\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\) [17]

Optimal result

Integrand size = 35, antiderivative size = 448 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 d^2 n^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d n^2 (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 n^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B d n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \] Output:

-2*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^3/g^4/(b*x+a)+1/2*b*B^2*d*n^2*(d*x+c)^2/ 
(-a*d+b*c)^3/g^4/(b*x+a)^2-2/27*b^2*B^2*n^2*(d*x+c)^3/(-a*d+b*c)^3/g^4/(b* 
x+a)^3-2*B*d^2*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^4/ 
(b*x+a)+b*B*d*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^4 
/(b*x+a)^2-2/9*b^2*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c 
)^3/g^4/(b*x+a)^3-d^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c) 
^3/g^4/(b*x+a)+b*d*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^ 
3/g^4/(b*x+a)^2-1/3*b^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+ 
b*c)^3/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 = 0.48 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.37 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {18 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n \left (12 A (b c-a d)^3+4 B (b c-a d)^3 n-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 n (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) n (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 n (a+b x)^3 \log (a+b x)-18 B d^3 n (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-18 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 n (a+b x)^3 \log (c+d x)+36 B d^3 n (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)-18 B d^3 n (a+b x)^3 \log ^2(c+d x)+36 B d^3 n (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 n (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 n (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{54 b g^4 (a+b x)^3} \] Input:

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

Output:

-1/54*(18*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^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] + 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*Lo 
g[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)]))/(b*c - a*d)^3)/(b*g^4*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2949, 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.

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

\(\Big \downarrow \) 2949

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-2*B^2*d^2*n^2*(c + d*x))/(a + b*x) + (b*B^2*d*n^2*(c + d*x)^2)/(2*(a + 
b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (2*B*d^2*n*(c + d 
*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (b*B*d*n*(c + d*x) 
^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 - (2*b^2*B*n*(c + d 
*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^3) - (d^2*(c + 
d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) + (b*d*(c + d*x)^ 
2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x)^2 - (b^2*(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)^3 
*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 2949

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 
1)*(g/b)^m   Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x], x, 
 (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && Ne 
Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt 
Q[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1122\) vs. \(2(440)=880\).

Time = 10.93 (sec) , antiderivative size = 1123, normalized size of antiderivative = 2.51

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1123\)

Input:

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

Output:

-1/54*(-36*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^4*n-54*B^2*x^2*ln(e*((b 
*x+a)/(d*x+c))^n)^2*a*b^6*d^4*n-162*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^ 
6*d^4*n^2-36*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c*d^3*n^2+36*A*B*x^2*a* 
b^6*d^4*n^2-36*A*B*x^2*b^7*c*d^3*n^2-54*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2* 
a^2*b^5*d^4*n-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*d^4*n^2+18*B^2*x 
*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^2*d^2*n^2-108*A*B*a^2*b^5*c*d^3*n^2+54*A* 
B*a*b^6*c^2*d^2*n^2-162*B^2*x*a*b^6*c*d^3*n^3+90*A*B*x*a^2*b^5*d^4*n^2+18* 
A*B*x*b^7*c^2*d^2*n^2-54*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^5*c*d^3*n+5 
4*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^6*c^2*d^2*n-108*B^2*ln(e*((b*x+a)/(d 
*x+c))^n)*a^2*b^5*c*d^3*n^2+54*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c^2*d^2 
*n^2-36*A*B*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^3*d*n+85*B^2*a^3*b^4*d^4*n^3-4 
*B^2*b^7*c^3*d*n^3+18*A^2*a^3*b^4*d^4*n-18*A^2*b^7*c^3*d*n-108*B^2*a^2*b^5 
*c*d^3*n^3+27*B^2*a*b^6*c^2*d^2*n^3+66*A*B*a^3*b^4*d^4*n^2-12*A*B*b^7*c^3* 
d*n^2-54*A^2*a^2*b^5*c*d^3*n+54*A^2*a*b^6*c^2*d^2*n-18*B^2*x^3*ln(e*((b*x+ 
a)/(d*x+c))^n)^2*b^7*d^4*n-66*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^4*n^ 
2+66*B^2*x^2*a*b^6*d^4*n^3-66*B^2*x^2*b^7*c*d^3*n^3+147*B^2*x*a^2*b^5*d^4* 
n^3+15*B^2*x*b^7*c^2*d^2*n^3-18*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^7*c^3*d* 
n-12*B^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^3*d*n^2-108*A*B*x^2*ln(e*((b*x+a) 
/(d*x+c))^n)*a*b^6*d^4*n-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c*d^3*n 
^2-108*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*d^4*n-108*A*B*x*a*b^6*c*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1164 vs. \(2 (440) = 880\).

Time = 0.11 (sec) , antiderivative size = 1164, normalized size of antiderivative = 2.60 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a 
^3*d^3 + (4*B^2*b^3*c^3 - 27*B^2*a*b^2*c^2*d + 108*B^2*a^2*b*c*d^2 - 85*B^ 
2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 + 6*(A*B*b^3*c* 
d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2* 
a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^ 
2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 
 3*B^2*a^2*b*c*d^2)*n^2)*log((b*x + a)/(d*x + c))^2 + 6*(2*A*B*b^3*c^3 - 9 
*A*B*a*b^2*c^2*d + 18*A*B*a^2*b*c*d^2 - 11*A*B*a^3*d^3)*n - 3*((5*B^2*b^3* 
c^2*d - 54*B^2*a*b^2*c*d^2 + 49*B^2*a^2*b*d^3)*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)*n)*x + 6*(6*A*B*b^3*c^3 - 18*A*B*a*b^2* 
c^2*d + 18*A*B*a^2*b*c*d^2 - 6*A*B*a^3*d^3 + 6*(B^2*b^3*c*d^2 - B^2*a*b^2* 
d^3)*n*x^2 - 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)*n*x + 
 (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2 - 11*B^2*a^3*d^3) 
*n + 6*(B^2*b^3*d^3*n*x^3 + 3*B^2*a*b^2*d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + 
(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n)*log((b*x + a)/(d* 
x + c)))*log(e) + 6*((11*B^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*x^3 + (2*B^2*b 
^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3* 
n + (2*B^2*b^3*c*d^2 + 9*B^2*a*b^2*d^3)*n^2)*x^2 + 6*(A*B*b^3*c^3 - 3*A*B* 
a*b^2*c^2*d + 3*A*B*a^2*b*c*d^2)*n + 3*(6*A*B*a^2*b*d^3*n - (B^2*b^3*c^2*d 
 - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n^2)*x)*log((b*x + a)/(d*x + c)...
 

Sympy [F]

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

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

Output:

(Integral(A**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(B**2*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) + Inte 
gral(2*A*B*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. 1432 vs. \(2 (440) = 880\).

Time = 0.13 (sec) , antiderivative size = 1432, normalized size of antiderivative = 3.20 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*A*B*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)) - 1/54*(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 - 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))*log(e*(b*x/(d*x + c) + a/(d*x + c 
))^n) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b 
^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^ 
3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2* 
b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a 
^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3) 
*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11 
*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*...
 

Giac [A] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 840, normalized size of antiderivative = 1.88 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(18*(B^2*b^2*n^2 - 3*(b*x + a)*B^2*b*d*n^2/(d*x + c) + 3*(b*x + a)^2 
*B^2*d^2*n^2/(d*x + c)^2)*log((b*x + a)/(d*x + c))^2/((b*x + a)^3*b^2*c^2* 
g^4/(d*x + c)^3 - 2*(b*x + a)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*x + a)^3*a^2* 
d^2*g^4/(d*x + c)^3) + 6*(2*B^2*b^2*n^2 - 9*(b*x + a)*B^2*b*d*n^2/(d*x + c 
) + 18*(b*x + a)^2*B^2*d^2*n^2/(d*x + c)^2 + 6*B^2*b^2*n*log(e) - 18*(b*x 
+ a)*B^2*b*d*n*log(e)/(d*x + c) + 18*(b*x + a)^2*B^2*d^2*n*log(e)/(d*x + c 
)^2 + 6*A*B*b^2*n - 18*(b*x + a)*A*B*b*d*n/(d*x + c) + 18*(b*x + a)^2*A*B* 
d^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/((b*x + a)^3*b^2*c^2*g^4/(d*x 
+ c)^3 - 2*(b*x + a)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*x + a)^3*a^2*d^2*g^4/( 
d*x + c)^3) + (4*B^2*b^2*n^2 - 27*(b*x + a)*B^2*b*d*n^2/(d*x + c) + 108*(b 
*x + a)^2*B^2*d^2*n^2/(d*x + c)^2 + 12*B^2*b^2*n*log(e) - 54*(b*x + a)*B^2 
*b*d*n*log(e)/(d*x + c) + 108*(b*x + a)^2*B^2*d^2*n*log(e)/(d*x + c)^2 + 1 
8*B^2*b^2*log(e)^2 - 54*(b*x + a)*B^2*b*d*log(e)^2/(d*x + c) + 54*(b*x + a 
)^2*B^2*d^2*log(e)^2/(d*x + c)^2 + 12*A*B*b^2*n - 54*(b*x + a)*A*B*b*d*n/( 
d*x + c) + 108*(b*x + a)^2*A*B*d^2*n/(d*x + c)^2 + 36*A*B*b^2*log(e) - 108 
*(b*x + a)*A*B*b*d*log(e)/(d*x + c) + 108*(b*x + a)^2*A*B*d^2*log(e)/(d*x 
+ c)^2 + 18*A^2*b^2 - 54*(b*x + a)*A^2*b*d/(d*x + c) + 54*(b*x + a)^2*A^2* 
d^2/(d*x + c)^2)/((b*x + a)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*x + a)^3*a*b* 
c*d*g^4/(d*x + c)^3 + (b*x + a)^3*a^2*d^2*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 = 29.08 (sec) , antiderivative size = 1038, normalized size of antiderivative = 2.32 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((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 + 18*A^2*b^2*c^2 + 85*B^2*a^2*d^2*n^2 + 4*B^2*b^2*c^2*n^2 
 - 36*A^2*a*b*c*d + 66*A*B*a^2*d^2*n + 12*A*B*b^2*c^2*n - 23*B^2*a*b*c*d*n 
^2 - 42*A*B*a*b*c*d*n)/(6*(a*d - b*c)) + (x*(49*B^2*a*b*d^2*n^2 - 5*B^2*b^ 
2*c*d*n^2 + 30*A*B*a*b*d^2*n - 6*A*B*b^2*c*d*n))/(2*(a*d - b*c)) + (d*x^2* 
(11*B^2*b^2*d*n^2 + 6*A*B*b^2*d*n))/(a*d - b*c))/(x*(27*a^2*b^3*c*g^4 - 27 
*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d*g^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g 
^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*c*g^4 - 9*a^4*b*d*g^4) - log(e*((a + b*x)/ 
(c + d*x))^n)*((2*A*B)/(3*a^3*b*g^4 + 3*b^4*g^4*x^3 + 9*a^2*b^2*g^4*x + 9* 
a*b^3*g^4*x^2) + (2*B^2*d^3*(x*(b*((b*g^4*n*(a*d - b*c)*(3*a*d - b*c))/(2* 
d^2) + (a*b*g^4*n*(a*d - b*c))/d) + (2*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))/d^2) + a*((b*g^4*n*(a*d - b*c)*(3*a*d - b* 
c))/(2*d^2) + (a*b*g^4*n*(a*d - b*c))/d) + (b*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^3*g^4*n*x^2*(a*d - b*c))/d))/(3*b*g^4* 
(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(3*a^3*b*g^4 + 3*b^4*g 
^4*x^3 + 9*a^2*b^2*g^4*x + 9*a*b^3*g^4*x^2))) - log(e*((a + b*x)/(c + d*x) 
)^n)^2*(B^2/(3*b*(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)/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2) 
)) - (B*d^3*n*atan((B*d^3*n*(6*A + 11*B*n)*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - 
 a*b^3*c^2*d*g^4 - a^2*b^2*c*d^2*g^4)/(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b 
^2*c*d*g^4) + 2*b*d*x)*(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*d*g^4)*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1598, normalized size of antiderivative = 3.57 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(36*log(a + b*x)*a**5*b*d**3*n + 54*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 + 
 162*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 + 162*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 + 54*log(a + b*x)*a*b**5*d**3*n**2*x**3 + 12*lo 
g(a + b*x)*b**6*c*d**2*n**2*x**3 - 36*log(c + d*x)*a**5*b*d**3*n - 54*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*lo 
g(c + d*x)*a**3*b**3*c*d**2*n**2 - 162*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 - 162*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 - 54*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*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**3*c*d**2 + 54*log(((a + b*x)** 
n*e)/(c + d*x)**n)**2*a**3*b**3*d**3*x - 54*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a**2*b**4*c**2*d + 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b 
**4*d**3*x**2 + 18*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*c**3 + 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 + 108*log(((a + b*x)**n*e)/(c + d*x)**n)* 
a**4*b**2*c*d**2 - 54*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b**2*d**3...