\(\int (a g+b g x)^2 (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [56]

Optimal result

Integrand size = 40, antiderivative size = 450 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=-\frac {B^2 (b c-a d)^3 g^2 i x}{3 b d^2}+\frac {B^2 (b c-a d)^2 g^2 i (c+d x)^2}{12 d^3}-\frac {B (b c-a d)^2 g^2 i (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d}-\frac {B (b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^2}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{12 b^2}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b}+\frac {B (b c-a d)^3 g^2 i (a+b x) \left (2 A+B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d^2}+\frac {B (b c-a d)^4 g^2 i \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (2 A+3 B+2 B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{12 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i \log (c+d x)}{6 b^2 d^3}+\frac {B^2 (b c-a d)^4 g^2 i \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.51 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {g^2 i \left (4 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+3 d (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {4 B (b c-a d)^2 \left (2 A b d (b c-a d) x+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}-\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x+6 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-6 B (b c-a d)^3 \log (c+d x)-6 (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{12 b^2} \] Input:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2962, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2773

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a*g + b*g*x)^2*(c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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)])^2)/ 
(4*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^4) - (B*(((a + b*x)^3*(A + 
B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + 
d*x))^3) - (B*(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)])^2)/(3*b*(c 
+ d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*(((a + b*x)^2*(A + B*Log[ 
(e*(a + b*x))/(c + d*x)]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x))/(c + d*x))^ 
2) - (((a + b*x)*(2*A + B + 2*B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(c + d*x 
)*(b - (d*(a + b*x))/(c + d*x))) - (-(((2*A + 3*B + 2*B*Log[(e*(a + b*x))/ 
(c + d*x)])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (2*B*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 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [F]

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

Input:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2243 vs. \(2 (429) = 858\).

Time = 0.16 (sec) , antiderivative size = 2243, normalized size of antiderivative = 4.98 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(g**2*i*( - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + 
b*d*x**2),x)*a**4*b**2*d**5 + 8*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c 
+ a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4 - 12*int((log((a*e + b*e*x 
)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**4*c**2*d**3 + 
8*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x) 
*a*b**5*c**3*d**2 - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + 
b*c*x + b*d*x**2),x)*b**6*c**4*d - 2*log(a + b*x)*a**5*d**4 + 8*log(a + b* 
x)*a**4*b*c*d**3 - log(a + b*x)*a**4*b*d**4 - 12*log(a + b*x)*a**3*b**2*c* 
*2*d**2 + 4*log(a + b*x)*a**3*b**2*c*d**3 + 8*log(a + b*x)*a**2*b**3*c**3* 
d - 6*log(a + b*x)*a**2*b**3*c**2*d**2 - 2*log(a + b*x)*a*b**4*c**4 + 4*lo 
g(a + b*x)*a*b**4*c**3*d - log(a + b*x)*b**5*c**4 + log((a*e + b*e*x)/(c + 
 d*x))**2*a**3*b**2*c*d**3 + 3*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c 
**2*d**2 + 12*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**3*c*d**3*x + 6*log(( 
a*e + b*e*x)/(c + d*x))**2*a**2*b**3*d**4*x**2 - log((a*e + b*e*x)/(c + d* 
x))**2*a*b**4*c**3*d + 12*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*c*d**3*x* 
*2 + 8*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*d**4*x**3 + 4*log((a*e + b*e 
*x)/(c + d*x))**2*b**5*c*d**3*x**3 + 3*log((a*e + b*e*x)/(c + d*x))**2*b** 
5*d**4*x**4 + 12*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c**2*d**2 + 24*log 
((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d**3*x + 2*log((a*e + b*e*x)/(c + d* 
x))*a**3*b**2*c*d**3 + 12*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*d**4*x...