\(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [179]

Optimal result

Integrand size = 45, antiderivative size = 976 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Output:

-7/180*B^2*(-a*d+b*c)^5*g^2*i^3*n^2*x/b^3/d^2-7/360*B^2*(-a*d+b*c)^4*g^2*i 
^3*n^2*(d*x+c)^2/b^2/d^3-1/60*B^2*(-a*d+b*c)^3*g^2*i^3*n^2*(d*x+c)^3/b/d^3 
+1/60*B^2*(-a*d+b*c)^2*g^2*i^3*n^2*(d*x+c)^4/d^3-1/60*B*(-a*d+b*c)^4*g^2*i 
^3*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d-1/30*B*(-a*d+b*c)^3*g 
^2*i^3*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4-1/10*B*(-a*d+b*c)^4 
*g^2*i^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/d^3+1/45*B*(-a*d+ 
b*c)^3*g^2*i^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^3+7/60*B*(- 
a*d+b*c)^2*g^2*i^3*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/15*b* 
B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/60* 
(-a*d+b*c)^3*g^2*i^3*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4+1/20* 
(-a*d+b*c)^2*g^2*i^3*(b*x+a)^3*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b 
^3+1/10*(-a*d+b*c)*g^2*i^3*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c)) 
^n))^2/b^2+1/6*g^2*i^3*(b*x+a)^3*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)) 
^2/b+1/60*B*(-a*d+b*c)^5*g^2*i^3*n*(b*x+a)*(2*A+B*n+2*B*ln(e*((b*x+a)/(d*x 
+c))^n))/b^4/d^2+1/60*B*(-a*d+b*c)^6*g^2*i^3*n*(2*A+3*B*n+2*B*ln(e*((b*x+a 
)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^4/d^3+1/36*B^2*(-a*d+b*c)^6*g^2* 
i^3*n^2*ln((b*x+a)/(d*x+c))/b^4/d^3+11/180*B^2*(-a*d+b*c)^6*g^2*i^3*n^2*ln 
(d*x+c)/b^4/d^3+1/30*B^2*(-a*d+b*c)^6*g^2*i^3*n^2*polylog(2,d*(b*x+a)/b/(d 
*x+c))/b^4/d^3
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 1627, normalized size of antiderivative = 1.67 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g^2*i^3*(15*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n])^2 - 24*b*(b*c - a*d)*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n] 
)^2 + 10*b^2*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (5*B*( 
b*c - a*d)^3*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)*Log[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)])))/b^4 + (2*B*(b*c - a*d)^2*n*(24*A*b*d*(b*c - a*d) 
^3*x - 12*B*(b*c - a*d)^3*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c 
- a*d)^2*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]) - B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + 
d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - 
 a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 12*b^2*(b*c - a*d)^2*(c 
 + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*b^3*(b*c - a*d)*(c + 
d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*b^4*(c + d*x)^4*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n]) + 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*L...
 

Rubi [A] (verified)

Time = 3.28 (sec) , antiderivative size = 1266, normalized size of antiderivative = 1.30, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2961, 2783, 2782, 27, 1195, 2009, 2783, 2782, 27, 1195, 2009, 2783, 2773, 49, 2009, 2781, 2784, 2784, 2754, 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.

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

\(\Big \downarrow \) 2961

\(\displaystyle g^2 i^3 (b c-a d)^6 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2782

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \left (-B n \int \frac {(c+d x) \left (b^2-\frac {5 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{30 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{3 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \left (-\frac {B n \int \frac {(c+d x) \left (b^2-\frac {5 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{30 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{3 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \left (-\frac {B n \int \left (\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {9 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {6 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {c+d x}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{30 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{3 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{2 b}-\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{30 d^3}\right )}{3 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 2783

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {-\frac {2 B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{2 b}-\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{30 d^3}\right )}{3 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 2782

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {-\frac {2 B n \left (-B n \int \frac {(c+d x) \left (b^2-\frac {4 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{12 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{2 b}-\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \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 {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{30 d^3}\right )}{3 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {-\frac {2 B n \left (-\frac {B n \int \frac {(c+d x) \left (b^2-\frac {4 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{12 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{2 b}-\frac {B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^3 \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 {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{30 d^3}\right )}{3 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \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 {5 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{12 d^3}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2783

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2773

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \frac {(a+b x)^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}\right )}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \left (\frac {b^2}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{3 b}\right )}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2781

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {2 B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}}{2 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle (b c-a d)^6 g^2 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{6 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {3 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\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}\right )}{30 d^3}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\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}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}-\frac {2 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}}{2 b}\right )\)

Input:

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

Output:

(b*c - a*d)^6*g^2*i^3*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
^2)/(6*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^6) - (B*n*((b^2*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3*(b - (d*(a + b*x))/(c + d*x))^5) - 
 (b*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d^3*(b - (d*(a + b*x))/(c + 
 d*x))^4) + (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*d^3*(b - (d*(a + b*x 
))/(c + d*x))^3) - (B*n*((3*b)/(2*(b - (d*(a + b*x))/(c + d*x))^4) - 3/(b 
- (d*(a + b*x))/(c + d*x))^3 + 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))/(30*d^3)))/(3*b) + (((a + b*x)^3*(A + B 
*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b*(c + d*x)^3*(b - (d*(a + b*x))/(c 
 + d*x))^5) - (2*B*n*((b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^3* 
(b - (d*(a + b*x))/(c + d*x))^4) - (2*b*(A + B*Log[e*((a + b*x)/(c + d*x)) 
^n]))/(3*d^3*(b - (d*(a + b*x))/(c + d*x))^3) + (A + B*Log[e*((a + b*x)/(c 
 + d*x))^n])/(2*d^3*(b - (d*(a + b*x))/(c + d*x))^2) - (B*n*(b/(b - (d*(a 
+ b*x))/(c + d*x))^3 - 5/(2*(b - (d*(a + b*x))/(c + d*x))^2) + 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))/(12*d^3)))/(5*b) + (2*(((a + b*x)^3*(A + B*Log[e*(( 
a + b*x)/(c + d*x))^n])^2)/(4*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^ 
4) - (B*n*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*(c + 
d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (B*n*(b^2/(2*d^3*(b - (d*(a +...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

rule 2754

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

rule 2773

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

rule 2781

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

rule 2782

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q 
_), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* 
x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
 

rule 2783

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a 
+ b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Simp[(m + q + 2)/(d*(q + 1))   Int[ 
(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Simp[b*n*(p/(d*(q 
+ 1)))   Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) 
/; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L 
tQ[q, -1] && GtQ[m, 0]
 

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

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 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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(A^2*b^2*d^3*g^2*i^3*x^5 + A^2*a^2*c^3*g^2*i^3 + (3*A^2*b^2*c*d^2 
+ 2*A^2*a*b*d^3)*g^2*i^3*x^4 + (3*A^2*b^2*c^2*d + 6*A^2*a*b*c*d^2 + A^2*a^ 
2*d^3)*g^2*i^3*x^3 + (A^2*b^2*c^3 + 6*A^2*a*b*c^2*d + 3*A^2*a^2*c*d^2)*g^2 
*i^3*x^2 + (2*A^2*a*b*c^3 + 3*A^2*a^2*c^2*d)*g^2*i^3*x + (B^2*b^2*d^3*g^2* 
i^3*x^5 + B^2*a^2*c^3*g^2*i^3 + (3*B^2*b^2*c*d^2 + 2*B^2*a*b*d^3)*g^2*i^3* 
x^4 + (3*B^2*b^2*c^2*d + 6*B^2*a*b*c*d^2 + B^2*a^2*d^3)*g^2*i^3*x^3 + (B^2 
*b^2*c^3 + 6*B^2*a*b*c^2*d + 3*B^2*a^2*c*d^2)*g^2*i^3*x^2 + (2*B^2*a*b*c^3 
 + 3*B^2*a^2*c^2*d)*g^2*i^3*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b 
^2*d^3*g^2*i^3*x^5 + A*B*a^2*c^3*g^2*i^3 + (3*A*B*b^2*c*d^2 + 2*A*B*a*b*d^ 
3)*g^2*i^3*x^4 + (3*A*B*b^2*c^2*d + 6*A*B*a*b*c*d^2 + A*B*a^2*d^3)*g^2*i^3 
*x^3 + (A*B*b^2*c^3 + 6*A*B*a*b*c^2*d + 3*A*B*a^2*c*d^2)*g^2*i^3*x^2 + (2* 
A*B*a*b*c^3 + 3*A*B*a^2*c^2*d)*g^2*i^3*x)*log(e*((b*x + a)/(d*x + c))^n), 
x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5931 vs. \(2 (937) = 1874\).

Time = 0.66 (sec) , antiderivative size = 5931, normalized size of antiderivative = 6.08 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*A*B*b^2*d^3*g^2*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A 
^2*b^2*d^3*g^2*i^3*x^6 + 6/5*A*B*b^2*c*d^2*g^2*i^3*x^5*log(e*(b*x/(d*x + c 
) + a/(d*x + c))^n) + 4/5*A*B*a*b*d^3*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a 
/(d*x + c))^n) + 3/5*A^2*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A^2*a*b*d^3*g^2*i^3*x 
^5 + 3/2*A*B*b^2*c^2*d*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) 
+ 3*A*B*a*b*c*d^2*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2 
*A*B*a^2*d^3*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A^2* 
b^2*c^2*d*g^2*i^3*x^4 + 3/2*A^2*a*b*c*d^2*g^2*i^3*x^4 + 1/4*A^2*a^2*d^3*g^ 
2*i^3*x^4 + 2/3*A*B*b^2*c^3*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c) 
)^n) + 4*A*B*a*b*c^2*d*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) 
+ 2*A*B*a^2*c*d^2*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3 
*A^2*b^2*c^3*g^2*i^3*x^3 + 2*A^2*a*b*c^2*d*g^2*i^3*x^3 + A^2*a^2*c*d^2*g^2 
*i^3*x^3 + 2*A*B*a*b*c^3*g^2*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n 
) + 3*A*B*a^2*c^2*d*g^2*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A 
^2*a*b*c^3*g^2*i^3*x^2 + 3/2*A^2*a^2*c^2*d*g^2*i^3*x^2 - 1/180*A*B*b^2*d^3 
*g^2*i^3*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c 
*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d 
^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5 
*d^5)*x)/(b^5*d^5)) + 1/10*A*B*b^2*c*d^2*g^2*i^3*n*(12*a^5*log(b*x + a)/b^ 
5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**2*i*(12*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**6*b**2*d**7*n - 72*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**5*b**3*c*d**6*n + 180*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**4*c**2*d**5*n - 240*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**5*c**3*d**4*n + 180*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** 
6*c**4*d**3*n - 72*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**7*c**5*d**2*n + 12*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**8*c**6*d*n + 12*lo 
g(a + b*x)*a**7*d**6*n - 72*log(a + b*x)*a**6*b*c*d**5*n - 4*log(a + b*x)* 
a**6*b*d**6*n**2 + 180*log(a + b*x)*a**5*b**2*c**2*d**4*n + 24*log(a + b*x 
)*a**5*b**2*c*d**5*n**2 - 240*log(a + b*x)*a**4*b**3*c**3*d**3*n - 60*log( 
a + b*x)*a**4*b**3*c**2*d**4*n**2 + 180*log(a + b*x)*a**3*b**4*c**4*d**2*n 
 + 80*log(a + b*x)*a**3*b**4*c**3*d**3*n**2 - 72*log(a + b*x)*a**2*b**5*c* 
*5*d*n - 60*log(a + b*x)*a**2*b**5*c**4*d**2*n**2 + 12*log(a + b*x)*a*b**6 
*c**6*n + 24*log(a + b*x)*a*b**6*c**5*d*n**2 - 4*log(a + b*x)*b**7*c**6*n* 
*2 - 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**5*b**2*c*d**5 + 30*log(((a 
 + b*x)**n*e)/(c + d*x)**n)**2*a**4*b**3*c**2*d**4 - 60*log(((a + b*x)**n* 
e)/(c + d*x)**n)**2*a**3*b**4*c**3*d**3 - 30*log(((a + b*x)**n*e)/(c + ...