\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^6} \, dx\) [177]

Optimal result

Integrand size = 45, antiderivative size = 493 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=-\frac {2 B^2 d^2 i^2 n^2 (c+d x)^3}{27 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B^2 d i^2 n^2 (c+d x)^4}{16 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B^2 i^2 n^2 (c+d x)^5}{125 (b c-a d)^3 g^6 (a+b x)^5}-\frac {2 B d^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B i^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 (b c-a d)^3 g^6 (a+b x)^5} \] Output:

-2/27*B^2*d^2*i^2*n^2*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/16*b*B^2*d*i^ 
2*n^2*(d*x+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-2/125*b^2*B^2*i^2*n^2*(d*x+c)^5 
/(-a*d+b*c)^3/g^6/(b*x+a)^5-2/9*B*d^2*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/( 
d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/4*b*B*d*i^2*n*(d*x+c)^4*(A+B*ln(e 
*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^4-2/25*b^2*B*i^2*n*(d*x+c) 
^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^5-1/3*d^2*i^2* 
(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/2 
*b*d*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^6/(b*x 
+a)^4-1/5*b^2*i^2*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3 
/g^6/(b*x+a)^5
 

Mathematica [C] (verified)

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

Time = 3.46 (sec) , antiderivative size = 2112, normalized size of antiderivative = 4.28 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Result too large to show} \] Input:

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g 
+ b*g*x)^6,x]
 

Output:

-1/54000*(i^2*(10800*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^ 
2 + 27000*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
^2 - 18000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d 
*x))^n])^2 + 1000*B*d^2*n*(a + b*x)^2*(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] - 18*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)]) + 
 375*B*d*n*(a + b*x)*(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*...
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.74, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

\(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^6} \, dx\)

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^2 \left (-\frac {b^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 (a+b x)^5}-\frac {2 b^2 B n (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{25 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 B d^2 n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^4}+\frac {b B d n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (a+b x)^4}-\frac {2 b^2 B^2 n^2 (c+d x)^5}{125 (a+b x)^5}-\frac {2 B^2 d^2 n^2 (c+d x)^3}{27 (a+b x)^3}+\frac {b B^2 d n^2 (c+d x)^4}{16 (a+b x)^4}\right )}{g^6 (b c-a d)^3}\)

Input:

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g* 
x)^6,x]
 

Output:

(i^2*((-2*B^2*d^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) + (b*B^2*d*n^2*(c + d* 
x)^4)/(16*(a + b*x)^4) - (2*b^2*B^2*n^2*(c + d*x)^5)/(125*(a + b*x)^5) - ( 
2*B*d^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x) 
^3) + (b*B*d*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(a + 
 b*x)^4) - (2*b^2*B*n*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/ 
(25*(a + b*x)^5) - (d^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
^2)/(3*(a + b*x)^3) + (b*d*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n])^2)/(2*(a + b*x)^4) - (b^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x 
))^n])^2)/(5*(a + b*x)^5)))/((b*c - a*d)^3*g^6)
 

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. \(2527\) vs. \(2(475)=950\).

Time = 61.60 (sec) , antiderivative size = 2528, normalized size of antiderivative = 5.13

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2528\)

Input:

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x,method 
=_RETURNVERBOSE)
 

Output:

-1/54000*(-4000*B^2*a^2*b^7*c^3*d^3*i^2*n^3+9000*A*B*x^2*a*b^8*c^2*d^4*i^2 
*n^2-54000*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^7*c^2*d^4*i^2*n+72000*B 
^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^8*c^3*d^3*i^2*n-36000*B^2*x*ln(e*((b* 
x+a)/(d*x+c))^n)*a^2*b^7*c^2*d^4*i^2*n^2+1489*B^2*a^5*b^4*d^6*i^2*n^3-864* 
B^2*b^9*c^5*d*i^2*n^3+1800*A^2*a^5*b^4*d^6*i^2*n-10800*A^2*b^9*c^5*d*i^2*n 
+13500*A*B*a*b^8*c^4*d^2*i^2*n^2+30000*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b 
^8*c^3*d^3*i^2*n^2-54000*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^4*d^2*i^2*n 
-36000*A*B*x*a^2*b^7*c^2*d^4*i^2*n^2+30000*A*B*x*a*b^8*c^3*d^3*i^2*n^2-180 
00*A*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*d^6*i^2*n-18000*B^2*x^3*ln(e*(( 
b*x+a)/(d*x+c))^n)*a*b^8*c*d^5*i^2*n^2-36000*A*B*x^3*ln(e*((b*x+a)/(d*x+c) 
)^n)*a^2*b^7*d^6*i^2*n-18000*A*B*x^3*a*b^8*c*d^5*i^2*n^2-54000*B^2*x^2*ln( 
e*((b*x+a)/(d*x+c))^n)^2*a^2*b^7*c*d^5*i^2*n+54000*B^2*x^2*ln(e*((b*x+a)/( 
d*x+c))^n)^2*a*b^8*c^2*d^4*i^2*n-36000*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a 
^2*b^7*c*d^5*i^2*n^2+9000*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^2*d^4* 
i^2*n^2-36000*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^3*d^3*i^2*n-36000*A* 
B*x^2*a^2*b^7*c*d^5*i^2*n^2-36000*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c^ 
3*d^3*i^2*n+54000*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^4*d^2*i^2*n+9000*A 
^2*x*a^4*b^5*d^6*i^2*n-27000*A^2*x*b^9*c^4*d^2*i^2*n-12000*A*B*a^2*b^7*c^3 
*d^3*i^2*n^2+3375*B^2*a*b^8*c^4*d^2*i^2*n^3+2820*A*B*a^5*b^4*d^6*i^2*n^2-4 
320*A*B*b^9*c^5*d*i^2*n^2-18000*A^2*a^2*b^7*c^3*d^3*i^2*n+27000*A^2*a*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2633 vs. \(2 (475) = 950\).

Time = 0.19 (sec) , antiderivative size = 2633, normalized size of antiderivative = 5.34 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x 
, algorithm="fricas")
 

Output:

-1/54000*((864*B^2*b^5*c^5 - 3375*B^2*a*b^4*c^4*d + 4000*B^2*a^2*b^3*c^3*d 
^2 - 1489*B^2*a^5*d^5)*i^2*n^2 + 60*(47*(B^2*b^5*c*d^4 - B^2*a*b^4*d^5)*i^ 
2*n^2 + 60*(A*B*b^5*c*d^4 - A*B*a*b^4*d^5)*i^2*n)*x^4 + 60*(72*A*B*b^5*c^5 
 - 225*A*B*a*b^4*c^4*d + 200*A*B*a^2*b^3*c^3*d^2 - 47*A*B*a^5*d^5)*i^2*n + 
 30*((13*B^2*b^5*c^2*d^3 + 350*B^2*a*b^4*c*d^4 - 363*B^2*a^2*b^3*d^5)*i^2* 
n^2 - 60*(A*B*b^5*c^2*d^3 - 10*A*B*a*b^4*c*d^4 + 9*A*B*a^2*b^3*d^5)*i^2*n) 
*x^3 + 1800*(6*A^2*b^5*c^5 - 15*A^2*a*b^4*c^4*d + 10*A^2*a^2*b^3*c^3*d^2 - 
 A^2*a^5*d^5)*i^2 - 10*((86*B^2*b^5*c^3*d^2 - 375*B^2*a*b^4*c^2*d^3 - 1200 
*B^2*a^2*b^3*c*d^4 + 1489*B^2*a^3*b^2*d^5)*i^2*n^2 - 60*(2*A*B*b^5*c^3*d^2 
 - 15*A*B*a*b^4*c^2*d^3 + 60*A*B*a^2*b^3*c*d^4 - 47*A*B*a^3*b^2*d^5)*i^2*n 
 - 1800*(A^2*b^5*c^3*d^2 - 3*A^2*a*b^4*c^2*d^3 + 3*A^2*a^2*b^3*c*d^4 - A^2 
*a^3*b^2*d^5)*i^2)*x^2 + 1800*(10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 
 3*B^2*a^2*b^3*c*d^4 - B^2*a^3*b^2*d^5)*i^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B 
^2*a*b^4*c^3*d^2 + 6*B^2*a^2*b^3*c^2*d^3 - B^2*a^4*b*d^5)*i^2*x + (6*B^2*b 
^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3*c^3*d^2 - B^2*a^5*d^5)*i^2)*l 
og(e)^2 + 1800*(B^2*b^5*d^5*i^2*n^2*x^5 + 5*B^2*a*b^4*d^5*i^2*n^2*x^4 + 10 
*B^2*a^2*b^3*d^5*i^2*n^2*x^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 
 3*B^2*a^2*b^3*c*d^4)*i^2*n^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B^2*a*b^4*c^3*d 
^2 + 6*B^2*a^2*b^3*c^2*d^3)*i^2*n^2*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4* 
d + 10*B^2*a^2*b^3*c^3*d^2)*i^2*n^2)*log((b*x + a)/(d*x + c))^2 + 5*((1...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)** 
6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10936 vs. \(2 (475) = 950\).

Time = 0.77 (sec) , antiderivative size = 10936, normalized size of antiderivative = 22.18 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x 
, algorithm="maxima")
 

Output:

-1/150*A*B*c^2*i^2*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137* 
a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3* 
d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3 
*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10 
*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)* 
g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c 
*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^ 
6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a 
^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 
+ 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + 
 a^8*b^2*d^4)*g^6*x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 
 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b 
^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5 
*b*d^5)*g^6) - 60*d^5*log(d*x + c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4* 
c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6)) - 1/900* 
A*B*d^2*i^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 
 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b 
^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a 
^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 
 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^...
 

Giac [A] (verification not implemented)

Time = 5.97 (sec) , antiderivative size = 931, normalized size of antiderivative = 1.89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x 
, algorithm="giac")
 

Output:

-1/54000*(1800*(6*B^2*b^2*i^2*n^2 - 15*(b*x + a)*B^2*b*d*i^2*n^2/(d*x + c) 
 + 10*(b*x + a)^2*B^2*d^2*i^2*n^2/(d*x + c)^2)*log((b*x + a)/(d*x + c))^2/ 
((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c) 
^5 + (b*x + a)^5*a^2*d^2*g^6/(d*x + c)^5) + 60*(72*B^2*b^2*i^2*n^2 - 225*( 
b*x + a)*B^2*b*d*i^2*n^2/(d*x + c) + 200*(b*x + a)^2*B^2*d^2*i^2*n^2/(d*x 
+ c)^2 + 360*B^2*b^2*i^2*n*log(e) - 900*(b*x + a)*B^2*b*d*i^2*n*log(e)/(d* 
x + c) + 600*(b*x + a)^2*B^2*d^2*i^2*n*log(e)/(d*x + c)^2 + 360*A*B*b^2*i^ 
2*n - 900*(b*x + a)*A*B*b*d*i^2*n/(d*x + c) + 600*(b*x + a)^2*A*B*d^2*i^2* 
n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/((b*x + a)^5*b^2*c^2*g^6/(d*x + c) 
^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^5*a^2*d^2*g^6/(d*x 
+ c)^5) + (864*B^2*b^2*i^2*n^2 - 3375*(b*x + a)*B^2*b*d*i^2*n^2/(d*x + c) 
+ 4000*(b*x + a)^2*B^2*d^2*i^2*n^2/(d*x + c)^2 + 4320*B^2*b^2*i^2*n*log(e) 
 - 13500*(b*x + a)*B^2*b*d*i^2*n*log(e)/(d*x + c) + 12000*(b*x + a)^2*B^2* 
d^2*i^2*n*log(e)/(d*x + c)^2 + 10800*B^2*b^2*i^2*log(e)^2 - 27000*(b*x + a 
)*B^2*b*d*i^2*log(e)^2/(d*x + c) + 18000*(b*x + a)^2*B^2*d^2*i^2*log(e)^2/ 
(d*x + c)^2 + 4320*A*B*b^2*i^2*n - 13500*(b*x + a)*A*B*b*d*i^2*n/(d*x + c) 
 + 12000*(b*x + a)^2*A*B*d^2*i^2*n/(d*x + c)^2 + 21600*A*B*b^2*i^2*log(e) 
- 54000*(b*x + a)*A*B*b*d*i^2*log(e)/(d*x + c) + 36000*(b*x + a)^2*A*B*d^2 
*i^2*log(e)/(d*x + c)^2 + 10800*A^2*b^2*i^2 - 27000*(b*x + a)*A^2*b*d*i^2/ 
(d*x + c) + 18000*(b*x + a)^2*A^2*d^2*i^2/(d*x + c)^2)/((b*x + a)^5*b^2...
 

Mupad [B] (verification not implemented)

Time = 32.53 (sec) , antiderivative size = 3296, normalized size of antiderivative = 6.69 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g* 
x)^6,x)
 

Output:

((1800*A^2*a^4*d^4*i^2 + 10800*A^2*b^4*c^4*i^2 + 1489*B^2*a^4*d^4*i^2*n^2 
+ 864*B^2*b^4*c^4*i^2*n^2 - 16200*A^2*a*b^3*c^3*d*i^2 + 1800*A^2*a^3*b*c*d 
^3*i^2 + 2820*A*B*a^4*d^4*i^2*n + 4320*A*B*b^4*c^4*i^2*n + 1800*A^2*a^2*b^ 
2*c^2*d^2*i^2 + 1489*B^2*a^2*b^2*c^2*d^2*i^2*n^2 - 2511*B^2*a*b^3*c^3*d*i^ 
2*n^2 + 1489*B^2*a^3*b*c*d^3*i^2*n^2 + 2820*A*B*a^2*b^2*c^2*d^2*i^2*n - 91 
80*A*B*a*b^3*c^3*d*i^2*n + 2820*A*B*a^3*b*c*d^3*i^2*n)/(60*(a*d - b*c)) + 
(x*(1800*A^2*a^3*b*d^4*i^2 + 5400*A^2*b^4*c^3*d*i^2 - 9000*A^2*a*b^3*c^2*d 
^2*i^2 + 1800*A^2*a^2*b^2*c*d^3*i^2 + 1489*B^2*a^3*b*d^4*i^2*n^2 + 189*B^2 
*b^4*c^3*d*i^2*n^2 + 1620*A*B*b^4*c^3*d*i^2*n - 911*B^2*a*b^3*c^2*d^2*i^2* 
n^2 + 1489*B^2*a^2*b^2*c*d^3*i^2*n^2 + 2820*A*B*a^3*b*d^4*i^2*n - 4380*A*B 
*a*b^3*c^2*d^2*i^2*n + 2820*A*B*a^2*b^2*c*d^3*i^2*n))/(12*(a*d - b*c)) + ( 
x^2*(1800*A^2*a^2*b^2*d^4*i^2 + 1800*A^2*b^4*c^2*d^2*i^2 - 3600*A^2*a*b^3* 
c*d^3*i^2 + 1489*B^2*a^2*b^2*d^4*i^2*n^2 - 86*B^2*b^4*c^2*d^2*i^2*n^2 + 28 
20*A*B*a^2*b^2*d^4*i^2*n + 120*A*B*b^4*c^2*d^2*i^2*n + 289*B^2*a*b^3*c*d^3 
*i^2*n^2 - 780*A*B*a*b^3*c*d^3*i^2*n))/(6*(a*d - b*c)) + (x^3*(363*B^2*a*b 
^3*d^4*i^2*n^2 + 13*B^2*b^4*c*d^3*i^2*n^2 - 60*A*B*b^4*c*d^3*i^2*n + 540*A 
*B*a*b^3*d^4*i^2*n))/(2*(a*d - b*c)) + (d*x^4*(47*B^2*b^4*d^3*i^2*n^2 + 60 
*A*B*b^4*d^3*i^2*n))/(a*d - b*c))/(x*(4500*a^4*b^5*c*g^6 - 4500*a^5*b^4*d* 
g^6) - x^4*(4500*a^2*b^7*d*g^6 - 4500*a*b^8*c*g^6) + x^5*(900*b^9*c*g^6 - 
900*a*b^8*d*g^6) + x^2*(9000*a^3*b^6*c*g^6 - 9000*a^4*b^5*d*g^6) + x^3*...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 3287, normalized size of antiderivative = 6.67 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x)
 

Output:

( - 3600*log(a + b*x)*a**7*b*d**5*n - 2100*log(a + b*x)*a**6*b**2*d**5*n** 
2 - 18000*log(a + b*x)*a**6*b**2*d**5*n*x - 720*log(a + b*x)*a**5*b**3*c*d 
**4*n**2 - 10500*log(a + b*x)*a**5*b**3*d**5*n**2*x - 36000*log(a + b*x)*a 
**5*b**3*d**5*n*x**2 - 3600*log(a + b*x)*a**4*b**4*c*d**4*n**2*x - 21000*l 
og(a + b*x)*a**4*b**4*d**5*n**2*x**2 - 36000*log(a + b*x)*a**4*b**4*d**5*n 
*x**3 - 7200*log(a + b*x)*a**3*b**5*c*d**4*n**2*x**2 - 21000*log(a + b*x)* 
a**3*b**5*d**5*n**2*x**3 - 18000*log(a + b*x)*a**3*b**5*d**5*n*x**4 - 7200 
*log(a + b*x)*a**2*b**6*c*d**4*n**2*x**3 - 10500*log(a + b*x)*a**2*b**6*d* 
*5*n**2*x**4 - 3600*log(a + b*x)*a**2*b**6*d**5*n*x**5 - 3600*log(a + b*x) 
*a*b**7*c*d**4*n**2*x**4 - 2100*log(a + b*x)*a*b**7*d**5*n**2*x**5 - 720*l 
og(a + b*x)*b**8*c*d**4*n**2*x**5 + 3600*log(c + d*x)*a**7*b*d**5*n + 2100 
*log(c + d*x)*a**6*b**2*d**5*n**2 + 18000*log(c + d*x)*a**6*b**2*d**5*n*x 
+ 720*log(c + d*x)*a**5*b**3*c*d**4*n**2 + 10500*log(c + d*x)*a**5*b**3*d* 
*5*n**2*x + 36000*log(c + d*x)*a**5*b**3*d**5*n*x**2 + 3600*log(c + d*x)*a 
**4*b**4*c*d**4*n**2*x + 21000*log(c + d*x)*a**4*b**4*d**5*n**2*x**2 + 360 
00*log(c + d*x)*a**4*b**4*d**5*n*x**3 + 7200*log(c + d*x)*a**3*b**5*c*d**4 
*n**2*x**2 + 21000*log(c + d*x)*a**3*b**5*d**5*n**2*x**3 + 18000*log(c + d 
*x)*a**3*b**5*d**5*n*x**4 + 7200*log(c + d*x)*a**2*b**6*c*d**4*n**2*x**3 + 
 10500*log(c + d*x)*a**2*b**6*d**5*n**2*x**4 + 3600*log(c + d*x)*a**2*b**6 
*d**5*n*x**5 + 3600*log(c + d*x)*a*b**7*c*d**4*n**2*x**4 + 2100*log(c +...