\(\int (a g+b g x)^4 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2 \, dx\) [210]

Optimal result

Integrand size = 34, antiderivative size = 515 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {10 B^2 (b c-a d)^5 g^4 \log (a+b x)}{3 b d^5}-\frac {26 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{15 b d^5}+\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}-\frac {4 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \] Output:

26/15*B^2*(-a*d+b*c)^4*g^4*x/d^4-7/15*B^2*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3 
+2/15*B^2*(-a*d+b*c)^2*g^4*(b*x+a)^3/b/d^2-10/3*B^2*(-a*d+b*c)^5*g^4*ln(b* 
x+a)/b/d^5-26/15*B^2*(-a*d+b*c)^5*g^4*ln((d*x+c)/(b*x+a))/b/d^5+2/5*B*(-a* 
d+b*c)^3*g^4*(b*x+a)^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d^3-4/15*B*(-a*d+ 
b*c)^2*g^4*(b*x+a)^3*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d^2+1/5*B*(-a*d+b*c 
)*g^4*(b*x+a)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d-4/5*B*(-a*d+b*c)^4*g^4 
*(d*x+c)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/d^5+1/5*g^4*(b*x+a)^5*(A+B*ln(e*( 
d*x+c)^2/(b*x+a)^2))^2/b-4/5*B*(-a*d+b*c)^5*g^4*(A+B*ln(e*(d*x+c)^2/(b*x+a 
)^2))*ln(1-d*(b*x+a)/b/(d*x+c))/b/d^5+8/5*B^2*(-a*d+b*c)^5*g^4*polylog(2,d 
*(b*x+a)/b/(d*x+c))/b/d^5
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.02 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {B (b c-a d) \left (12 A b d (b c-a d)^3 x+24 B (b c-a d)^4 \log (c+d x)-4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )-12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-6 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+4 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-3 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-12 (b c-a d)^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^5}\right )}{5 b} \] Input:

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

Output:

(g^4*((a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 - (B*(b*c - a 
*d)*(12*A*b*d*(b*c - a*d)^3*x + 24*B*(b*c - a*d)^4*Log[c + d*x] - 4*B*(b*c 
 - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + 
 d*x]) - B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + 
b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) - 12*B*(b*c - a 
*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*d*(b*c - a*d)^3*(a + b* 
x)*Log[(e*(c + d*x)^2)/(a + b*x)^2] - 6*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + 
 B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) + 4*d^3*(b*c - a*d)*(a + b*x)^3*(A + 
B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 3*d^4*(a + b*x)^4*(A + B*Log[(e*(c + 
 d*x)^2)/(a + b*x)^2]) - 12*(b*c - a*d)^4*Log[c + d*x]*(A + B*Log[(e*(c + 
d*x)^2)/(a + b*x)^2]) - 12*B*(b*c - a*d)^4*((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^5)))/(5*b)
 

Rubi [A] (verified)

Time = 1.98 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.38, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.559, Rules used = {2952, 2756, 2789, 2756, 54, 2009, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((b*c - a*d)^5*g^4*((A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(5*b*(d - 
(b*(c + d*x))/(a + b*x))^5) - (4*B*((b*((A + B*Log[(e*(c + d*x)^2)/(a + b* 
x)^2])/(4*b*(d - (b*(c + d*x))/(a + b*x))^4) - (B*(1/(3*d*(d - (b*(c + d*x 
))/(a + b*x))^3) + 1/(2*d^2*(d - (b*(c + d*x))/(a + b*x))^2) + 1/(d^3*(d - 
 (b*(c + d*x))/(a + b*x))) + Log[(c + d*x)/(a + b*x)]/d^4 - Log[d - (b*(c 
+ d*x))/(a + b*x)]/d^4))/(2*b)))/d + ((b*((A + B*Log[(e*(c + d*x)^2)/(a + 
b*x)^2])/(3*b*(d - (b*(c + d*x))/(a + b*x))^3) - (2*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)^2)/(a + b*x)^2])/(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))/b))/d + ((b*(((c + d*x)*(A 
+ B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(d*(a + b*x)*(d - (b*(c + d*x))/(a 
+ b*x))) + (2*B*Log[d - (b*(c + d*x))/(a + b*x)])/(b*d)))/d + (-(((A + B*L 
og[(e*(c + d*x)^2)/(a + b*x)^2])*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)/d)/d)/d))/(5*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]

\[\int \left (b g x +a g \right )^{4} {\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )}^{2}d x\]

Input:

int((b*g*x+a*g)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Output:

int((b*g*x+a*g)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Fricas [F]

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

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="f 
ricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((b*g*x+a*g)**4*(A+B*ln(e*(d*x+c)**2/(b*x+a)**2))**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2660 vs. \(2 (490) = 980\).

Time = 0.20 (sec) , antiderivative size = 2660, normalized size of antiderivative = 5.17 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="m 
axima")
 

Output:

1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^ 
3*b*g^4*x^2 + 2*(x*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^ 
2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 2*a*log(b*x + 
a)/b + 2*c*log(d*x + c)/d)*A*B*a^4*g^4 + 4*(x^2*log(d^2*e*x^2/(b^2*x^2 + 2 
*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a 
*b*x + a^2)) + 2*a^2*log(b*x + a)/b^2 - 2*c^2*log(d*x + c)/d^2 + 2*(b*c - 
a*d)*x/(b*d))*A*B*a^3*b*g^4 + 4*(x^3*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^ 
2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2) 
) - 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^2*b^2*g^4 + 2/3*(3*x^4*lo 
g(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2 
) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) + 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*a*b^3*g^4 + 1/15*(6*x^5*log( 
d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) 
+ c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 12*a^5*log(b*x + a)/b^5 + 12*c^5*log( 
d*x + c)/d^5 + (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d 
^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d 
^4))*A*B*b^4*g^4 + A^2*a^4*g^4*x + 2/15*((6*g^4*log(e) - 25*g^4)*b^4*c^5 - 
 (30*g^4*log(e) - 113*g^4)*a*b^3*c^4*d + 4*(15*g^4*log(e) - 49*g^4)*a^2...
 

Giac [F]

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

integrate((b*g*x+a*g)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="g 
iac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**4*( - 12*int((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + 
 b**2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**5*b**2*d**6 + 60*in 
t((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x) 
/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**4*b**3*c*d**5 - 120*int((log((c**2 
*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d* 
x + b*c*x + b*d*x**2),x)*a**3*b**4*c**2*d**4 + 120*int((log((c**2*e + 2*c* 
d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x + b*c*x 
 + b*d*x**2),x)*a**2*b**5*c**3*d**3 - 60*int((log((c**2*e + 2*c*d*e*x + d* 
*2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x** 
2),x)*a*b**6*c**4*d**2 + 12*int((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a 
**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**7*c* 
*5*d - 12*log(c + d*x)*a**6*d**5 + 60*log(c + d*x)*a**5*b*c*d**4 + 50*log( 
c + d*x)*a**5*b*d**5 - 120*log(c + d*x)*a**4*b**2*c**2*d**3 - 250*log(c + 
d*x)*a**4*b**2*c*d**4 + 120*log(c + d*x)*a**3*b**3*c**3*d**2 + 500*log(c + 
 d*x)*a**3*b**3*c**2*d**3 - 60*log(c + d*x)*a**2*b**4*c**4*d - 500*log(c + 
 d*x)*a**2*b**4*c**3*d**2 + 12*log(c + d*x)*a*b**5*c**5 + 250*log(c + d*x) 
*a*b**5*c**4*d - 50*log(c + d*x)*b**6*c**5 + 12*log((c**2*e + 2*c*d*e*x + 
d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a**4*b**2*c*d**4 + 15*log((c 
**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a**4*b** 
2*d**5*x - 18*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + ...