3.3.8 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx\) [208]

3.3.8.1 Optimal result

Integrand size = 45, antiderivative size = 732 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\frac {B^2 d^4 n^2 (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {8 A b B d^3 n (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}-\frac {8 b B^2 d^3 n^2 (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {8 b^3 B^2 d n^2 (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B^2 n^2 (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {8 b B^2 d^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}-\frac {B d^4 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {8 b^3 B d n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {2 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{B (b c-a d)^5 g^3 i^3 n} \]

output

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

3.3.8.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1653\) vs. \(2(732)=1464\).

Time = 1.17 (sec) , antiderivative size = 1653, normalized size of antiderivative = 2.26 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx =\text {Too large to display} \]

input

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^3*(c*i + 
 d*i*x)^3),x]
 

output

(8*b^2*B^2*d^2*n^2*(a + b*x)^2*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]^3 + 2* 
B*n*Log[(a + b*x)/(c + d*x)]^2*(12*a^2*A*b^2*c^2*d^2 - b^4*B*c^4*n + 8*a*b 
^3*B*c^3*d*n - 8*a^3*b*B*c*d^3*n + a^4*B*d^4*n + 24*a*A*b^3*c^2*d^2*x + 24 
*a^2*A*b^2*c*d^3*x + 4*b^4*B*c^3*d*n*x + 24*a*b^3*B*c^2*d^2*n*x - 24*a^2*b 
^2*B*c*d^3*n*x - 4*a^3*b*B*d^4*n*x + 12*A*b^4*c^2*d^2*x^2 + 48*a*A*b^3*c*d 
^3*x^2 + 12*a^2*A*b^2*d^4*x^2 + 18*b^4*B*c^2*d^2*n*x^2 - 18*a^2*b^2*B*d^4* 
n*x^2 + 24*A*b^4*c*d^3*x^3 + 24*a*A*b^3*d^4*x^3 + 12*b^4*B*c*d^3*n*x^3 - 1 
2*a*b^3*B*d^4*n*x^3 + 12*A*b^4*d^4*x^4 + 12*b^2*B*d^2*(a + b*x)^2*(c + d*x 
)^2*Log[e*((a + b*x)/(c + d*x))^n] - 12*b^2*B*d^2*n*(a + b*x)^2*(c + d*x)^ 
2*Log[(a + b*x)/(c + d*x)]) + 12*b^2*d^2*(a + b*x)^2*(c + d*x)^2*Log[a + b 
*x]*(2*A^2 + 5*B^2*n^2 + 4*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a 
+ b*x)/(c + d*x)]) + 2*B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b* 
x)/(c + d*x)])^2) + 2*b^2*d*(b*c - a*d)*(a + b*x)*(c + d*x)^2*(6*A^2 + 14* 
A*B*n + 15*B^2*n^2 + 6*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(6*A + 
 7*B*n)*Log[(a + b*x)/(c + d*x)] + 6*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 
2*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 7*B*n - 6*B*n*Log[(a + b*x)/(c + 
 d*x)])) - b^2*(b*c - a*d)^2*(c + d*x)^2*(2*A^2 + 2*A*B*n + B^2*n^2 + 2*B^ 
2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(2*A + B*n)*Log[(a + b*x)/(c + 
d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d 
*x))^n]*(2*A + B*n - 2*B*n*Log[(a + b*x)/(c + d*x)])) + 2*B*(b*c - a*d)...
 

3.3.8.3 Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.72, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

input

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^3*(c*i + d*i*x 
)^3),x]
 

output

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

3.3.8.3.1 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]
 

3.3.8.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3005\) vs. \(2(720)=1440\).

Time = 54.53 (sec) , antiderivative size = 3006, normalized size of antiderivative = 4.11

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(3006\)

input

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x,method 
=_RETURNVERBOSE)
 

output

-1/4*(-32*B^2*a^3*b^5*c*d^7*n^3+32*B^2*a*b^7*c^3*d^5*n^3-2*A*B*a^4*b^4*d^8 
*n^2-2*A*B*b^8*c^4*d^4*n^2-16*A^2*a^3*b^5*c*d^7*n+16*A^2*a*b^7*c^3*d^5*n+1 
6*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^3*b^8*c*d^7-60*B^2*x^3*a*b^7*d^8*n^3+6 
0*B^2*x^3*b^8*c*d^7*n^3+8*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^6*d^8+ 
8*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3*b^8*c^2*d^6-90*B^2*x^2*a^2*b^6*d^8*n 
^3+90*B^2*x^2*b^8*c^2*d^6*n^3+48*A^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^7*d 
^8+48*A^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^8*c*d^7-24*A^2*x^3*a*b^7*d^8*n+2 
4*A^2*x^3*b^8*c*d^7*n-28*B^2*x*a^3*b^5*d^8*n^3+28*B^2*x*b^8*c^3*d^5*n^3+24 
*A^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^6*d^8+24*A^2*x^2*ln(e*((b*x+a)/(d 
*x+c))^n)*b^8*c^2*d^6-36*A^2*x^2*a^2*b^6*d^8*n+36*A^2*x^2*b^8*c^2*d^6*n+8* 
B^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^6*c^2*d^6+2*B^2*ln(e*((b*x+a)/(d*x+c 
))^n)^2*a^4*b^4*d^8*n-2*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^8*c^4*d^4*n-2*B^ 
2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^4*d^8*n^2-2*B^2*ln(e*((b*x+a)/(d*x+c))^n 
)*b^8*c^4*d^4*n^2-8*A^2*x*a^3*b^5*d^8*n+8*A^2*x*b^8*c^3*d^5*n+24*A^2*ln(e* 
((b*x+a)/(d*x+c))^n)*a^2*b^6*c^2*d^6+60*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)* 
b^8*d^8*n^2+24*A*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*b^8*d^8+16*B^2*x^3*ln(e 
*((b*x+a)/(d*x+c))^n)^3*a*b^7*d^8-96*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b 
^6*c*d^7*n+96*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^7*c^2*d^6*n-48*A*B*x^3*l 
n(e*((b*x+a)/(d*x+c))^n)*a*b^7*d^8*n+48*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)* 
b^8*c*d^7*n+192*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^7*c*d^7*n^2+96*A*...
 

3.3.8.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3062 vs. \(2 (720) = 1440\).

Time = 0.39 (sec) , antiderivative size = 3062, normalized size of antiderivative = 4.18 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x 
, algorithm="fricas")
 

output

-1/4*(2*A^2*b^4*c^4 - 16*A^2*a*b^3*c^3*d + 16*A^2*a^3*b*c*d^3 - 2*A^2*a^4* 
d^4 - 12*(2*A^2*b^4*c*d^3 - 2*A^2*a*b^3*d^4 + 5*(B^2*b^4*c*d^3 - B^2*a*b^3 
*d^4)*n^2)*x^3 - 8*(B^2*b^4*d^4*n^2*x^4 + B^2*a^2*b^2*c^2*d^2*n^2 + 2*(B^2 
*b^4*c*d^3 + B^2*a*b^3*d^4)*n^2*x^3 + (B^2*b^4*c^2*d^2 + 4*B^2*a*b^3*c*d^3 
 + B^2*a^2*b^2*d^4)*n^2*x^2 + 2*(B^2*a*b^3*c^2*d^2 + B^2*a^2*b^2*c*d^3)*n^ 
2*x)*log((b*x + a)/(d*x + c))^3 + (B^2*b^4*c^4 - 32*B^2*a*b^3*c^3*d + 32*B 
^2*a^3*b*c*d^3 - B^2*a^4*d^4)*n^2 - 6*(6*A^2*b^4*c^2*d^2 - 6*A^2*a^2*b^2*d 
^4 + 15*(B^2*b^4*c^2*d^2 - B^2*a^2*b^2*d^4)*n^2 + 4*(A*B*b^4*c^2*d^2 - 2*A 
*B*a*b^3*c*d^3 + A*B*a^2*b^2*d^4)*n)*x^2 + 2*(B^2*b^4*c^4 - 8*B^2*a*b^3*c^ 
3*d + 8*B^2*a^3*b*c*d^3 - B^2*a^4*d^4 - 12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4) 
*x^3 - 18*(B^2*b^4*c^2*d^2 - B^2*a^2*b^2*d^4)*x^2 - 4*(B^2*b^4*c^3*d + 6*B 
^2*a*b^3*c^2*d^2 - 6*B^2*a^2*b^2*c*d^3 - B^2*a^3*b*d^4)*x - 12*(B^2*b^4*d^ 
4*x^4 + B^2*a^2*b^2*c^2*d^2 + 2*(B^2*b^4*c*d^3 + B^2*a*b^3*d^4)*x^3 + (B^2 
*b^4*c^2*d^2 + 4*B^2*a*b^3*c*d^3 + B^2*a^2*b^2*d^4)*x^2 + 2*(B^2*a*b^3*c^2 
*d^2 + B^2*a^2*b^2*c*d^3)*x)*log((b*x + a)/(d*x + c)))*log(e)^2 - 2*(12*A* 
B*b^4*d^4*n*x^4 + 12*A*B*a^2*b^2*c^2*d^2*n + 12*((B^2*b^4*c*d^3 - B^2*a*b^ 
3*d^4)*n^2 + 2*(A*B*b^4*c*d^3 + A*B*a*b^3*d^4)*n)*x^3 - (B^2*b^4*c^4 - 8*B 
^2*a*b^3*c^3*d + 8*B^2*a^3*b*c*d^3 - B^2*a^4*d^4)*n^2 + 6*(3*(B^2*b^4*c^2* 
d^2 - B^2*a^2*b^2*d^4)*n^2 + 2*(A*B*b^4*c^2*d^2 + 4*A*B*a*b^3*c*d^3 + A*B* 
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 - 6*B^2*a...
 

3.3.8.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Timed out} \]

input

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**3/(d*i*x+c*i)** 
3,x)
 

output

Timed out
 

3.3.8.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5594 vs. \(2 (720) = 1440\).

Time = 0.66 (sec) , antiderivative size = 5594, normalized size of antiderivative = 7.64 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x 
, algorithm="maxima")
 

output

1/2*B^2*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d 
^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b 
*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c 
*d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b 
^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)*g^3*i^3*x^3 
+ (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + 
a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 
+ 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*g^3*i^3*x + (a^2*b^4*c^ 
6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)*g 
^3*i^3) + 12*b^2*d^2*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c 
^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^3*i^3) - 12*b^2*d 
^2*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^ 
2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^3*i^3))*log(e*(b*x/(d*x + c) + a/(d 
*x + c))^n)^2 + A*B*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c 
*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c 
*d^2 + a^2*b*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 
 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d 
^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5*b*d^6)* 
g^3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^ 
2*c^2*d^4 + a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a...
 

3.3.8.8 Giac [F]

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

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x 
, algorithm="giac")
 

output

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

3.3.8.9 Mupad [B] (verification not implemented)

Time = 8.43 (sec) , antiderivative size = 2419, normalized size of antiderivative = 3.30 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx=\text {Too large to display} \]

input

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

output

((3*x^2*(6*A^2*a*b^2*d^3 + 6*A^2*b^3*c*d^2 + 15*B^2*a*b^2*d^3*n^2 + 15*B^2 
*b^3*c*d^2*n^2 - 4*A*B*a*b^2*d^3*n + 4*A*B*b^3*c*d^2*n))/(a*d - b*c) - (2* 
A^2*a^3*d^3 + 2*A^2*b^3*c^3 + B^2*a^3*d^3*n^2 + B^2*b^3*c^3*n^2 - 14*A^2*a 
*b^2*c^2*d - 14*A^2*a^2*b*c*d^2 - 2*A*B*a^3*d^3*n + 2*A*B*b^3*c^3*n - 31*B 
^2*a*b^2*c^2*d*n^2 - 31*B^2*a^2*b*c*d^2*n^2 - 30*A*B*a*b^2*c^2*d*n + 30*A* 
B*a^2*b*c*d^2*n)/(2*(a*d - b*c)) + (2*x*(2*A^2*a^2*b*d^3 + 2*A^2*b^3*c^2*d 
 + 14*A^2*a*b^2*c*d^2 + 7*B^2*a^2*b*d^3*n^2 + 7*B^2*b^3*c^2*d*n^2 + 31*B^2 
*a*b^2*c*d^2*n^2 - 6*A*B*a^2*b*d^3*n + 6*A*B*b^3*c^2*d*n))/(a*d - b*c) + ( 
6*x^3*(2*A^2*b^3*d^3 + 5*B^2*b^3*d^3*n^2))/(a*d - b*c))/(x^4*(2*a^3*b^2*d^ 
5*g^3*i^3 - 2*b^5*c^3*d^2*g^3*i^3 + 6*a*b^4*c^2*d^3*g^3*i^3 - 6*a^2*b^3*c* 
d^4*g^3*i^3) - x*(4*a*b^4*c^5*g^3*i^3 - 4*a^5*c*d^4*g^3*i^3 - 8*a^2*b^3*c^ 
4*d*g^3*i^3 + 8*a^4*b*c^2*d^3*g^3*i^3) + x^3*(4*a^4*b*d^5*g^3*i^3 - 4*b^5* 
c^4*d*g^3*i^3 + 8*a*b^4*c^3*d^2*g^3*i^3 - 8*a^3*b^2*c*d^4*g^3*i^3) + x^2*( 
2*a^5*d^5*g^3*i^3 - 2*b^5*c^5*g^3*i^3 - 2*a*b^4*c^4*d*g^3*i^3 + 2*a^4*b*c* 
d^4*g^3*i^3 + 16*a^2*b^3*c^3*d^2*g^3*i^3 - 16*a^3*b^2*c^2*d^3*g^3*i^3) - 2 
*a^2*b^3*c^5*g^3*i^3 + 2*a^5*c^2*d^3*g^3*i^3 + 6*a^3*b^2*c^4*d*g^3*i^3 - 6 
*a^4*b*c^3*d^2*g^3*i^3) + log(e*((a + b*x)/(c + d*x))^n)^2*((x*((3*B^2*b*d 
*(a*d + b*c)^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 - (B^2*b*d)/(a^2*d^2 + b 
^2*c^2 - 2*a*b*c*d) + (6*B^2*a*b^2*c*d^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^ 
2) - (B^2*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (6*B^2*b^3...