\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^5} \, dx\) [18]

Optimal result

Integrand size = 35, antiderivative size = 615 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d^3 n^2 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 n^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d n^2 (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 n^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}+\frac {2 B d^3 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B d n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \] Output:

2*B^2*d^3*n^2*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3/4*b*B^2*d^2*n^2*(d*x+c)^2 
/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/9*b^2*B^2*d*n^2*(d*x+c)^3/(-a*d+b*c)^4/g^5/( 
b*x+a)^3-1/32*b^3*B^2*n^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+2*B*d^3*n*( 
d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*B*d^ 
2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(b*x+a)^2+2 
/3*b^2*B*d*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(b 
*x+a)^3-1/8*b^3*B*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4 
/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/ 
g^5/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b* 
c)^4/g^5/(b*x+a)^2+b^2*d*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d 
+b*c)^4/g^5/(b*x+a)^3-1/4*b^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/ 
(-a*d+b*c)^4/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 = 0.64 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.14 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {72 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n \left (36 A (b c-a d)^4+9 B (b c-a d)^4 n+48 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 n (a+b x)+72 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 n (a+b x)^2+144 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) n (a+b x)^3-144 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 n (a+b x)^4 \log (a+b x)+72 B d^4 n (a+b x)^4 \log ^2(a+b x)+36 B (b c-a d)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+48 B d (-b c+a d)^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+72 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 B d^3 (-b c+a d) (a+b x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 n (a+b x)^4 \log (c+d x)-144 B d^4 n (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)+72 B d^4 n (a+b x)^4 \log ^2(c+d x)-144 B d^4 n (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 n (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 n (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{288 b g^5 (a+b x)^4} \] Input:

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

Output:

-1/288*(72*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (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 + 300* 
B*d^3*(-(b*c) + a*d)*n*(a + b*x)^3 - 144*A*d^4*(a + b*x)^4*Log[a + b*x] - 
300*B*d^4*n*(a + b*x)^4*Log[a + b*x] + 72*B*d^4*n*(a + b*x)^4*Log[a + b*x] 
^2 + 36*B*(b*c - a*d)^4*Log[e*((a + b*x)/(c + d*x))^n] + 48*B*d*(-(b*c) + 
a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 72*B*d^2*(b*c - a*d)^2*( 
a + b*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 144*B*d^3*(-(b*c) + a*d)*(a + 
b*x)^3*Log[e*((a + b*x)/(c + d*x))^n] - 144*B*d^4*(a + b*x)^4*Log[a + b*x] 
*Log[e*((a + b*x)/(c + d*x))^n] + 144*A*d^4*(a + b*x)^4*Log[c + d*x] + 300 
*B*d^4*n*(a + b*x)^4*Log[c + d*x] - 144*B*d^4*n*(a + b*x)^4*Log[(d*(a + b* 
x))/(-(b*c) + a*d)]*Log[c + d*x] + 144*B*d^4*(a + b*x)^4*Log[e*((a + b*x)/ 
(c + d*x))^n]*Log[c + d*x] + 72*B*d^4*n*(a + b*x)^4*Log[c + d*x]^2 - 144*B 
*d^4*n*(a + b*x)^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 144*B*d^4 
*n*(a + b*x)^4*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 144*B*d^4*n*(a + 
 b*x)^4*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(b*c - a*d)^4)/(b*g^5*(a + 
 b*x)^4)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 473, 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.

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

\(\Big \downarrow \) 2949

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((2*B^2*d^3*n^2*(c + d*x))/(a + b*x) - (3*b*B^2*d^2*n^2*(c + d*x)^2)/(4*(a 
 + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3)/(9*(a + b*x)^3) - (b^3*B^2*n^2* 
(c + d*x)^4)/(32*(a + b*x)^4) + (2*B*d^3*n*(c + d*x)*(A + B*Log[e*((a + b* 
x)/(c + d*x))^n]))/(a + b*x) - (3*b*B*d^2*n*(c + d*x)^2*(A + B*Log[e*((a + 
 b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + (2*b^2*B*d*n*(c + d*x)^3*(A + B*Lo 
g[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) - (b^3*B*n*(c + d*x)^4*(A + 
 B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(a + b*x)^4) + (d^3*(c + d*x)*(A + 
B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + 
 B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3 
*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x)^3 - (b^3*(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)^4* 
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 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. \(2325\) vs. \(2(599)=1198\).

Time = 27.96 (sec) , antiderivative size = 2326, normalized size of antiderivative = 3.78

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2326\)

Input:

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

Output:

1/288*(72*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b^3*c*d^4*n+300*B^2*x^4* 
ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c*d^4*n^2+300*A*B*x^4*a^6*b^3*c*d^4*n^2- 
576*A*B*x^4*a^5*b^4*c^2*d^3*n^2+432*A*B*x^4*a^4*b^5*c^3*d^2*n^2-192*A*B*x^ 
4*a^3*b^6*c^4*d*n^2+288*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b^2*c*d^4* 
n+1056*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c*d^4*n^2+144*B^2*x^3*ln( 
e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^2*d^3*n^2+1056*A*B*x^3*a^7*b^2*c*d^4*n^2- 
2160*A*B*x^3*a^6*b^3*c^2*d^3*n^2+1728*A*B*x^3*a^5*b^4*c^3*d^2*n^2-768*A*B* 
x^3*a^4*b^5*c^4*d*n^2+432*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*b*c*d^4* 
n+9*B^2*x^4*a^2*b^7*c^5*n^3+36*B^2*x^3*a^3*b^6*c^5*n^3+72*A^2*x^4*a^2*b^7* 
c^5*n+54*B^2*x^2*a^4*b^5*c^5*n^3+288*A^2*x^3*a^3*b^6*c^5*n+576*B^2*x*a^9*c 
*d^4*n^3+36*B^2*x*a^5*b^4*c^5*n^3+432*A^2*x^2*a^4*b^5*c^5*n+288*B^2*ln(e*( 
(b*x+a)/(d*x+c))^n)^2*a^9*c^2*d^3*n-72*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6 
*b^3*c^5*n+576*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^9*c^2*d^3*n^2-36*B^2*ln(e*( 
(b*x+a)/(d*x+c))^n)*a^6*b^3*c^5*n^2+288*A^2*x*a^9*c*d^4*n+288*A^2*x*a^5*b^ 
4*c^5*n+144*A*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c*d^4*n+576*A*B*x^3* 
ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c*d^4*n+864*A*B*x^2*ln(e*((b*x+a)/(d*x+c 
))^n)*a^8*b*c*d^4*n+1296*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^8*b*c*d^4*n^2 
+576*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c^2*d^3*n^2-72*B^2*x^2*ln(e 
*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^3*d^2*n^2+1296*A*B*x^2*a^8*b*c*d^4*n^2-288 
0*A*B*x^2*a^7*b^2*c^2*d^3*n^2+2520*A*B*x^2*a^6*b^3*c^3*d^2*n^2-1152*A*B...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1762 vs. \(2 (599) = 1198\).

Time = 0.13 (sec) , antiderivative size = 1762, normalized size of antiderivative = 2.87 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/288*(72*A^2*b^4*c^4 - 288*A^2*a*b^3*c^3*d + 432*A^2*a^2*b^2*c^2*d^2 - 2 
88*A^2*a^3*b*c*d^3 + 72*A^2*a^4*d^4 - 12*(25*(B^2*b^4*c*d^3 - B^2*a*b^3*d^ 
4)*n^2 + 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*n)*x^3 + (9*B^2*b^4*c^4 - 64*B 
^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 576*B^2*a^3*b*c*d^3 + 415*B^2*a 
^4*d^4)*n^2 + 6*((13*B^2*b^4*c^2*d^2 - 176*B^2*a*b^3*c*d^3 + 163*B^2*a^2*b 
^2*d^4)*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) 
*n)*x^2 + 72*(B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4* 
B^2*a^3*b*c*d^3 + B^2*a^4*d^4)*log(e)^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2* 
a*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*a^3*b*d^4*n^2*x - (B 
^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3 
)*n^2)*log((b*x + a)/(d*x + c))^2 + 12*(3*A*B*b^4*c^4 - 16*A*B*a*b^3*c^3*d 
 + 36*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 25*A*B*a^4*d^4)*n - 4*((7 
*B^2*b^4*c^3*d - 60*B^2*a*b^3*c^2*d^2 + 324*B^2*a^2*b^2*c*d^3 - 271*B^2*a^ 
3*b*d^4)*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)*n)*x + 12*(12*A*B*b^4*c^4 - 48*A*B*a*b^3*c^3*d + 7 
2*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 12*A*B*a^4*d^4 - 12*(B^2*b^4* 
c*d^3 - B^2*a*b^3*d^4)*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)*n*x^2 - 4*(B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a 
^2*b^2*c*d^3 - 13*B^2*a^3*b*d^4)*n*x + (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d 
 + 36*B^2*a^2*b^2*c^2*d^2 - 48*B^2*a^3*b*c*d^3 + 25*B^2*a^4*d^4)*n - 12...
 

Sympy [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {\int \frac {A^{2}}{a^{5} + 5 a^{4} b x + 10 a^{3} b^{2} x^{2} + 10 a^{2} b^{3} x^{3} + 5 a b^{4} x^{4} + b^{5} x^{5}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{5} + 5 a^{4} b x + 10 a^{3} b^{2} x^{2} + 10 a^{2} b^{3} x^{3} + 5 a b^{4} x^{4} + b^{5} x^{5}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{5} + 5 a^{4} b x + 10 a^{3} b^{2} x^{2} + 10 a^{2} b^{3} x^{3} + 5 a b^{4} x^{4} + b^{5} x^{5}}\, dx}{g^{5}} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**5,x)
 

Output:

(Integral(A**2/(a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x**3 
+ 5*a*b**4*x**4 + b**5*x**5), x) + Integral(B**2*log(e*(a/(c + d*x) + b*x/ 
(c + d*x))**n)**2/(a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x* 
*3 + 5*a*b**4*x**4 + b**5*x**5), x) + Integral(2*A*B*log(e*(a/(c + d*x) + 
b*x/(c + d*x))**n)/(a**5 + 5*a**4*b*x + 10*a**3*b**2*x**2 + 10*a**2*b**3*x 
**3 + 5*a*b**4*x**4 + b**5*x**5), x))/g**5
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2136 vs. \(2 (599) = 1198\).

Time = 0.19 (sec) , antiderivative size = 2136, normalized size of antiderivative = 3.47 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*A*B*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)) 
+ 1/288*(12*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)*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1206 vs. \(2 (599) = 1198\).

Time = 1.51 (sec) , antiderivative size = 1206, normalized size of antiderivative = 1.96 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/288*(72*(B^2*b^3*n^2 - 4*(b*x + a)*B^2*b^2*d*n^2/(d*x + c) + 6*(b*x + a 
)^2*B^2*b*d^2*n^2/(d*x + c)^2 - 4*(b*x + a)^3*B^2*d^3*n^2/(d*x + c)^3)*log 
((b*x + a)/(d*x + c))^2/((b*x + a)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*x + a) 
^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 
 - (b*x + a)^4*a^3*d^3*g^5/(d*x + c)^4) + 12*(3*B^2*b^3*n^2 - 16*(b*x + a) 
*B^2*b^2*d*n^2/(d*x + c) + 36*(b*x + a)^2*B^2*b*d^2*n^2/(d*x + c)^2 - 48*( 
b*x + a)^3*B^2*d^3*n^2/(d*x + c)^3 + 12*B^2*b^3*n*log(e) - 48*(b*x + a)*B^ 
2*b^2*d*n*log(e)/(d*x + c) + 72*(b*x + a)^2*B^2*b*d^2*n*log(e)/(d*x + c)^2 
 - 48*(b*x + a)^3*B^2*d^3*n*log(e)/(d*x + c)^3 + 12*A*B*b^3*n - 48*(b*x + 
a)*A*B*b^2*d*n/(d*x + c) + 72*(b*x + a)^2*A*B*b*d^2*n/(d*x + c)^2 - 48*(b* 
x + a)^3*A*B*d^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/((b*x + a)^4*b^3* 
c^3*g^5/(d*x + c)^4 - 3*(b*x + a)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + 
 a)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x + a)^4*a^3*d^3*g^5/(d*x + c)^4) + 
 (9*B^2*b^3*n^2 - 64*(b*x + a)*B^2*b^2*d*n^2/(d*x + c) + 216*(b*x + a)^2*B 
^2*b*d^2*n^2/(d*x + c)^2 - 576*(b*x + a)^3*B^2*d^3*n^2/(d*x + c)^3 + 36*B^ 
2*b^3*n*log(e) - 192*(b*x + a)*B^2*b^2*d*n*log(e)/(d*x + c) + 432*(b*x + a 
)^2*B^2*b*d^2*n*log(e)/(d*x + c)^2 - 576*(b*x + a)^3*B^2*d^3*n*log(e)/(d*x 
 + c)^3 + 72*B^2*b^3*log(e)^2 - 288*(b*x + a)*B^2*b^2*d*log(e)^2/(d*x + c) 
 + 432*(b*x + a)^2*B^2*b*d^2*log(e)^2/(d*x + c)^2 - 288*(b*x + a)^3*B^2*d^ 
3*log(e)^2/(d*x + c)^3 + 36*A*B*b^3*n - 192*(b*x + a)*A*B*b^2*d*n/(d*x ...
 

Mupad [B] (verification not implemented)

Time = 30.71 (sec) , antiderivative size = 1769, normalized size of antiderivative = 2.88 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

(B*d^4*n*atan((B*d^4*n*(12*A + 25*B*n)*(24*b^5*c^4*g^5 - 24*a^4*b*d^4*g^5 
- 48*a*b^4*c^3*d*g^5 + 48*a^3*b^2*c*d^3*g^5)*1i)/(24*b*g^5*(25*B^2*d^4*n^2 
 + 12*A*B*d^4*n)*(a*d - b*c)^4) + (B*d^5*n*x*(12*A + 25*B*n)*(b^4*c^3*g^5 
- a^3*b*d^3*g^5 - 3*a*b^3*c^2*d*g^5 + 3*a^2*b^2*c*d^2*g^5)*2i)/(g^5*(25*B^ 
2*d^4*n^2 + 12*A*B*d^4*n)*(a*d - b*c)^4))*(12*A + 25*B*n)*1i)/(12*b*g^5*(a 
*d - b*c)^4) - ((72*A^2*a^3*d^3 - 72*A^2*b^3*c^3 + 415*B^2*a^3*d^3*n^2 - 9 
*B^2*b^3*c^3*n^2 + 216*A^2*a*b^2*c^2*d - 216*A^2*a^2*b*c*d^2 + 300*A*B*a^3 
*d^3*n - 36*A*B*b^3*c^3*n + 55*B^2*a*b^2*c^2*d*n^2 - 161*B^2*a^2*b*c*d^2*n 
^2 + 156*A*B*a*b^2*c^2*d*n - 276*A*B*a^2*b*c*d^2*n)/(12*(a*d - b*c)) + (x^ 
2*(163*B^2*a*b^2*d^3*n^2 - 13*B^2*b^3*c*d^2*n^2 + 84*A*B*a*b^2*d^3*n - 12* 
A*B*b^3*c*d^2*n))/(2*(a*d - b*c)) + (x*(271*B^2*a^2*b*d^3*n^2 + 7*B^2*b^3* 
c^2*d*n^2 - 53*B^2*a*b^2*c*d^2*n^2 + 156*A*B*a^2*b*d^3*n + 12*A*B*b^3*c^2* 
d*n - 60*A*B*a*b^2*c*d^2*n))/(3*(a*d - b*c)) + (d*x^3*(25*B^2*b^3*d^2*n^2 
+ 12*A*B*b^3*d^2*n))/(a*d - b*c))/(x*(96*a^3*b^4*c^2*g^5 + 96*a^5*b^2*d^2* 
g^5 - 192*a^4*b^3*c*d*g^5) + x^3*(96*a*b^6*c^2*g^5 + 96*a^3*b^4*d^2*g^5 - 
192*a^2*b^5*c*d*g^5) + x^4*(24*b^7*c^2*g^5 + 24*a^2*b^5*d^2*g^5 - 48*a*b^6 
*c*d*g^5) + x^2*(144*a^2*b^5*c^2*g^5 + 144*a^4*b^3*d^2*g^5 - 288*a^3*b^4*c 
*d*g^5) + 24*a^6*b*d^2*g^5 + 24*a^4*b^3*c^2*g^5 - 48*a^5*b^2*c*d*g^5) - lo 
g(e*((a + b*x)/(c + d*x))^n)^2*(B^2/(4*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^2*d^4)/(4*b*g^5*(a^4...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 2328, normalized size of antiderivative = 3.79 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(144*log(a + b*x)*a**6*b*d**4*n + 264*log(a + b*x)*a**5*b**2*d**4*n**2 + 5 
76*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 
 + 1056*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 + 1584*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*log 
(a + b*x)*a**2*b**5*c*d**3*n**2*x**2 + 1056*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 + 264*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 - 264*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 - 1056*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 - 1584*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 - 1056*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 - 264*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 + 288*log(((a + b*x)**n 
*e)/(c + d*x)**n)**2*a**4*b**3*c*d**3 + 288*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a**4*b**3*d**4*x - 432*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3* 
b**4*c**2*d**2 + 432*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**4*d*...