\(\int (a g+b g x)^2 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [160]

Optimal result

Integrand size = 43, antiderivative size = 487 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B^2 (b c-a d)^3 g^2 i n^2 x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i n^2 (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.47 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {g^2 i \left (4 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+3 d (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 n (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{12 b^2} \] Input:

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

Output:

(g^2*i*(4*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 
 + 3*d*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (4*B*(b*c - 
a*d)^2*n*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)*Log[e*((a + 
b*x)/(c + d*x))^n] - d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] 
) - 2*B*(b*c - a*d)^2*n*Log[c + d*x] - 2*(b*c - a*d)^2*(A + B*Log[e*((a + 
b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(b*d*x + (-(b*c) + a*d) 
*Log[c + d*x]) + B*(b*c - a*d)^2*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - 
 Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d 
^3 - (B*(b*c - a*d)*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + 
b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A 
+ B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3*(A + B*Log[e*((a + 
 b*x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 6*(b*c - a*d)^3* 
(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(2*b 
*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*( 
b*c - a*d)^2*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*n 
*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Po 
lyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^3))/(12*b^2)
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2961, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2773

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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]

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

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

g**2*i*(Integral(A**2*a**2*c, x) + Integral(A**2*a**2*d*x, x) + Integral(A 
**2*b**2*c*x**2, x) + Integral(A**2*b**2*d*x**3, x) + Integral(B**2*a**2*c 
*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2, x) + Integral(2*A*B*a**2*c*lo 
g(e*(a/(c + d*x) + b*x/(c + d*x))**n), x) + Integral(2*A**2*a*b*c*x, x) + 
Integral(2*A**2*a*b*d*x**2, x) + Integral(B**2*a**2*d*x*log(e*(a/(c + d*x) 
 + b*x/(c + d*x))**n)**2, x) + Integral(B**2*b**2*c*x**2*log(e*(a/(c + d*x 
) + b*x/(c + d*x))**n)**2, x) + Integral(B**2*b**2*d*x**3*log(e*(a/(c + d* 
x) + b*x/(c + d*x))**n)**2, x) + Integral(2*A*B*a**2*d*x*log(e*(a/(c + d*x 
) + b*x/(c + d*x))**n), x) + Integral(2*A*B*b**2*c*x**2*log(e*(a/(c + d*x) 
 + b*x/(c + d*x))**n), x) + Integral(2*A*B*b**2*d*x**3*log(e*(a/(c + d*x) 
+ b*x/(c + d*x))**n), x) + Integral(2*B**2*a*b*c*x*log(e*(a/(c + d*x) + b* 
x/(c + d*x))**n)**2, x) + Integral(2*B**2*a*b*d*x**2*log(e*(a/(c + d*x) + 
b*x/(c + d*x))**n)**2, x) + Integral(4*A*B*a*b*c*x*log(e*(a/(c + d*x) + b* 
x/(c + d*x))**n), x) + Integral(4*A*B*a*b*d*x**2*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. 2691 vs. \(2 (466) = 932\).

Time = 0.60 (sec) , antiderivative size = 2691, normalized size of antiderivative = 5.53 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*A*B*b^2*d*g^2*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*b 
^2*d*g^2*i*x^4 + 2/3*A*B*b^2*c*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c 
))^n) + 4/3*A*B*a*b*d*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1 
/3*A^2*b^2*c*g^2*i*x^3 + 2/3*A^2*a*b*d*g^2*i*x^3 + 2*A*B*a*b*c*g^2*i*x^2*l 
og(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*B*a^2*d*g^2*i*x^2*log(e*(b*x/(d* 
x + c) + a/(d*x + c))^n) + A^2*a*b*c*g^2*i*x^2 + 1/2*A^2*a^2*d*g^2*i*x^2 - 
 1/12*A*B*b^2*d*g^2*i*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)) + 1/3*A*B*b^2*c*g^2*i*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)) + 2/3*A*B*a*b*d*g^2*i*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)) - 2*A*B*a*b*c*g^2*i*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 
 + (b*c - a*d)*x/(b*d)) - A*B*a^2*d*g^2*i*n*(a^2*log(b*x + a)/b^2 - c^2*lo 
g(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*a^2*c*g^2*i*n*(a*log(b*x + a 
)/b - c*log(d*x + c)/d) + 2*A*B*a^2*c*g^2*i*x*log(e*(b*x/(d*x + c) + a/(d* 
x + c))^n) + A^2*a^2*c*g^2*i*x - 1/12*(2*a^3*c*d^3*g^2*i*n^2 + (g^2*i*n^2 
+ 2*g^2*i*n*log(e))*b^3*c^4 - 2*(g^2*i*n^2 + 4*g^2*i*n*log(e))*a*b^2*c^3*d 
 - (g^2*i*n^2 - 12*g^2*i*n*log(e))*a^2*b*c^2*d^2)*B^2*log(d*x + c)/(b*d^3) 
 - 1/6*(b^4*c^4*g^2*i*n^2 - 4*a*b^3*c^3*d*g^2*i*n^2 + 6*a^2*b^2*c^2*d^2...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(g**2*i*( - 2*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 + 8*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 - 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**2*b**4*c**2*d**3*n + 8*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 - 2*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 - 
 2*log(a + b*x)*a**5*d**4*n + 8*log(a + b*x)*a**4*b*c*d**3*n - log(a + b*x 
)*a**4*b*d**4*n**2 - 12*log(a + b*x)*a**3*b**2*c**2*d**2*n + 4*log(a + b*x 
)*a**3*b**2*c*d**3*n**2 + 8*log(a + b*x)*a**2*b**3*c**3*d*n - 6*log(a + b* 
x)*a**2*b**3*c**2*d**2*n**2 - 2*log(a + b*x)*a*b**4*c**4*n + 4*log(a + b*x 
)*a*b**4*c**3*d*n**2 - log(a + b*x)*b**5*c**4*n**2 + log(((a + b*x)**n*e)/ 
(c + d*x)**n)**2*a**3*b**2*c*d**3 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)** 
2*a**2*b**3*c**2*d**2 + 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3 
*c*d**3*x + 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3*d**4*x**2 - 
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*a*b**4*c*d**3*x**2 + 8*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a*b**4*d**4*x**3 + 4*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 + 12*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2*c**2*d**2 + 2*log(((a + b*x)...