\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{a g+b g x} \, dx\) [68]

Optimal result

Integrand size = 42, antiderivative size = 535 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=-\frac {B d (b c-a d) i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g}+\frac {2 B (b c-a d)^2 i^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g}+\frac {d (b c-a d) i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^3 g}+\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b g}+\frac {B^2 (b c-a d)^2 i^2 \log (c+d x)}{b^3 g}+\frac {B (b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B^2 (b c-a d)^2 i^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g}-\frac {B^2 (b c-a d)^2 i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B (b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B^2 (b c-a d)^2 i^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2615\) vs. \(2(535)=1070\).

Time = 4.37 (sec) , antiderivative size = 2615, normalized size of antiderivative = 4.89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=\text {Result too large to show} \] Input:

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

Output:

(i^2*(12*A^2*b*d*(2*b*c - a*d)*x + 6*A^2*b^2*d^2*x^2 + 12*A^2*(b*c - a*d)^ 
2*Log[a + b*x] - 24*A*b*B*c*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + 
Log[a + b*x]) + 2*(-(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a 
*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + (-(b*d*x) + a*d*Log[a + b*x])*Log[ 
(e*(a + b*x))/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 
 12*A*b^2*B*c^2*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x] - Log[c/d + 
 x] - Log[(e*(a + b*x))/(c + d*x)]) - 2*(Log[c/d + x]*Log[(d*(a + b*x))/(- 
(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 6*A*B*(-4*a*d^2* 
(a + b*x)*(-1 + Log[a/b + x]) + 2*a^2*d^2*Log[a/b + x]^2 + 4*a*b*d*(c + d* 
x)*(-1 + Log[c/d + x]) + d^2*(b*x*(2*a - b*x) + 2*b^2*x^2*Log[a/b + x] - 2 
*a^2*Log[a + b*x]) - 2*d^2*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x])*(Log[a/ 
b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d*x)]) + b^2*(d*x*(-2*c + d 
*x) - 2*d^2*x^2*Log[c/d + x] + 2*c^2*Log[c + d*x]) - 4*a^2*d^2*(Log[c/d + 
x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c - a*d 
)])) - 8*b*B^2*c*(a*d*Log[a/b + x]^3 - 3*d*(2*b*x - 2*(a + b*x)*Log[a/b + 
x] + (a + b*x)*Log[a/b + x]^2) - 3*b*(2*d*x - 2*(c + d*x)*Log[c/d + x] + ( 
c + d*x)*Log[c/d + x]^2) - 3*d*(b*x - a*Log[a + b*x])*(-Log[a/b + x] + Log 
[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^2 + 6*(a*d + 2*b*d*x - b*d*x*Log 
[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*x)* 
Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d...
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2962, 2789, 2756, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2755

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 2751

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

rule 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 2755

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy 
mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) 
 Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, 
n, p}, x] && GtQ[p, 0]
 

rule 2756

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

rule 2779

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

rule 2789

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

rule 2821

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

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

i**2*(Integral(A**2*c**2/(a + b*x), x) + Integral(A**2*d**2*x**2/(a + b*x) 
, x) + Integral(B**2*c**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))**2/(a + b*x 
), x) + Integral(2*A*B*c**2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x) 
, x) + Integral(2*A**2*c*d*x/(a + b*x), x) + Integral(B**2*d**2*x**2*log(a 
*e/(c + d*x) + b*e*x/(c + d*x))**2/(a + b*x), x) + Integral(2*A*B*d**2*x** 
2*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x) + Integral(2*B**2*c*d 
*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))**2/(a + b*x), x) + Integral(4*A*B* 
c*d*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(a + b*x), x))/g
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a g+b g x} \, dx=\frac {-2 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2}}{b x +a}d x \right ) b^{5} c^{2}-4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )}{b x +a}d x \right ) a \,b^{4} c^{2}-2 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} x^{2}}{b x +a}d x \right ) b^{5} d^{2}-4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} x}{b x +a}d x \right ) b^{5} c d -4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x^{2}}{b x +a}d x \right ) a \,b^{4} d^{2}-8 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{b x +a}d x \right ) a \,b^{4} c d -2 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{2}+4 \,\mathrm {log}\left (b x +a \right ) a^{3} b c d -2 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c^{2}+2 a^{3} b \,d^{2} x -4 a^{2} b^{2} c d x -a^{2} b^{2} d^{2} x^{2}}{2 b^{3} g} \] Input:

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

Output:

( - 2*int(log((a*e + b*e*x)/(c + d*x))**2/(a + b*x),x)*b**5*c**2 - 4*int(l 
og((a*e + b*e*x)/(c + d*x))/(a + b*x),x)*a*b**4*c**2 - 2*int((log((a*e + b 
*e*x)/(c + d*x))**2*x**2)/(a + b*x),x)*b**5*d**2 - 4*int((log((a*e + b*e*x 
)/(c + d*x))**2*x)/(a + b*x),x)*b**5*c*d - 4*int((log((a*e + b*e*x)/(c + d 
*x))*x**2)/(a + b*x),x)*a*b**4*d**2 - 8*int((log((a*e + b*e*x)/(c + d*x))* 
x)/(a + b*x),x)*a*b**4*c*d - 2*log(a + b*x)*a**4*d**2 + 4*log(a + b*x)*a** 
3*b*c*d - 2*log(a + b*x)*a**2*b**2*c**2 + 2*a**3*b*d**2*x - 4*a**2*b**2*c* 
d*x - a**2*b**2*d**2*x**2)/(2*b**3*g)