\(\int (a g+b g x)^4 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2 \, dx\) [128]

Optimal result

Integrand size = 34, antiderivative size = 377 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=-\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{5 b}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3 \left (2 A+B+2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^2}-\frac {B (b c-a d)^3 g^4 (a+b x)^2 \left (6 A+7 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^3}+\frac {2 B (b c-a d)^4 g^4 (a+b x) \left (6 A+13 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{15 b d^4}+\frac {2 B (b c-a d)^5 g^4 \left (6 A+25 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{15 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \] Output:

-1/5*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d+1/5*g^ 
4*(b*x+a)^5*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/b+2/15*B*(-a*d+b*c)^2*g^4*(b 
*x+a)^3*(2*A+B+2*B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d^2-1/15*B*(-a*d+b*c)^3*g^ 
4*(b*x+a)^2*(6*A+7*B+6*B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d^3+2/15*B*(-a*d+b*c 
)^4*g^4*(b*x+a)*(6*A+13*B+6*B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/d^4+2/15*B*(-a* 
d+b*c)^5*g^4*(6*A+25*B+6*B*ln(e*(b*x+a)^2/(d*x+c)^2))*ln((-a*d+b*c)/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 = 0.51 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.39 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {B (b c-a d) \left (12 A b d (b c-a d)^3 x+12 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-6 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+4 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-3 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-24 B (b c-a d)^4 \log (c+d x)-12 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)+4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )+B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^5}\right )}{5 b} \] Input:

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

Output:

(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(b*c - a 
*d)*(12*A*b*d*(b*c - a*d)^3*x + 12*B*d*(b*c - a*d)^3*(a + b*x)*Log[(e*(a + 
 b*x)^2)/(c + d*x)^2] - 6*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + 
 b*x)^2)/(c + d*x)^2]) + 4*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(a + 
b*x)^2)/(c + d*x)^2]) - 3*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + 
d*x)^2]) - 24*B*(b*c - a*d)^4*Log[c + d*x] - 12*(b*c - a*d)^4*(A + B*Log[( 
e*(a + b*x)^2)/(c + d*x)^2])*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*(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 = 1.04 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2950, 2781, 2784, 27, 2784, 2784, 27, 2784, 2754, 2838}

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

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2784

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (6 A+7 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}}{2 d}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {2 (a+b x) \left (6 A+13 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}}{2 d}\right )}{5 b}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+13 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{d}}{3 d}}{2 d}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+25 B+6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{d}}{3 d}}{2 d}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {12 B \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+25 B\right )}{d}}{d}}{d}}{3 d}}{2 d}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle g^4 (b c-a d)^5 \left (\frac {(a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{5 b (c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {4 B \left (\frac {(a+b x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 d (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {\frac {(a+b x)^3 \left (2 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+2 A+B\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+7 B\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+13 B\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+6 A+25 B\right )}{d}-\frac {12 B \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{d}}{3 d}}{2 d}\right )}{5 b}\right )\)

Input:

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

Output:

(b*c - a*d)^5*g^4*(((a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 
)/(5*b*(c + d*x)^5*(b - (d*(a + b*x))/(c + d*x))^5) - (4*B*(((a + b*x)^4*( 
A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*d*(c + d*x)^4*(b - (d*(a + b*x 
))/(c + d*x))^4) - (((a + b*x)^3*(2*A + B + 2*B*Log[(e*(a + b*x)^2)/(c + d 
*x)^2]))/(3*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (((a + b*x)^2 
*(6*A + 7*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(2*d*(c + d*x)^2*(b - 
 (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)*(6*A + 13*B + 6*B*Log[(e*(a + b 
*x)^2)/(c + d*x)^2]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((6 
*A + 25*B + 6*B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[1 - (d*(a + b*x))/(b 
*(c + d*x))])/d) - (12*B*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/d) 
/(3*d))/(2*d)))/(5*b))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2781

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

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

rule 2838

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

rule 2950

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/b)^m   Subst[Int[x^m*((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] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2650 vs. \(2 (362) = 724\).

Time = 0.20 (sec) , antiderivative size = 2650, normalized size of antiderivative = 7.03 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**4*(12*int((log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d* 
*2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**5*b**2*d**6 - 60*int(( 
log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**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((a**2*e 
+ 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**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((a**2*e + 2*a*b*e 
*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**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((a**2*e + 2*a*b*e*x + b**2* 
e*x**2)/(c**2 + 2*c*d*x + d**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((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 
 + 2*c*d*x + d**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((a**2*e + 2*a*b*e*x + b** 
2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a**4*b**2*c*d**4 + 15*log((a**2 
*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a**4*b**2*d 
**5*x - 18*log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**...