\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx\) [209]

Optimal result

Integrand size = 45, antiderivative size = 908 \[ \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 (c i+d i x)^3} \, dx=-\frac {B^2 d^5 n^2 (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {10 A b B d^4 n (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {10 b B^2 d^4 n^2 (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {20 b^3 B^2 d^2 n^2 (c+d x)}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B^2 d n^2 (c+d x)^2}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {2 b^5 B^2 n^2 (c+d x)^3}{27 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {B d^5 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)^6 g^4 i^3 (c+d x)^2}-\frac {20 b^3 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)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 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 )}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {2 b^5 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)^6 g^4 i^3 (a+b x)^3}-\frac {d^5 (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)^6 g^4 i^3 (c+d x)^2}+\frac {5 b d^4 (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)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 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)^6 g^4 i^3 (a+b x)}+\frac {5 b^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}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 (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)^6 g^4 i^3 (a+b x)^3}-\frac {10 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^6 g^4 i^3 n} \] Output:

-1/4*B^2*d^5*n^2*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-10*A*b*B*d^4*n*( 
b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+10*b*B^2*d^4*n^2*(b*x+a)/(-a*d+b*c)^6/ 
g^4/i^3/(d*x+c)-20*b^3*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/ 
4*b^4*B^2*d*n^2*(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-2/27*b^5*B^2*n^2* 
(d*x+c)^3/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-10*b*B^2*d^4*n*(b*x+a)*ln(e*((b*x 
+a)/(d*x+c))^n)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+1/2*B*d^5*n*(b*x+a)^2*(A+B*ln 
(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-20*b^3*B*d^2*n*(d* 
x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^4* 
B*d*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+ 
a)^2-2/9*b^5*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^ 
4/i^3/(b*x+a)^3-1/2*d^5*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+ 
b*c)^6/g^4/i^3/(d*x+c)^2+5*b*d^4*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2 
/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*d^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c 
))^n))^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^4*d*(d*x+c)^2*(A+B*ln(e*((b*x+ 
a)/(d*x+c))^n))^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-1/3*b^5*(d*x+c)^3*(A+B*ln 
(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-10/3*b^2*d^3*(A+ 
B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^6/g^4/i^3/n
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2138\) vs. \(2(908)=1816\).

Time = 3.52 (sec) , antiderivative size = 2138, normalized size of antiderivative = 2.35 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

-1/108*(360*b^2*B^2*d^3*n^2*(a + b*x)^3*(c + d*x)^2*Log[(a + b*x)/(c + d*x 
)]^3 + 18*B*n*Log[(a + b*x)/(c + d*x)]^2*(60*a^3*A*b^2*c^2*d^3 + 2*b^5*B*c 
^5*n - 15*a*b^4*B*c^4*d*n + 60*a^2*b^3*B*c^3*d^2*n - 30*a^4*b*B*c*d^4*n + 
3*a^5*B*d^5*n + 180*a^2*A*b^3*c^2*d^3*x + 120*a^3*A*b^2*c*d^4*x - 5*b^5*B* 
c^4*d*n*x + 60*a*b^4*B*c^3*d^2*n*x + 180*a^2*b^3*B*c^2*d^3*n*x - 120*a^3*b 
^2*B*c*d^4*n*x - 15*a^4*b*B*d^5*n*x + 180*a*A*b^4*c^2*d^3*x^2 + 360*a^2*A* 
b^3*c*d^4*x^2 + 60*a^3*A*b^2*d^5*x^2 + 20*b^5*B*c^3*d^2*n*x^2 + 270*a*b^4* 
B*c^2*d^3*n*x^2 - 90*a^3*b^2*B*d^5*n*x^2 + 60*A*b^5*c^2*d^3*x^3 + 360*a*A* 
b^4*c*d^4*x^3 + 180*a^2*A*b^3*d^5*x^3 + 110*b^5*B*c^2*d^3*n*x^3 + 180*a*b^ 
4*B*c*d^4*n*x^3 - 90*a^2*b^3*B*d^5*n*x^3 + 120*A*b^5*c*d^4*x^4 + 180*a*A*b 
^4*d^5*x^4 + 100*b^5*B*c*d^4*n*x^4 + 60*A*b^5*d^5*x^5 + 20*b^5*B*d^5*n*x^5 
 + 60*b^2*B*d^3*(a + b*x)^3*(c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n] - 6 
0*b^2*B*d^3*n*(a + b*x)^3*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]) + 6*b^2*d^ 
2*(b*c - a*d)*(a + b*x)^2*(c + d*x)^2*(108*A^2 + 282*A*B*n + 319*B^2*n^2 + 
 108*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(36*A + 47*B*n)*Log[(a + 
 b*x)/(c + d*x)] + 108*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a 
+ b*x)/(c + d*x))^n]*(36*A + 47*B*n - 36*B*n*Log[(a + b*x)/(c + d*x)])) - 
3*b^2*d*(b*c - a*d)^2*(a + b*x)*(c + d*x)^2*(54*A^2 + 66*A*B*n + 37*B^2*n^ 
2 + 54*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(18*A + 11*B*n)*Log[(a 
 + b*x)/(c + d*x)] + 54*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*...
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 653, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

\(\displaystyle \frac {\int \left (\frac {(c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^5}{(a+b x)^4}-\frac {5 d (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^4}{(a+b x)^3}+\frac {10 d^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^3}{(a+b x)^2}-\frac {10 d^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^2}{a+b x}+5 d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b-\frac {d^5 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x}\right )d\frac {a+b x}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^5 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b^5 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}+\frac {5 b^4 B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {10 b^3 d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {20 b^3 B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {10 b^2 d^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B n}-\frac {d^5 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (c+d x)^2}+\frac {B d^5 n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {10 A b B d^4 n (a+b x)}{c+d x}-\frac {2 b^5 B^2 n^2 (c+d x)^3}{27 (a+b x)^3}+\frac {5 b^4 B^2 d n^2 (c+d x)^2}{4 (a+b x)^2}-\frac {20 b^3 B^2 d^2 n^2 (c+d x)}{a+b x}-\frac {B^2 d^5 n^2 (a+b x)^2}{4 (c+d x)^2}-\frac {10 b B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {10 b B^2 d^4 n^2 (a+b x)}{c+d x}}{g^4 i^3 (b c-a d)^6}\)

Input:

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

Output:

(-1/4*(B^2*d^5*n^2*(a + b*x)^2)/(c + d*x)^2 - (10*A*b*B*d^4*n*(a + b*x))/( 
c + d*x) + (10*b*B^2*d^4*n^2*(a + b*x))/(c + d*x) - (20*b^3*B^2*d^2*n^2*(c 
 + d*x))/(a + b*x) + (5*b^4*B^2*d*n^2*(c + d*x)^2)/(4*(a + b*x)^2) - (2*b^ 
5*B^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (10*b*B^2*d^4*n*(a + b*x)*Log[e* 
((a + b*x)/(c + d*x))^n])/(c + d*x) + (B*d^5*n*(a + b*x)^2*(A + B*Log[e*(( 
a + b*x)/(c + d*x))^n]))/(2*(c + d*x)^2) - (20*b^3*B*d^2*n*(c + d*x)*(A + 
B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (5*b^4*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^5*B*n*(c + d* 
x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^3) - (d^5*(a + b 
*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(c + d*x)^2) + (5*b*d^4 
*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(c + d*x) - (10*b^3*d 
^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) + (5*b^4* 
d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a + b*x)^2) - 
(b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(a + b*x)^3) 
 - (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*n))/((b*c - 
a*d)^6*g^4*i^3)
 

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. \(4485\) vs. \(2(888)=1776\).

Time = 210.38 (sec) , antiderivative size = 4486, normalized size of antiderivative = 4.94

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(4486\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x,method 
=_RETURNVERBOSE)
 

Output:

-1/108*(27*B^2*a^5*b^5*d^9*n^3+8*B^2*b^10*c^5*d^4*n^3+54*A^2*a^5*b^5*d^9*n 
+36*A^2*b^10*c^5*d^4*n+6480*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^8*c^2*d^ 
7*n^2+1080*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^10*c^2*d^7+1320*B^2*x^2*l 
n(e*((b*x+a)/(d*x+c))^n)*b^10*c^3*d^6*n^2-5880*B^2*x^2*a^2*b^8*c*d^8*n^3+9 
210*B^2*x^2*a*b^9*c^2*d^7*n^3+6480*A^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^9 
*c*d^8+1080*A^2*x^3*a*b^9*c*d^8*n+1080*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2 
*a^3*b^7*d^9+840*A*B*x^2*a^3*b^7*d^9*n^2+1320*A*B*x^2*b^10*c^3*d^6*n^2+720 
*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^3*b^7*c*d^8+1080*B^2*x*ln(e*((b*x+a)/ 
(d*x+c))^n)^3*a^2*b^8*c^2*d^7-270*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b^ 
6*d^9*n-90*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^10*c^4*d^5*n+810*B^2*x*ln(e 
*((b*x+a)/(d*x+c))^n)*a^4*b^6*d^9*n^2-150*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)* 
b^10*c^4*d^5*n^2-6280*B^2*x*a^3*b^7*c*d^8*n^3+3540*B^2*x*a^2*b^8*c^2*d^7*n 
^3+3780*B^2*x*a*b^9*c^3*d^6*n^3+6480*A^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2 
*b^8*c*d^8+3240*A^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^9*c^2*d^7-2160*A^2*x 
^2*a^2*b^8*c*d^8*n+3780*A^2*x^2*a*b^9*c^2*d^7*n+810*A*B*x*a^4*b^6*d^9*n^2- 
150*A*B*x*b^10*c^4*d^5*n^2-540*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b^6*c*d 
^8*n+1080*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^8*c^3*d^6*n-270*B^2*ln(e*( 
(b*x+a)/(d*x+c))^n)^2*a*b^9*c^4*d^5*n+1080*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a 
^4*b^6*c*d^8*n^2+2160*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^8*c^3*d^6*n^2-27 
0*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^9*c^4*d^5*n^2+2160*A^2*x*ln(e*((b*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4725 vs. \(2 (888) = 1776\).

Time = 0.25 (sec) , antiderivative size = 4725, normalized size of antiderivative = 5.20 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x 
, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9293 vs. \(2 (888) = 1776\).

Time = 0.81 (sec) , antiderivative size = 9293, normalized size of antiderivative = 10.23 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x 
, algorithm="maxima")
 

Output:

-1/6*B^2*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^ 
2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*( 
2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a 
*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7* 
c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5* 
b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^3 
+ 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d 
^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5 
*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7* 
b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25 
*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - 
 a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b^4*c^5 
*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c* 
d^6)*g^4*i^3*x + (a^3*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10* 
a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2*d^3*log 
(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3* 
d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3* 
log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c 
^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(e*(b* 
x/(d*x + c) + a/(d*x + c))^n)^2 - 1/3*A*B*((60*b^4*d^4*x^4 + 2*b^4*c^4 ...
 

Giac [F]

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

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

Mupad [B] (verification not implemented)

Time = 35.04 (sec) , antiderivative size = 4649, normalized size of antiderivative = 5.12 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log(e*((a + b*x)/(c + d*x))^n)*((x*((a*d + b*c)*(20*A*B*a*b*d^2 + 10*A*B*b 
^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*B^2*b^2*c*d*n)/3) + a*c*(30*A*B*b^2*d^ 
2 - 20*B^2*b^2*d^2*n) + (5*B^2*a^2*b*d^3*n)/6 + (5*B^2*b^3*c^2*d*n)/6 - 5* 
A*B*a^2*b*d^3 - 5*A*B*b^3*c^2*d + 10*A*B*a*b^2*c*d^2 - (5*B^2*a*b^2*c*d^2* 
n)/3) + x^2*((a*d + b*c)*(30*A*B*b^2*d^2 - 20*B^2*b^2*d^2*n) + b*d*(20*A*B 
*a*b*d^2 + 10*A*B*b^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*B^2*b^2*c*d*n)/3)) 
+ a*c*(20*A*B*a*b*d^2 + 10*A*B*b^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*B^2*b^ 
2*c*d*n)/3) - 3*A*B*a^3*d^3 - 2*A*B*b^3*c^3 + b*d*x^3*(30*A*B*b^2*d^2 - 20 
*B^2*b^2*d^2*n) + (3*B^2*a^3*d^3*n)/2 - (2*B^2*b^3*c^3*n)/3 + A*B*a*b^2*c^ 
2*d + 4*A*B*a^2*b*c*d^2 + (17*B^2*a*b^2*c^2*d*n)/6 - (11*B^2*a^2*b*c*d^2*n 
)/3)/(x^5*(3*a^4*b^3*d^6*g^4*i^3 + 3*b^7*c^4*d^2*g^4*i^3 - 12*a*b^6*c^3*d^ 
3*g^4*i^3 - 12*a^3*b^4*c*d^5*g^4*i^3 + 18*a^2*b^5*c^2*d^4*g^4*i^3) + x*(9* 
a^2*b^5*c^6*g^4*i^3 + 6*a^7*c*d^5*g^4*i^3 - 30*a^3*b^4*c^5*d*g^4*i^3 - 15* 
a^6*b*c^2*d^4*g^4*i^3 + 30*a^4*b^3*c^4*d^2*g^4*i^3) + x^2*(3*a^7*d^6*g^4*i 
^3 + 9*a*b^6*c^6*g^4*i^3 + 6*a^6*b*c*d^5*g^4*i^3 - 18*a^2*b^5*c^5*d*g^4*i^ 
3 - 15*a^3*b^4*c^4*d^2*g^4*i^3 + 60*a^4*b^3*c^3*d^3*g^4*i^3 - 45*a^5*b^2*c 
^2*d^4*g^4*i^3) + x^3*(3*b^7*c^6*g^4*i^3 + 9*a^6*b*d^6*g^4*i^3 + 6*a*b^6*c 
^5*d*g^4*i^3 - 18*a^5*b^2*c*d^5*g^4*i^3 - 45*a^2*b^5*c^4*d^2*g^4*i^3 + 60* 
a^3*b^4*c^3*d^3*g^4*i^3 - 15*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(9*a^5*b^2*d^6 
*g^4*i^3 + 6*b^7*c^5*d*g^4*i^3 - 15*a*b^6*c^4*d^2*g^4*i^3 - 30*a^4*b^3*...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 7917, normalized size of antiderivative = 8.72 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 3240*log(a + b*x)*a**6*b**2*c**2*d**4*n - 6480*log(a + b*x)*a**6*b* 
*2*c*d**5*n*x - 3240*log(a + b*x)*a**6*b**2*d**6*n*x**2 - 2160*log(a + b*x 
)*a**5*b**3*c**3*d**3*n - 14040*log(a + b*x)*a**5*b**3*c**2*d**4*n*x - 216 
00*log(a + b*x)*a**5*b**3*c*d**5*n*x**2 - 9720*log(a + b*x)*a**5*b**3*d**6 
*n*x**3 - 3600*log(a + b*x)*a**4*b**4*c**3*d**3*n**2 - 6480*log(a + b*x)*a 
**4*b**4*c**3*d**3*n*x - 8100*log(a + b*x)*a**4*b**4*c**2*d**4*n**3 - 7200 
*log(a + b*x)*a**4*b**4*c**2*d**4*n**2*x - 22680*log(a + b*x)*a**4*b**4*c* 
*2*d**4*n*x**2 - 16200*log(a + b*x)*a**4*b**4*c*d**5*n**3*x - 3600*log(a + 
 b*x)*a**4*b**4*c*d**5*n**2*x**2 - 25920*log(a + b*x)*a**4*b**4*c*d**5*n*x 
**3 - 8100*log(a + b*x)*a**4*b**4*d**6*n**3*x**2 - 9720*log(a + b*x)*a**4* 
b**4*d**6*n*x**4 - 6600*log(a + b*x)*a**3*b**5*c**3*d**3*n**3 - 10800*log( 
a + b*x)*a**3*b**5*c**3*d**3*n**2*x - 6480*log(a + b*x)*a**3*b**5*c**3*d** 
3*n*x**2 - 37500*log(a + b*x)*a**3*b**5*c**2*d**4*n**3*x - 21600*log(a + b 
*x)*a**3*b**5*c**2*d**4*n**2*x**2 - 16200*log(a + b*x)*a**3*b**5*c**2*d**4 
*n*x**3 - 55200*log(a + b*x)*a**3*b**5*c*d**5*n**3*x**2 - 10800*log(a + b* 
x)*a**3*b**5*c*d**5*n**2*x**3 - 12960*log(a + b*x)*a**3*b**5*c*d**5*n*x**4 
 - 24300*log(a + b*x)*a**3*b**5*d**6*n**3*x**3 - 3240*log(a + b*x)*a**3*b* 
*5*d**6*n*x**5 - 19800*log(a + b*x)*a**2*b**6*c**3*d**3*n**3*x - 10800*log 
(a + b*x)*a**2*b**6*c**3*d**3*n**2*x**2 - 2160*log(a + b*x)*a**2*b**6*c**3 
*d**3*n*x**3 - 63900*log(a + b*x)*a**2*b**6*c**2*d**4*n**3*x**2 - 21600...