\(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\) [166]

Optimal result

Integrand size = 43, antiderivative size = 307 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {B^2 d i n^2 (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B^2 i n^2 (c+d x)^3}{27 (b c-a d)^2 g^4 (a+b x)^3}+\frac {B d i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {2 b B i n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^2 g^4 (a+b x)^3} \] Output:

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

Mathematica [C] (verified)

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

Time = 1.14 (sec) , antiderivative size = 1082, normalized size of antiderivative = 3.52 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/108*(i*(36*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 54* 
d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 2*B*n 
*(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] + 6 
6*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*Pol 
yLog[2, (b*(c + d*x))/(b*c - a*d)]) + 27*B*d*n*(a + b*x)*(2*(b*c - a*d)^2* 
(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(-(b*c) + a*d)*(a + b*x)*(A + 
 B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B 
*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d^2*(a + b*x)^2*(A + B*Log[e*((a + b* 
x)/(c + d*x))^n])*Log[c + d*x] - 4*B*d*n*(a + b*x)*(b*c - a*d + d*(a + ...
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.78, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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. \(1098\) vs. \(2(295)=590\).

Time = 10.21 (sec) , antiderivative size = 1099, normalized size of antiderivative = 3.58

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1099\)

Input:

int((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x,method=_ 
RETURNVERBOSE)
 

Output:

-1/108*(18*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^2*i*n^2+8*B^2*b^6*c^3 
*d*i*n^3+18*A^2*a^3*b^3*d^4*i*n+36*A^2*b^6*c^3*d*i*n-54*B^2*x*a*b^5*c*d^3* 
i*n^3+90*A*B*x*a^2*b^4*d^4*i*n^2+18*A*B*x*b^6*c^2*d^2*i*n^2-54*B^2*ln(e*(( 
b*x+a)/(d*x+c))^n)^2*a*b^5*c^2*d^2*i*n-54*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a* 
b^5*c^2*d^2*i*n^2-108*A^2*x*a*b^5*c*d^3*i*n+72*A*B*ln(e*((b*x+a)/(d*x+c))^ 
n)*b^6*c^3*d*i*n-27*B^2*a*b^5*c^2*d^2*i*n^3+30*A*B*a^3*b^3*d^4*i*n^2+24*A* 
B*b^6*c^3*d*i*n^2-54*A^2*a*b^5*c^2*d^2*i*n-36*A*B*x^3*ln(e*((b*x+a)/(d*x+c 
))^n)*b^6*d^4*i*n-54*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*d^4*i*n-54* 
B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^4*i*n^2-36*B^2*x^2*ln(e*((b*x+a) 
/(d*x+c))^n)*b^6*c*d^3*i*n^2+36*A*B*x^2*a*b^5*d^4*i*n^2-36*A*B*x^2*b^6*c*d 
^3*i*n^2+54*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*c^2*d^2*i*n-54*A*B*a*b^5 
*c^2*d^2*i*n^2+19*B^2*a^3*b^3*d^4*i*n^3-108*A*B*x^2*ln(e*((b*x+a)/(d*x+c)) 
^n)*a*b^5*d^4*i*n-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c*d^3*i*n-10 
8*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^3*i*n^2+108*A*B*x*ln(e*((b*x+a 
)/(d*x+c))^n)*b^6*c^2*d^2*i*n-108*A*B*x*a*b^5*c*d^3*i*n^2-108*A*B*ln(e*((b 
*x+a)/(d*x+c))^n)*a*b^5*c^2*d^2*i*n-18*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2 
*b^6*d^4*i*n-30*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^4*i*n^2+30*B^2*x^2 
*a*b^5*d^4*i*n^3-30*B^2*x^2*b^6*c*d^3*i*n^3+57*B^2*x*a^2*b^4*d^4*i*n^3-3*B 
^2*x*b^6*c^2*d^2*i*n^3+36*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*c^3*d*i*n+24 
*B^2*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d*i*n^2+54*A^2*x*a^2*b^4*d^4*i*n...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (295) = 590\).

Time = 0.12 (sec) , antiderivative size = 1167, normalized size of antiderivative = 3.80 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, 
algorithm="fricas")
 

Output:

-1/108*((8*B^2*b^3*c^3 - 27*B^2*a*b^2*c^2*d + 19*B^2*a^3*d^3)*i*n^2 + 6*(4 
*A*B*b^3*c^3 - 9*A*B*a*b^2*c^2*d + 5*A*B*a^3*d^3)*i*n - 6*(5*(B^2*b^3*c*d^ 
2 - B^2*a*b^2*d^3)*i*n^2 + 6*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i*n)*x^2 + 18 
*(3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*i*x + (2*B^2*b^3*c 
^3 - 3*B^2*a*b^2*c^2*d + B^2*a^3*d^3)*i)*log(e)^2 - 18*(B^2*b^3*d^3*i*n^2* 
x^3 + 3*B^2*a*b^2*d^3*i*n^2*x^2 - 3*(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i* 
n^2*x - (2*B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d)*i*n^2)*log((b*x + a)/(d*x + c) 
)^2 + 18*(2*A^2*b^3*c^3 - 3*A^2*a*b^2*c^2*d + A^2*a^3*d^3)*i - 3*((B^2*b^3 
*c^2*d + 18*B^2*a*b^2*c*d^2 - 19*B^2*a^2*b*d^3)*i*n^2 - 6*(A*B*b^3*c^2*d - 
 6*A*B*a*b^2*c*d^2 + 5*A*B*a^2*b*d^3)*i*n - 18*(A^2*b^3*c^2*d - 2*A^2*a*b^ 
2*c*d^2 + A^2*a^2*b*d^3)*i)*x - 6*(6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i*n*x 
^2 - (4*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 5*B^2*a^3*d^3)*i*n - 6*(2*A*B*b^ 
3*c^3 - 3*A*B*a*b^2*c^2*d + A*B*a^3*d^3)*i - 3*((B^2*b^3*c^2*d - 6*B^2*a*b 
^2*c*d^2 + 5*B^2*a^2*b*d^3)*i*n + 6*(A*B*b^3*c^2*d - 2*A*B*a*b^2*c*d^2 + A 
*B*a^2*b*d^3)*i)*x + 6*(B^2*b^3*d^3*i*n*x^3 + 3*B^2*a*b^2*d^3*i*n*x^2 - 3* 
(B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^2)*i*n*x - (2*B^2*b^3*c^3 - 3*B^2*a*b^2*c 
^2*d)*i*n)*log((b*x + a)/(d*x + c)))*log(e) + 6*((4*B^2*b^3*c^3 - 9*B^2*a* 
b^2*c^2*d)*i*n^2 - (5*B^2*b^3*d^3*i*n^2 + 6*A*B*b^3*d^3*i*n)*x^3 + 6*(2*A* 
B*b^3*c^3 - 3*A*B*a*b^2*c^2*d)*i*n - 3*(6*A*B*a*b^2*d^3*i*n + (2*B^2*b^3*c 
*d^2 + 3*B^2*a*b^2*d^3)*i*n^2)*x^2 + 3*((B^2*b^3*c^2*d - 6*B^2*a*b^2*c*...
 

Sympy [F]

\[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {i \left (\int \frac {A^{2} c}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {A^{2} d x}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 A B c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 A B d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx\right )}{g^{4}} \] Input:

integrate((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**4,x 
)
 

Output:

i*(Integral(A**2*c/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + 
 b**4*x**4), x) + Integral(A**2*d*x/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 
+ 4*a*b**3*x**3 + b**4*x**4), x) + Integral(B**2*c*log(e*(a/(c + d*x) + b* 
x/(c + d*x))**n)**2/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 
+ b**4*x**4), x) + Integral(2*A*B*c*log(e*(a/(c + d*x) + b*x/(c + d*x))**n 
)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + 
 Integral(B**2*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(a**4 + 4*a* 
*3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(2*A* 
B*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**4 + 4*a**3*b*x + 6*a**2* 
b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x))/g**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3312 vs. \(2 (295) = 590\).

Time = 0.23 (sec) , antiderivative size = 3312, normalized size of antiderivative = 10.79 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, 
algorithm="maxima")
 

Output:

-1/9*A*B*c*i*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b 
^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3* 
(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3 
*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)* 
g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^ 
3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c 
*d^2 - a^3*b*d^3)*g^4)) - 1/18*A*B*d*i*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5* 
a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5* 
a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^ 
2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d 
+ a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 
6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c 
*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 
3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/6*(3*b*x + a)*B^2 
*d*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x 
^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/54*(6*n*((6*b^2*d^2*x^2 + 2*b^2*c^ 
2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^ 
5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2) 
*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3* 
c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - ...
 

Giac [A] (verification not implemented)

Time = 2.15 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.63 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{108} \, {\left (\frac {18 \, {\left (2 \, B^{2} b i n^{2} - \frac {3 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {6 \, {\left (4 \, B^{2} b i n^{2} - \frac {9 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c} + 12 \, B^{2} b i n \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B^{2} d i n \log \left (e\right )}{d x + c} + 12 \, A B b i n - \frac {18 \, {\left (b x + a\right )} A B d i n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {8 \, B^{2} b i n^{2} - \frac {27 \, {\left (b x + a\right )} B^{2} d i n^{2}}{d x + c} + 24 \, B^{2} b i n \log \left (e\right ) - \frac {54 \, {\left (b x + a\right )} B^{2} d i n \log \left (e\right )}{d x + c} + 36 \, B^{2} b i \log \left (e\right )^{2} - \frac {54 \, {\left (b x + a\right )} B^{2} d i \log \left (e\right )^{2}}{d x + c} + 24 \, A B b i n - \frac {54 \, {\left (b x + a\right )} A B d i n}{d x + c} + 72 \, A B b i \log \left (e\right ) - \frac {108 \, {\left (b x + a\right )} A B d i \log \left (e\right )}{d x + c} + 36 \, A^{2} b i - \frac {54 \, {\left (b x + a\right )} A^{2} d i}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, 
algorithm="giac")
 

Output:

-1/108*(18*(2*B^2*b*i*n^2 - 3*(b*x + a)*B^2*d*i*n^2/(d*x + c))*log((b*x + 
a)/(d*x + c))^2/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d* 
x + c)^3) + 6*(4*B^2*b*i*n^2 - 9*(b*x + a)*B^2*d*i*n^2/(d*x + c) + 12*B^2* 
b*i*n*log(e) - 18*(b*x + a)*B^2*d*i*n*log(e)/(d*x + c) + 12*A*B*b*i*n - 18 
*(b*x + a)*A*B*d*i*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^3*b*c* 
g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3) + (8*B^2*b*i*n^2 - 27*( 
b*x + a)*B^2*d*i*n^2/(d*x + c) + 24*B^2*b*i*n*log(e) - 54*(b*x + a)*B^2*d* 
i*n*log(e)/(d*x + c) + 36*B^2*b*i*log(e)^2 - 54*(b*x + a)*B^2*d*i*log(e)^2 
/(d*x + c) + 24*A*B*b*i*n - 54*(b*x + a)*A*B*d*i*n/(d*x + c) + 72*A*B*b*i* 
log(e) - 108*(b*x + a)*A*B*d*i*log(e)/(d*x + c) + 36*A^2*b*i - 54*(b*x + a 
)*A^2*d*i/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^ 
4/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 28.89 (sec) , antiderivative size = 993, normalized size of antiderivative = 3.23 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int(((c*i + d*i*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g*x) 
^4,x)
 

Output:

- ((18*A^2*a^2*d^2*i - 36*A^2*b^2*c^2*i + 19*B^2*a^2*d^2*i*n^2 - 8*B^2*b^2 
*c^2*i*n^2 + 30*A*B*a^2*d^2*i*n - 24*A*B*b^2*c^2*i*n + 18*A^2*a*b*c*d*i + 
19*B^2*a*b*c*d*i*n^2 + 30*A*B*a*b*c*d*i*n)/(6*(a*d - b*c)) + (x*(18*A^2*a* 
b*d^2*i - 18*A^2*b^2*c*d*i + 19*B^2*a*b*d^2*i*n^2 + B^2*b^2*c*d*i*n^2 + 30 
*A*B*a*b*d^2*i*n - 6*A*B*b^2*c*d*i*n))/(2*(a*d - b*c)) + (x^2*(5*B^2*b^2*d 
^2*i*n^2 + 6*A*B*b^2*d^2*i*n))/(a*d - b*c))/(18*a^3*b^2*g^4 + 18*b^5*g^4*x 
^3 + 54*a^2*b^3*g^4*x + 54*a*b^4*g^4*x^2) - log(e*((a + b*x)/(c + d*x))^n) 
^2*(((B^2*c*i)/(3*b) + (B^2*d*i*x)/(2*b) + (B^2*a*d*i)/(6*b^2))/(a^3*g^4 + 
 b^3*g^4*x^3 + 3*a*b^2*g^4*x^2 + 3*a^2*b*g^4*x) - (B^2*d^3*i)/(6*b^2*g^4*( 
a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - log(e*((a + b*x)/(c + d*x))^n)*((A*B*a* 
d*i + 2*A*B*b*c*i - B^2*a*d*i*n + B^2*b*c*i*n + 3*A*B*b*d*i*x)/(3*a^3*b^2* 
g^4 + 3*b^5*g^4*x^3 + 9*a^2*b^3*g^4*x + 9*a*b^4*g^4*x^2) + (B^2*d^3*i*(x*( 
b*((a*b^2*g^4*n*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c)*(3*a*d - b*c))/(2* 
d^2)) + (2*a*b^3*g^4*n*(a*d - b*c))/d + (b^3*g^4*n*(a*d - b*c)*(3*a*d - b* 
c))/d^2) + a*((a*b^2*g^4*n*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c)*(3*a*d 
- b*c))/(2*d^2)) + (3*b^4*g^4*n*x^2*(a*d - b*c))/d + (b^2*g^4*n*(a*d - b*c 
)*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3))/(3*b^2*g^4*(a^2*d^2 + b^2*c^2 - 
 2*a*b*c*d)*(3*a^3*b^2*g^4 + 3*b^5*g^4*x^3 + 9*a^2*b^3*g^4*x + 9*a*b^4*g^4 
*x^2))) - (B*d^3*i*n*atan((B*d^3*i*n*(6*A + 5*B*n)*(2*b*d*x - (b^4*c^2*g^4 
 - a^2*b^2*d^2*g^4)/(b^2*g^4*(a*d - b*c)))*1i)/((a*d - b*c)*(5*B^2*d^3*...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1624, normalized size of antiderivative = 5.29 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x)
 

Output:

(i*(36*log(a + b*x)*a**5*b*d**3*n + 18*log(a + b*x)*a**4*b**2*d**3*n**2 + 
108*log(a + b*x)*a**4*b**2*d**3*n*x + 12*log(a + b*x)*a**3*b**3*c*d**2*n** 
2 + 54*log(a + b*x)*a**3*b**3*d**3*n**2*x + 108*log(a + b*x)*a**3*b**3*d** 
3*n*x**2 + 36*log(a + b*x)*a**2*b**4*c*d**2*n**2*x + 54*log(a + b*x)*a**2* 
b**4*d**3*n**2*x**2 + 36*log(a + b*x)*a**2*b**4*d**3*n*x**3 + 36*log(a + b 
*x)*a*b**5*c*d**2*n**2*x**2 + 18*log(a + b*x)*a*b**5*d**3*n**2*x**3 + 12*l 
og(a + b*x)*b**6*c*d**2*n**2*x**3 - 36*log(c + d*x)*a**5*b*d**3*n - 18*log 
(c + d*x)*a**4*b**2*d**3*n**2 - 108*log(c + d*x)*a**4*b**2*d**3*n*x - 12*l 
og(c + d*x)*a**3*b**3*c*d**2*n**2 - 54*log(c + d*x)*a**3*b**3*d**3*n**2*x 
- 108*log(c + d*x)*a**3*b**3*d**3*n*x**2 - 36*log(c + d*x)*a**2*b**4*c*d** 
2*n**2*x - 54*log(c + d*x)*a**2*b**4*d**3*n**2*x**2 - 36*log(c + d*x)*a**2 
*b**4*d**3*n*x**3 - 36*log(c + d*x)*a*b**5*c*d**2*n**2*x**2 - 18*log(c + d 
*x)*a*b**5*d**3*n**2*x**3 - 12*log(c + d*x)*b**6*c*d**2*n**2*x**3 + 54*log 
(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**4*c**2*d + 108*log(((a + b*x)** 
n*e)/(c + d*x)**n)**2*a**2*b**4*c*d**2*x + 54*log(((a + b*x)**n*e)/(c + d* 
x)**n)**2*a**2*b**4*d**3*x**2 - 36*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a 
*b**5*c**3 - 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*c**2*d*x + 18 
*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*d**3*x**3 - 36*log(((a + b*x 
)**n*e)/(c + d*x)**n)*a**5*b*d**3 - 18*log(((a + b*x)**n*e)/(c + d*x)**n)* 
a**4*b**2*d**3*n - 108*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b**2*d**...