\(\int (f+g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [67]

Optimal result

Integrand size = 32, antiderivative size = 923 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^3 g^3 n^2 x}{6 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-3 b c g-a d g) n^2 x}{4 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d^4}-\frac {B (b c-a d) g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d^4}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log (c+d x)}{4 b^4 d^4}+\frac {B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n^2 \log (c+d x)}{2 b^4 d^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b^4 d^4} \] Output:

1/6*B^2*(-a*d+b*c)^3*g^3*n^2*x/b^3/d^3+1/4*B^2*(-a*d+b*c)^2*g^2*(-a*d*g-3* 
b*c*g+4*b*d*f)*n^2*x/b^3/d^3+1/12*B^2*(-a*d+b*c)^2*g^3*n^2*(d*x+c)^2/b^2/d 
^4-1/2*B*(-a*d+b*c)*g*(a^2*d^2*g^2-2*a*b*d*g*(-c*g+2*d*f)+b^2*(3*c^2*g^2-8 
*c*d*f*g+6*d^2*f^2))*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d^3-1/4 
*B*(-a*d+b*c)*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/ 
(d*x+c))^n))/b^2/d^4-1/6*B*(-a*d+b*c)*g^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/( 
d*x+c))^n))/b/d^4-1/4*(-a*g+b*f)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4/g 
+1/4*(g*x+f)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/g-1/2*B*(-a*d+b*c)*(-a*d* 
g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2*d^2*f 
^2))*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^4/d^4+1/ 
6*B^2*(-a*d+b*c)^4*g^3*n^2*ln((b*x+a)/(d*x+c))/b^4/d^4+1/4*B^2*(-a*d+b*c)^ 
3*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*n^2*ln((b*x+a)/(d*x+c))/b^4/d^4+1/6*B^2*(-a 
*d+b*c)^4*g^3*n^2*ln(d*x+c)/b^4/d^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-a*d*g-3*b*c 
*g+4*b*d*f)*n^2*ln(d*x+c)/b^4/d^4+1/2*B^2*(-a*d+b*c)^2*g*(a^2*d^2*g^2-2*a* 
b*d*g*(-c*g+2*d*f)+b^2*(3*c^2*g^2-8*c*d*f*g+6*d^2*f^2))*n^2*ln(d*x+c)/b^4/ 
d^4-1/2*B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b 
^2*(c^2*g^2-2*c*d*f*g+2*d^2*f^2))*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/d 
^4
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 757, normalized size of antiderivative = 0.82 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (6 A b d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x+6 B d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 d^2 (b c-a d) g^3 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b^3 d^3 (b c-a d) g^4 x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^2 g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n \log (c+d x)-6 b^4 (d f-c g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) g^4 n \left (b d (b c-a d) x (2 b c+2 a d-b d x)+2 a^3 d^3 \log (a+b x)-2 b^3 c^3 \log (c+d x)\right )-3 B (b c-a d) g^3 (-4 b d f+b c g+a d g) n \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-3 B d^4 (b f-a g)^4 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+3 b^4 B (d f-c g)^4 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 b^4 d^4}}{4 g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 1100, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2953, 2798, 2804, 2009}

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.

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

\(\Big \downarrow \) 2953

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

\(\Big \downarrow \) 2798

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

\(\Big \downarrow \) 2804

\(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \left (\frac {(b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^4}{b d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(b c-a d)^3 (4 b d f-3 b c g-a d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^3}{b^2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {(b c-a d)^2 \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^2}{b^3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d) (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 a d^2 f g b+a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g}{b^4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b f-a g)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{2 g (b c-a d)}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d) \left (\frac {\left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d) g \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^4}{3 b d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B (b c-a d)^4 n \log \left (\frac {a+b x}{c+d x}\right ) g^4}{3 b^4 d^4}+\frac {B (b c-a d)^4 n \log \left (b-\frac {d (a+b x)}{c+d x}\right ) g^4}{3 b^4 d^4}-\frac {B (b c-a d)^4 n g^4}{3 b^3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B (b c-a d)^4 n g^4}{6 b^2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d)^3 (4 b d f-3 b c g-a d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^3}{2 b^2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B (b c-a d)^3 (4 b d f-3 b c g-a d g) n \log \left (\frac {a+b x}{c+d x}\right ) g^3}{2 b^4 d^4}+\frac {B (b c-a d)^3 (4 b d f-3 b c g-a d g) n \log \left (b-\frac {d (a+b x)}{c+d x}\right ) g^3}{2 b^4 d^4}-\frac {B (b c-a d)^3 (4 b d f-3 b c g-a d g) n g^3}{2 b^3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b c-a d)^2 \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) g^2}{b^4 d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B (b c-a d)^2 \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) n \log \left (b-\frac {d (a+b x)}{c+d x}\right ) g^2}{b^4 d^4}+\frac {(b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) g}{b^4 d^4}+\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) g}{b^4 d^4}+\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 B n}\right )}{2 (b c-a d) g}\right )\)

Input:

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

Output:

(b*c - a*d)*(((b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d*x))^4*(A + B*Log 
[e*((a + b*x)/(c + d*x))^n])^2)/(4*(b*c - a*d)*g*(b - (d*(a + b*x))/(c + d 
*x))^4) - (B*n*(-1/6*(B*(b*c - a*d)^4*g^4*n)/(b^2*d^4*(b - (d*(a + b*x))/( 
c + d*x))^2) - (B*(b*c - a*d)^4*g^4*n)/(3*b^3*d^4*(b - (d*(a + b*x))/(c + 
d*x))) - (B*(b*c - a*d)^3*g^3*(4*b*d*f - 3*b*c*g - a*d*g)*n)/(2*b^3*d^4*(b 
 - (d*(a + b*x))/(c + d*x))) + ((b*c - a*d)^4*g^4*(A + B*Log[e*((a + b*x)/ 
(c + d*x))^n]))/(3*b*d^4*(b - (d*(a + b*x))/(c + d*x))^3) + ((b*c - a*d)^3 
*g^3*(4*b*d*f - 3*b*c*g - a*d*g)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/( 
2*b^2*d^4*(b - (d*(a + b*x))/(c + d*x))^2) + ((b*c - a*d)^2*g^2*(a^2*d^2*g 
^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2*g^2))*(a 
 + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*d^3*(c + d*x)*(b - (d 
*(a + b*x))/(c + d*x))) + ((b*f - a*g)^4*(A + B*Log[e*((a + b*x)/(c + d*x) 
)^n])^2)/(2*b^4*B*n) - (B*(b*c - a*d)^4*g^4*n*Log[(a + b*x)/(c + d*x)])/(3 
*b^4*d^4) - (B*(b*c - a*d)^3*g^3*(4*b*d*f - 3*b*c*g - a*d*g)*n*Log[(a + b* 
x)/(c + d*x)])/(2*b^4*d^4) + (B*(b*c - a*d)^4*g^4*n*Log[b - (d*(a + b*x))/ 
(c + d*x)])/(3*b^4*d^4) + (B*(b*c - a*d)^3*g^3*(4*b*d*f - 3*b*c*g - a*d*g) 
*n*Log[b - (d*(a + b*x))/(c + d*x)])/(2*b^4*d^4) + (B*(b*c - a*d)^2*g^2*(a 
^2*d^2*g^2 - 2*a*b*d*g*(2*d*f - c*g) + b^2*(6*d^2*f^2 - 8*c*d*f*g + 3*c^2* 
g^2))*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b^4*d^4) + ((b*c - a*d)*g*(2*b* 
d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*...
 

Defintions of rubi rules used

rule 2009

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

rule 2798

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

rule 2804

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ 
u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / 
; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
 

rule 2953

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)   Sub 
st[Int[(b*f - a*g - (d*f - c*g)*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] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((A + B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n))**2*(f + g*x)**3, 
x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2651 vs. \(2 (892) = 1784\).

Time = 0.61 (sec) , antiderivative size = 2651, normalized size of antiderivative = 2.87 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*A*B*g^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*g^3*x^4 + 
 2*A*B*f*g^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*f*g^2*x^3 + 
3*A*B*f^2*g*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A^2*f^2*g*x^2 
 - 1/12*A*B*g^3*n*(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*f*g^2*n*(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) 
) - 3*A*B*f^2*g*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a* 
d)*x/(b*d)) + 2*A*B*f^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*f^ 
3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*f^3*x - 1/12*(6*a^3*c*d^3 
*g^3*n^2 - 3*(8*c*d^3*f*g^2*n^2 - c^2*d^2*g^3*n^2)*a^2*b + 2*(18*c*d^3*f^2 
*g*n^2 - 6*c^2*d^2*f*g^2*n^2 + c^3*d*g^3*n^2)*a*b^2 + (24*c*d^3*f^3*n*log( 
e) - (11*g^3*n^2 + 6*g^3*n*log(e))*c^4 + 12*(3*f*g^2*n^2 + 2*f*g^2*n*log(e 
))*c^3*d - 36*(f^2*g*n^2 + f^2*g*n*log(e))*c^2*d^2)*b^3)*B^2*log(d*x + c)/ 
(b^3*d^4) + 1/2*(4*a*b^3*d^4*f^3*n^2 - 6*a^2*b^2*d^4*f^2*g*n^2 + 4*a^3*b*d 
^4*f*g^2*n^2 - a^4*d^4*g^3*n^2 - (4*c*d^3*f^3*n^2 - 6*c^2*d^2*f^2*g*n^2 + 
4*c^3*d*f*g^2*n^2 - c^4*g^3*n^2)*b^4)*(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^4*d^4) + 1/12*(3* 
B^2*b^4*d^4*g^3*x^4*log(e)^2 + 6*(4*c*d^3*f^3*n^2 - 6*c^2*d^2*f^2*g*n^2 + 
4*c^3*d*f*g^2*n^2 - c^4*g^3*n^2)*B^2*b^4*log(b*x + a)*log(d*x + c) - 3*...
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 6*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b* 
d*x**2),x)*a**4*b**2*d**5*g**3*n + 24*int((log(((a + b*x)**n*e)/(c + d*x)* 
*n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**3*d**5*f*g**2*n - 36*in 
t((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2), 
x)*a**2*b**4*d**5*f**2*g*n + 24*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x) 
/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**5*d**5*f**3*n + 6*int((log(((a + 
 b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**4 
*d*g**3*n - 24*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b 
*c*x + b*d*x**2),x)*b**6*c**3*d**2*f*g**2*n + 36*int((log(((a + b*x)**n*e) 
/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**6*c**2*d**3*f**2* 
g*n - 24*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + 
 b*d*x**2),x)*b**6*c*d**4*f**3*n - 6*log(c + d*x)*a**5*d**4*g**3*n + 24*lo 
g(c + d*x)*a**4*b*d**4*f*g**2*n + 11*log(c + d*x)*a**4*b*d**4*g**3*n**2 - 
8*log(c + d*x)*a**3*b**2*c*d**3*g**3*n**2 - 36*log(c + d*x)*a**3*b**2*d**4 
*f**2*g*n - 36*log(c + d*x)*a**3*b**2*d**4*f*g**2*n**2 - 6*log(c + d*x)*a* 
*2*b**3*c**2*d**2*g**3*n**2 + 36*log(c + d*x)*a**2*b**3*c*d**3*f*g**2*n**2 
 + 24*log(c + d*x)*a**2*b**3*d**4*f**3*n + 36*log(c + d*x)*a**2*b**3*d**4* 
f**2*g*n**2 + 6*log(c + d*x)*a*b**4*c**4*g**3*n - 24*log(c + d*x)*a*b**4*c 
**3*d*f*g**2*n - 8*log(c + d*x)*a*b**4*c**3*d*g**3*n**2 + 36*log(c + d*x)* 
a*b**4*c**2*d**2*f**2*g*n + 36*log(c + d*x)*a*b**4*c**2*d**2*f*g**2*n**...