\(\int (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [181]

Optimal result

Integrand size = 35, antiderivative size = 454 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {5 B^2 (b c-a d)^3 i^3 n^2 x}{12 b^3}+\frac {B^2 (b c-a d)^2 i^3 n^2 (c+d x)^2}{12 b^2 d}-\frac {B (b c-a d)^3 i^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4}-\frac {B (b c-a d)^2 i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}+\frac {i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d}+\frac {5 B^2 (b c-a d)^4 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d}+\frac {11 B^2 (b c-a d)^4 i^3 n^2 \log (c+d x)}{12 b^4 d}+\frac {B (b c-a d)^4 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{2 b^4 d}-\frac {B^2 (b c-a d)^4 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{2 b^4 d} \] Output:

5/12*B^2*(-a*d+b*c)^3*i^3*n^2*x/b^3+1/12*B^2*(-a*d+b*c)^2*i^3*n^2*(d*x+c)^ 
2/b^2/d-1/2*B*(-a*d+b*c)^3*i^3*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b 
^4-1/4*B*(-a*d+b*c)^2*i^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/ 
d-1/6*B*(-a*d+b*c)*i^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d+1/4 
*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/d+5/12*B^2*(-a*d+b*c)^4*i 
^3*n^2*ln((b*x+a)/(d*x+c))/b^4/d+11/12*B^2*(-a*d+b*c)^4*i^3*n^2*ln(d*x+c)/ 
b^4/d+1/2*B*(-a*d+b*c)^4*i^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x 
+c)/d/(b*x+a))/b^4/d-1/2*B^2*(-a*d+b*c)^4*i^3*n^2*polylog(2,b*(d*x+c)/d/(b 
*x+a))/b^4/d
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.90 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {i^3 \left ((c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 n (b d x+(b c-a d) \log (a+b x))-B (b c-a d) n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-3 B (b c-a d)^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{3 b^4}\right )}{4 d} \] Input:

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

Output:

(i^3*((c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*(b*c - a*d 
)*n*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*n*(b*d*x + (b*c - a*d)*Lo 
g[a + b*x]) - B*(b*c - a*d)*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*( 
b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x 
)/(c + d*x))^n] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n]) + 2*b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 
6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*B* 
(b*c - a*d)^3*n*Log[c + d*x] - 3*B*(b*c - a*d)^3*n*(Log[a + b*x]*(Log[a + 
b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b 
*c) + a*d)])))/(3*b^4)))/(4*d)
 

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2951, 2756, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2951

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {c+d x}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 54

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2751

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B n \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle i^3 (b c-a d)^4 \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{3 d}\right )}{b}\right )}{2 d}\right )\)

Input:

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

Output:

(b*c - a*d)^4*i^3*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*d*(b - (d*( 
a + b*x))/(c + d*x))^4) - (B*n*((d*((A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
/(3*d*(b - (d*(a + b*x))/(c + d*x))^3) - (B*n*(1/(2*b*(b - (d*(a + b*x))/( 
c + d*x))^2) + 1/(b^2*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + 
d*x)]/b^3 - Log[b - (d*(a + b*x))/(c + d*x)]/b^3))/(3*d)))/b + ((d*((A + B 
*Log[e*((a + b*x)/(c + d*x))^n])/(2*d*(b - (d*(a + b*x))/(c + d*x))^2) - ( 
B*n*(1/(b*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^2 - 
Log[b - (d*(a + b*x))/(c + d*x)]/b^2))/(2*d)))/b + ((d*(((a + b*x)*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n]))/(b*(c + d*x)*(b - (d*(a + b*x))/(c + d*x) 
)) + (B*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b*d)))/b + (-(((A + B*Log[e*( 
(a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (B*n*P 
olyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b)/b)/b)/b))/(2*d))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2951

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/d)^m   Subst[Int[(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] && NeQ[b*c 
- a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, - 
1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2129 vs. \(2 (433) = 866\).

Time = 0.57 (sec) , antiderivative size = 2129, normalized size of antiderivative = 4.69 \[ \int (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*A*B*d^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*d^3*i 
^3*x^4 + 2*A*B*c*d^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2* 
c*d^2*i^3*x^3 + 3*A*B*c^2*d*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) 
 + 3/2*A^2*c^2*d*i^3*x^2 - 1/12*A*B*d^3*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6* 
c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2 
*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) + A*B*c*d^2*i^3*n*(2*a^3 
*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2* 
(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) - 3*A*B*c^2*d*i^3*n*(a^2*log(b*x + a)/b^ 
2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*c^3*i^3*n*(a*log(b 
*x + a)/b - c*log(d*x + c)/d) + 2*A*B*c^3*i^3*x*log(e*(b*x/(d*x + c) + a/( 
d*x + c))^n) + A^2*c^3*i^3*x - 1/12*(26*a*b^2*c^3*d*i^3*n^2 - 21*a^2*b*c^2 
*d^2*i^3*n^2 + 6*a^3*c*d^3*i^3*n^2 - (11*i^3*n^2 - 6*i^3*n*log(e))*b^3*c^4 
)*B^2*log(d*x + c)/(b^3*d) - 1/2*(b^4*c^4*i^3*n^2 - 4*a*b^3*c^3*d*i^3*n^2 
+ 6*a^2*b^2*c^2*d^2*i^3*n^2 - 4*a^3*b*c*d^3*i^3*n^2 + a^4*d^4*i^3*n^2)*(lo 
g(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c 
- a*d)))*B^2/(b^4*d) + 1/12*(3*B^2*b^4*d^4*i^3*x^4*log(e)^2 + 6*B^2*b^4*c^ 
4*i^3*n^2*log(b*x + a)*log(d*x + c) - 3*B^2*b^4*c^4*i^3*n^2*log(d*x + c)^2 
 + 2*(a*b^3*d^4*i^3*n*log(e) - (i^3*n*log(e) - 6*i^3*log(e)^2)*b^4*c*d^3)* 
B^2*x^3 + ((i^3*n^2 - 9*i^3*n*log(e) + 18*i^3*log(e)^2)*b^4*c^2*d^2 - 2*(i 
^3*n^2 - 6*i^3*n*log(e))*a*b^3*c*d^3 + (i^3*n^2 - 3*i^3*n*log(e))*a^2*b...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(i*(6*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b* 
d*x**2),x)*a**4*b**2*d**5*n - 24*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x 
)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4*n + 36*int((log(((a 
 + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b* 
*4*c**2*d**3*n - 24*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d* 
x + b*c*x + b*d*x**2),x)*a*b**5*c**3*d**2*n + 6*int((log(((a + b*x)**n*e)/ 
(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**4*d*n + 6*log 
(c + d*x)*a**5*d**4*n - 24*log(c + d*x)*a**4*b*c*d**3*n - 11*log(c + d*x)* 
a**4*b*d**4*n**2 + 36*log(c + d*x)*a**3*b**2*c**2*d**2*n + 44*log(c + d*x) 
*a**3*b**2*c*d**3*n**2 - 24*log(c + d*x)*a**2*b**3*c**3*d*n - 66*log(c + d 
*x)*a**2*b**3*c**2*d**2*n**2 + 6*log(c + d*x)*a*b**4*c**4*n + 44*log(c + d 
*x)*a*b**4*c**3*d*n**2 - 11*log(c + d*x)*b**5*c**4*n**2 - 3*log(((a + b*x) 
**n*e)/(c + d*x)**n)**2*a**3*b**2*c*d**3 + 9*log(((a + b*x)**n*e)/(c + d*x 
)**n)**2*a**2*b**3*c**2*d**2 - 9*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b 
**4*c**3*d - 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**5*c**3*d*x - 18*l 
og(((a + b*x)**n*e)/(c + d*x)**n)**2*b**5*c**2*d**2*x**2 - 12*log(((a + b* 
x)**n*e)/(c + d*x)**n)**2*b**5*c*d**3*x**3 - 3*log(((a + b*x)**n*e)/(c + d 
*x)**n)**2*b**5*d**4*x**4 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*d**4 
 - 24*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b*c*d**3 - 11*log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**4*b*d**4*n + 36*log(((a + b*x)**n*e)/(c + d*x)*...