(a + b*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^3*(a + b*x)^ 
3*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n] - 36*A*d^3*(a + b*x)^3*Log[c 
 + d*x] - 66*B*d^3*n*(a + b*x)^3*Log[c + d*x] + 36*B*d^3*n*(a + b*x)^3*Log 
[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 36*B*d^3*(a + b*x)^3*Log[e*( 
(a + b*x)/(c + d*x))^n]*Log[c + d*x] - 18*B*d^3*n*(a + b*x)^3*Log[c + d*x] 
^2 + 36*B*d^3*n*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 
36*B*d^3*n*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*B*d^3 
*n*(a + b*x)^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 3*B*n*(36*A*(b*c - 
 a*d)^4 + 9*B*(b*c - a*d)^4*n + 48*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d 
*(-(b*c) + a*d)^3*n*(a + b*x) + 72*A*d^2*(b*c - a*d)^2*(a + b*x)^2 + 78*B* 
d^2*(b*c - a*d)^2*n*(a + b*x)^2 + 144*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 3 
00*B*d^3*(-(b*c) + a*d)*n*(a + b*x)^3 - 144*A*d^4*(a + b*x)^4*Log[a + b...
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 2795, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i \left (-\frac {b^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^2 B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 (a+b x)^4}-\frac {d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}-\frac {B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {2 b d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}+\frac {4 b B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}-\frac {b^2 B^2 n^2 (c+d x)^4}{32 (a+b x)^4}-\frac {B^2 d^2 n^2 (c+d x)^2}{4 (a+b x)^2}+\frac {4 b B^2 d n^2 (c+d x)^3}{27 (a+b x)^3}\right )}{g^5 (b c-a d)^3}\)

Input:

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

Output:

(i*(-1/4*(B^2*d^2*n^2*(c + d*x)^2)/(a + b*x)^2 + (4*b*B^2*d*n^2*(c + d*x)^ 
3)/(27*(a + b*x)^3) - (b^2*B^2*n^2*(c + d*x)^4)/(32*(a + b*x)^4) - (B*d^2* 
n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + (4 
*b*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^ 
3) - (b^2*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(a + 
b*x)^4) - (d^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a 
 + b*x)^2) + (2*b*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/ 
(3*(a + b*x)^3) - (b^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^ 
2)/(4*(a + b*x)^4)))/((b*c - a*d)^3*g^5)
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

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. \(2219\) vs. \(2(457)=914\).

Time = 29.84 (sec) , antiderivative size = 2220, normalized size of antiderivative = 4.67

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2220\)

Input:

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

Output:

1/864*(72*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b^2*c*d^4*i*n+156*B^2*x^ 
4*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c*d^4*i*n^2+156*A*B*x^4*a^6*b^2*c*d^4* 
i*n^2+864*A^2*x*a^8*c^2*d^3*i*n-864*A^2*x*a^5*b^3*c^5*i*n+144*A*B*x^4*ln(e 
*((b*x+a)/(d*x+c))^n)*a^6*b^2*c*d^4*i*n+576*A*B*x^3*ln(e*((b*x+a)/(d*x+c)) 
^n)*a^7*b*c*d^4*i*n-1728*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^3*d^2*i*n 
+576*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^4*d*i*n+115*B^2*x^4*a^6*b^2 
*c*d^4*i*n^3-216*B^2*x^4*a^4*b^4*c^3*d^2*i*n^3+128*B^2*x^4*a^3*b^5*c^4*d*i 
*n^3-108*A*B*x^4*a^2*b^6*c^5*i*n^2+304*B^2*x^3*a^7*b*c*d^4*i*n^3-27*B^2*x^ 
4*a^2*b^6*c^5*i*n^3-108*B^2*x^3*a^3*b^5*c^5*i*n^3-216*A^2*x^4*a^2*b^6*c^5* 
i*n+1536*A*B*x^3*a^4*b^4*c^4*d*i*n^2+576*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n) 
*a^7*b*c^2*d^3*i*n^2-72*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^3*d^2* 
i*n^2+864*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^8*c*d^4*i*n+576*A*B*x^2*a^7* 
b*c^2*d^3*i*n^2-2664*A*B*x^2*a^6*b^2*c^3*d^2*i*n^2+2304*A*B*x^2*a^5*b^3*c^ 
4*d*i*n^2+216*B^2*x^2*a^8*c*d^4*i*n^3-162*B^2*x^2*a^4*b^4*c^5*i*n^3-864*A^ 
2*x^3*a^3*b^5*c^5*i*n+432*B^2*x*a^8*c^2*d^3*i*n^3-108*B^2*x*a^5*b^3*c^5*i* 
n^3+432*A^2*x^2*a^8*c*d^4*i*n-1296*A^2*x^2*a^4*b^4*c^5*i*n+432*B^2*ln(e*(( 
b*x+a)/(d*x+c))^n)^2*a^8*c^3*d^2*i*n+216*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a 
^6*b^2*c^5*i*n+432*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^8*c^3*d^2*i*n^2+108*B^2 
*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^5*i*n^2+432*A*B*ln(e*((b*x+a)/(d*x+c) 
)^n)*a^6*b^2*c^5*i*n-864*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b*c^3*d^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1868 vs. \(2 (457) = 914\).

Time = 0.15 (sec) , antiderivative size = 1868, normalized size of antiderivative = 3.93 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/864*((27*B^2*b^4*c^4 - 128*B^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 
115*B^2*a^4*d^4)*i*n^2 + 12*(13*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i*n^2 + 12 
*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*i*n)*x^3 + 12*(9*A*B*b^4*c^4 - 32*A*B*a*b 
^3*c^3*d + 36*A*B*a^2*b^2*c^2*d^2 - 13*A*B*a^4*d^4)*i*n - 6*((B^2*b^4*c^2* 
d^2 - 80*B^2*a*b^3*c*d^3 + 79*B^2*a^2*b^2*d^4)*i*n^2 + 12*(A*B*b^4*c^2*d^2 
 - 8*A*B*a*b^3*c*d^3 + 7*A*B*a^2*b^2*d^4)*i*n)*x^2 + 72*(4*(B^2*b^4*c^3*d 
- 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2*c*d^3 - B^2*a^3*b*d^4)*i*x + (3*B^2* 
b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - B^2*a^4*d^4)*i)*log( 
e)^2 + 72*(B^2*b^4*d^4*i*n^2*x^4 + 4*B^2*a*b^3*d^4*i*n^2*x^3 + 6*B^2*a^2*b 
^2*d^4*i*n^2*x^2 + 4*(B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2* 
c*d^3)*i*n^2*x + (3*B^2*b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^ 
2)*i*n^2)*log((b*x + a)/(d*x + c))^2 + 72*(3*A^2*b^4*c^4 - 8*A^2*a*b^3*c^3 
*d + 6*A^2*a^2*b^2*c^2*d^2 - A^2*a^4*d^4)*i - 4*((5*B^2*b^4*c^3*d - 12*B^2 
*a*b^3*c^2*d^2 - 108*B^2*a^2*b^2*c*d^3 + 115*B^2*a^3*b*d^4)*i*n^2 - 12*(A* 
B*b^4*c^3*d - 6*A*B*a*b^3*c^2*d^2 + 18*A*B*a^2*b^2*c*d^3 - 13*A*B*a^3*b*d^ 
4)*i*n - 72*(A^2*b^4*c^3*d - 3*A^2*a*b^3*c^2*d^2 + 3*A^2*a^2*b^2*c*d^3 - A 
^2*a^3*b*d^4)*i)*x + 12*(12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i*n*x^3 - 6*(B 
^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*i*n*x^2 + (9*B^2*b 
^4*c^4 - 32*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^2 - 13*B^2*a^4*d^4)*i*n 
 + 12*(3*A*B*b^4*c^4 - 8*A*B*a*b^3*c^3*d + 6*A*B*a^2*b^2*c^2*d^2 - A*B*...
 

Sympy [F(-1)]

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

integrate((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(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. 4838 vs. \(2 (457) = 914\).

Time = 0.36 (sec) , antiderivative size = 4838, normalized size of antiderivative = 10.19 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*A*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)) - 1/72*A*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 - 1 
3*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...
 

Giac [A] (verification not implemented)

Time = 2.84 (sec) , antiderivative size = 871, normalized size of antiderivative = 1.83 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/864*(72*(3*B^2*b^2*i*n^2 - 8*(b*x + a)*B^2*b*d*i*n^2/(d*x + c) + 6*(b*x 
 + a)^2*B^2*d^2*i*n^2/(d*x + c)^2)*log((b*x + a)/(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) + 12*(9*B^2*b^2*i*n^2 - 32*(b*x + a)*B^2*b*d 
*i*n^2/(d*x + c) + 36*(b*x + a)^2*B^2*d^2*i*n^2/(d*x + c)^2 + 36*B^2*b^2*i 
*n*log(e) - 96*(b*x + a)*B^2*b*d*i*n*log(e)/(d*x + c) + 72*(b*x + a)^2*B^2 
*d^2*i*n*log(e)/(d*x + c)^2 + 36*A*B*b^2*i*n - 96*(b*x + a)*A*B*b*d*i*n/(d 
*x + c) + 72*(b*x + a)^2*A*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) + (27*B^2*b^2*i*n^2 - 128*(b*x 
+ a)*B^2*b*d*i*n^2/(d*x + c) + 216*(b*x + a)^2*B^2*d^2*i*n^2/(d*x + c)^2 + 
 108*B^2*b^2*i*n*log(e) - 384*(b*x + a)*B^2*b*d*i*n*log(e)/(d*x + c) + 432 
*(b*x + a)^2*B^2*d^2*i*n*log(e)/(d*x + c)^2 + 216*B^2*b^2*i*log(e)^2 - 576 
*(b*x + a)*B^2*b*d*i*log(e)^2/(d*x + c) + 432*(b*x + a)^2*B^2*d^2*i*log(e) 
^2/(d*x + c)^2 + 108*A*B*b^2*i*n - 384*(b*x + a)*A*B*b*d*i*n/(d*x + c) + 4 
32*(b*x + a)^2*A*B*d^2*i*n/(d*x + c)^2 + 432*A*B*b^2*i*log(e) - 1152*(b*x 
+ a)*A*B*b*d*i*log(e)/(d*x + c) + 864*(b*x + a)^2*A*B*d^2*i*log(e)/(d*x + 
c)^2 + 216*A^2*b^2*i - 576*(b*x + a)*A^2*b*d*i/(d*x + c) + 432*(b*x + a)^2 
*A^2*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...
 

Mupad [B] (verification not implemented)

Time = 31.16 (sec) , antiderivative size = 1794, normalized size of antiderivative = 3.78 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

((72*A^2*a^3*d^3*i + 216*A^2*b^3*c^3*i + 115*B^2*a^3*d^3*i*n^2 + 27*B^2*b^ 
3*c^3*i*n^2 + 156*A*B*a^3*d^3*i*n + 108*A*B*b^3*c^3*i*n - 360*A^2*a*b^2*c^ 
2*d*i + 72*A^2*a^2*b*c*d^2*i - 101*B^2*a*b^2*c^2*d*i*n^2 + 115*B^2*a^2*b*c 
*d^2*i*n^2 - 276*A*B*a*b^2*c^2*d*i*n + 156*A*B*a^2*b*c*d^2*i*n)/(12*(a*d - 
 b*c)) + (x^2*(79*B^2*a*b^2*d^3*i*n^2 - B^2*b^3*c*d^2*i*n^2 + 84*A*B*a*b^2 
*d^3*i*n - 12*A*B*b^3*c*d^2*i*n))/(2*(a*d - b*c)) + (x*(72*A^2*a^2*b*d^3*i 
 + 72*A^2*b^3*c^2*d*i + 115*B^2*a^2*b*d^3*i*n^2 - 5*B^2*b^3*c^2*d*i*n^2 - 
144*A^2*a*b^2*c*d^2*i + 7*B^2*a*b^2*c*d^2*i*n^2 + 156*A*B*a^2*b*d^3*i*n + 
12*A*B*b^3*c^2*d*i*n - 60*A*B*a*b^2*c*d^2*i*n))/(3*(a*d - b*c)) + (d*x^3*( 
13*B^2*b^3*d^2*i*n^2 + 12*A*B*b^3*d^2*i*n))/(a*d - b*c))/(x*(288*a^3*b^4*c 
*g^5 - 288*a^4*b^3*d*g^5) - x^3*(288*a^2*b^5*d*g^5 - 288*a*b^6*c*g^5) + x^ 
4*(72*b^7*c*g^5 - 72*a*b^6*d*g^5) + x^2*(432*a^2*b^5*c*g^5 - 432*a^3*b^4*d 
*g^5) + 72*a^4*b^3*c*g^5 - 72*a^5*b^2*d*g^5) - log(e*((a + b*x)/(c + d*x)) 
^n)^2*(((B^2*c*i)/(4*b) + (B^2*d*i*x)/(3*b) + (B^2*a*d*i)/(12*b^2))/(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^2*d^4*i)/(12*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 
))) - log(e*((a + b*x)/(c + d*x))^n)*((A*B*a*d*i + 3*A*B*b*c*i - B^2*a*d*i 
*n + B^2*b*c*i*n + 4*A*B*b*d*i*x)/(6*a^4*b^2*g^5 + 6*b^6*g^5*x^4 + 24*a^3* 
b^3*g^5*x + 24*a*b^5*g^5*x^3 + 36*a^2*b^4*g^5*x^2) + (B^2*d^4*i*(x^2*(b*(b 
*((3*a*b^2*g^5*n*(a*d - b*c))/(2*d) + (b^2*g^5*n*(a*d - b*c)*(4*a*d - b...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 2426, normalized size of antiderivative = 5.11 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(144*log(a + b*x)*a**6*b*d**4*n + 120*log(a + b*x)*a**5*b**2*d**4*n**2 
+ 576*log(a + b*x)*a**5*b**2*d**4*n*x + 36*log(a + b*x)*a**4*b**3*c*d**3*n 
**2 + 480*log(a + b*x)*a**4*b**3*d**4*n**2*x + 864*log(a + b*x)*a**4*b**3* 
d**4*n*x**2 + 144*log(a + b*x)*a**3*b**4*c*d**3*n**2*x + 720*log(a + b*x)* 
a**3*b**4*d**4*n**2*x**2 + 576*log(a + b*x)*a**3*b**4*d**4*n*x**3 + 216*lo 
g(a + b*x)*a**2*b**5*c*d**3*n**2*x**2 + 480*log(a + b*x)*a**2*b**5*d**4*n* 
*2*x**3 + 144*log(a + b*x)*a**2*b**5*d**4*n*x**4 + 144*log(a + b*x)*a*b**6 
*c*d**3*n**2*x**3 + 120*log(a + b*x)*a*b**6*d**4*n**2*x**4 + 36*log(a + b* 
x)*b**7*c*d**3*n**2*x**4 - 144*log(c + d*x)*a**6*b*d**4*n - 120*log(c + d* 
x)*a**5*b**2*d**4*n**2 - 576*log(c + d*x)*a**5*b**2*d**4*n*x - 36*log(c + 
d*x)*a**4*b**3*c*d**3*n**2 - 480*log(c + d*x)*a**4*b**3*d**4*n**2*x - 864* 
log(c + d*x)*a**4*b**3*d**4*n*x**2 - 144*log(c + d*x)*a**3*b**4*c*d**3*n** 
2*x - 720*log(c + d*x)*a**3*b**4*d**4*n**2*x**2 - 576*log(c + d*x)*a**3*b* 
*4*d**4*n*x**3 - 216*log(c + d*x)*a**2*b**5*c*d**3*n**2*x**2 - 480*log(c + 
 d*x)*a**2*b**5*d**4*n**2*x**3 - 144*log(c + d*x)*a**2*b**5*d**4*n*x**4 - 
144*log(c + d*x)*a*b**6*c*d**3*n**2*x**3 - 120*log(c + d*x)*a*b**6*d**4*n* 
*2*x**4 - 36*log(c + d*x)*b**7*c*d**3*n**2*x**4 + 432*log(((a + b*x)**n*e) 
/(c + d*x)**n)**2*a**3*b**4*c**2*d**2 + 864*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a**3*b**4*c*d**3*x + 432*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a** 
3*b**4*d**4*x**2 - 576*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**5*...