\(\int (a g+b g x)^3 (A+B \log (\frac {e (c+d x)}{a+b x}))^2 \, dx\) [183]

Optimal result

Integrand size = 32, antiderivative size = 420 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=-\frac {5 B^2 (b c-a d)^3 g^3 x}{12 d^3}+\frac {B^2 (b c-a d)^2 g^3 (a+b x)^2}{12 b d^2}+\frac {11 B^2 (b c-a d)^4 g^3 \log (a+b x)}{12 b d^4}+\frac {5 B^2 (b c-a d)^4 g^3 \log \left (\frac {c+d x}{a+b x}\right )}{12 b d^4}-\frac {B (b c-a d)^2 g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b d^2}+\frac {B (b c-a d) g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{6 b d}+\frac {B (b c-a d)^3 g^3 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 d^4}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b}+\frac {B (b c-a d)^4 g^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac {B^2 (b c-a d)^4 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.93 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {g^3 \left ((a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B (b c-a d) \left (6 A b d (b c-a d)^2 x+6 B (b c-a d)^3 \log (a+b x)-B (b c-a d) \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-3 B (b c-a d)^2 (b d x+(-b c+a d) \log (c+d x))+6 b B (b c-a d)^2 (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )+3 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+2 d^3 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-6 (b c-a d)^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-3 B (b c-a d)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^4}\right )}{4 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.23, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {2952, 2756, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 54

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

rule 2009

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

rule 2751

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

rule 2756

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

rule 2779

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

rule 2789

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

rule 2838

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

rule 2952

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/d)^m   Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, ( 
a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[ 
n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1735 vs. \(2 (399) = 798\).

Time = 0.15 (sec) , antiderivative size = 1735, normalized size of antiderivative = 4.13 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*A^2*b^3*g^3*x^4 + A^2*a*b^2*g^3*x^3 + 3/2*A^2*a^2*b*g^3*x^2 + 2*(x*log 
(d*e*x/(b*x + a) + c*e/(b*x + a)) - a*log(b*x + a)/b + c*log(d*x + c)/d)*A 
*B*a^3*g^3 + 3*(x^2*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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*b*g^3 + (2*x^3 
*log(d*e*x/(b*x + a) + c*e/(b*x + a)) - 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^2*g^3 + 1/12*(6*x^4*log(d*e*x/(b*x + a) + c*e/(b*x + a)) + 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^3*g^3 + A^2*a^3*g^3*x - 1/12*((6*g^3*log(e) - 11*g^3)*b^3*c^4 - 2*( 
12*g^3*log(e) - 19*g^3)*a*b^2*c^3*d + 9*(4*g^3*log(e) - 5*g^3)*a^2*b*c^2*d 
^2 - 6*(4*g^3*log(e) - 3*g^3)*a^3*c*d^3)*B^2*log(d*x + c)/d^4 + 1/2*(b^4*c 
^4*g^3 - 4*a*b^3*c^3*d*g^3 + 6*a^2*b^2*c^2*d^2*g^3 - 4*a^3*b*c*d^3*g^3 + a 
^4*d^4*g^3)*(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*d^4) + 1/12*(3*B^2*b^4*d^4*g^3*x^4*log(e)^2 
 + 2*(b^4*c*d^3*g^3*log(e) + (6*g^3*log(e)^2 - g^3*log(e))*a*b^3*d^4)*B^2* 
x^3 - ((3*g^3*log(e) - g^3)*b^4*c^2*d^2 - 2*(6*g^3*log(e) - g^3)*a*b^3*c*d 
^3 - (18*g^3*log(e)^2 - 9*g^3*log(e) + g^3)*a^2*b^2*d^4)*B^2*x^2 + ((6*g^3 
*log(e) - 5*g^3)*b^4*c^3*d - (24*g^3*log(e) - 17*g^3)*a*b^3*c^2*d^2 + (36* 
g^3*log(e) - 19*g^3)*a^2*b^2*c*d^3 + (12*g^3*log(e)^2 - 18*g^3*log(e) +...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**3*( - 6*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b* 
d*x**2),x)*a**4*b**2*d**5 + 24*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + 
 a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*c*d**4 - 36*int((log((c*e + d*e*x) 
/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**4*c**2*d**3 + 2 
4*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x) 
*a*b**5*c**3*d**2 - 6*int((log((c*e + d*e*x)/(a + b*x))*x)/(a*c + a*d*x + 
b*c*x + b*d*x**2),x)*b**6*c**4*d - 6*log(c + d*x)*a**5*d**4 + 24*log(c + d 
*x)*a**4*b*c*d**3 + 11*log(c + d*x)*a**4*b*d**4 - 36*log(c + d*x)*a**3*b** 
2*c**2*d**2 - 44*log(c + d*x)*a**3*b**2*c*d**3 + 24*log(c + d*x)*a**2*b**3 
*c**3*d + 66*log(c + d*x)*a**2*b**3*c**2*d**2 - 6*log(c + d*x)*a*b**4*c**4 
 - 44*log(c + d*x)*a*b**4*c**3*d + 11*log(c + d*x)*b**5*c**4 + 9*log((c*e 
+ d*e*x)/(a + b*x))**2*a**3*b**2*c*d**3 + 12*log((c*e + d*e*x)/(a + b*x))* 
*2*a**3*b**2*d**4*x - 9*log((c*e + d*e*x)/(a + b*x))**2*a**2*b**3*c**2*d** 
2 + 18*log((c*e + d*e*x)/(a + b*x))**2*a**2*b**3*d**4*x**2 + 3*log((c*e + 
d*e*x)/(a + b*x))**2*a*b**4*c**3*d + 12*log((c*e + d*e*x)/(a + b*x))**2*a* 
b**4*d**4*x**3 + 3*log((c*e + d*e*x)/(a + b*x))**2*b**5*d**4*x**4 + 6*log( 
(c*e + d*e*x)/(a + b*x))*a**5*d**4 + 24*log((c*e + d*e*x)/(a + b*x))*a**4* 
b*d**4*x - 11*log((c*e + d*e*x)/(a + b*x))*a**4*b*d**4 + 26*log((c*e + d*e 
*x)/(a + b*x))*a**3*b**2*c*d**3 + 36*log((c*e + d*e*x)/(a + b*x))*a**3*b** 
2*d**4*x**2 - 18*log((c*e + d*e*x)/(a + b*x))*a**3*b**2*d**4*x - 21*log...