\(\int (a g+b g x) (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [161]

Optimal result

Integrand size = 41, antiderivative size = 372 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 g i n^2 x}{3 b d}-\frac {B (b c-a d)^2 g i n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac {B (b c-a d) g i n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac {(b c-a d) g i (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b}-\frac {B (b c-a d)^3 g i n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b^2 d^2}-\frac {B^2 (b c-a d)^3 g i n^2 \log (c+d x)}{3 b^2 d^2}-\frac {B^2 (b c-a d)^3 g i n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^2 d^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(937\) vs. \(2(372)=744\).

Time = 0.84 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.52 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g*i*(-6*A*b^2*B*c*d*(b*c - a*d)*n*x + 6*a*A*b*B*d^2*(-(b*c) + a*d)*n*x + 
4*A*b*B*d*(b*c - a*d)*(b*c + a*d)*n*x - 6*b*B^2*c*d*(b*c - a*d)*n*(a + b*x 
)*Log[e*((a + b*x)/(c + d*x))^n] + 6*a*B^2*d^2*(-(b*c) + a*d)*n*(a + b*x)* 
Log[e*((a + b*x)/(c + d*x))^n] + 4*B^2*d*(b*c - a*d)*(b*c + a*d)*n*(a + b* 
x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*b^2*B*d^2*(b*c - a*d)*n*x^2*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n]) + 6*a^2*b*B*c*d^2*n*Log[a + b*x]*(A + B*Log 
[e*((a + b*x)/(c + d*x))^n]) - 2*a^3*B*d^3*n*Log[a + b*x]*(A + B*Log[e*((a 
 + b*x)/(c + d*x))^n]) + 6*a*b^2*c*d^2*x*(A + B*Log[e*((a + b*x)/(c + d*x) 
)^n])^2 + 3*b^2*d^2*(b*c + a*d)*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
^2 + 2*b^3*d^3*x^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 6*b*B^2*c*(b 
*c - a*d)^2*n^2*Log[c + d*x] + 6*a*B^2*d*(b*c - a*d)^2*n^2*Log[c + d*x] - 
4*B^2*(b*c - a*d)^2*(b*c + a*d)*n^2*Log[c + d*x] + 2*b^3*B*c^3*n*(A + B*Lo 
g[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 6*a*b^2*B*c^2*d*n*(A + B*Log[ 
e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 2*B^2*(b*c - a*d)*n^2*(a^2*d^2* 
Log[a + b*x] - b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c + d*x])) - 3*a^2*b*B^2* 
c*d^2*n^2*(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)]) + a^3*B^2*d^3*n^2*(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)]) - b^3*B^2*c^3*n^2*((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 ...
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2961, 2783, 2773, 49, 2009, 2781, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2773

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2781

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

\(\Big \downarrow \) 2784

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

\(\Big \downarrow \) 2754

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 49

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

rule 2009

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

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 2773

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

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 2783

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[(m + q + 2)/(d*(q + 1))   Int[ 
(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], 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, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L 
tQ[q, -1] && GtQ[m, 0]
 

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1542 vs. \(2 (355) = 710\).

Time = 0.57 (sec) , antiderivative size = 1542, normalized size of antiderivative = 4.15 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

2/3*A*B*b*d*g*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A^2*b*d*g 
*i*x^3 + A*B*b*c*g*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*B*a*d* 
g*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A^2*b*c*g*i*x^2 + 1/2 
*A^2*a*d*g*i*x^2 + 1/3*A*B*b*d*g*i*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 
)) - A*B*b*c*g*i*n*(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*d*g*i*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*a*c*g*i*n*(a*log(b*x + a)/b - c*log(d*x + c 
)/d) + 2*A*B*a*c*g*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*c*g* 
i*x - 1/3*(a^2*c*d^2*g*i*n^2 - b^2*c^3*g*i*n*log(e) - (g*i*n^2 - 3*g*i*n*l 
og(e))*a*b*c^2*d)*B^2*log(d*x + c)/(b*d^2) + 1/3*(b^3*c^3*g*i*n^2 - 3*a*b^ 
2*c^2*d*g*i*n^2 + 3*a^2*b*c*d^2*g*i*n^2 - a^3*d^3*g*i*n^2)*(log(b*x + a)*l 
og((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2 
/(b^2*d^2) + 1/6*(2*B^2*b^3*d^3*g*i*x^3*log(e)^2 - ((2*g*i*n*log(e) - 3*g* 
i*log(e)^2)*b^3*c*d^2 - (2*g*i*n*log(e) + 3*g*i*log(e)^2)*a*b^2*d^3)*B^2*x 
^2 - (3*a^2*b*c*d^2*g*i*n^2 - a^3*d^3*g*i*n^2)*B^2*log(b*x + a)^2 - 2*(b^3 
*c^3*g*i*n^2 - 3*a*b^2*c^2*d*g*i*n^2)*B^2*log(b*x + a)*log(d*x + c) + (b^3 
*c^3*g*i*n^2 - 3*a*b^2*c^2*d*g*i*n^2)*B^2*log(d*x + c)^2 + 2*((g*i*n^2 - g 
*i*n*log(e))*b^3*c^2*d - (2*g*i*n^2 - 3*g*i*log(e)^2)*a*b^2*c*d^2 + (g*i*n 
^2 + g*i*n*log(e))*a^2*b*d^3)*B^2*x - 2*(a*b^2*c^2*d*g*i*n^2 + a^3*d^3*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g*i*( - 2*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**2*d**4*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)*a**2*b**3*c*d**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)*a*b* 
*4*c**2*d**2*n + 2*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**5*c**3*d*n - 2*log(a + b*x)*a**4*d**3*n + 6*log 
(a + b*x)*a**3*b*c*d**2*n - 6*log(a + b*x)*a**2*b**2*c**2*d*n + 2*log(a + 
b*x)*a*b**3*c**3*n + log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**2*c*d** 
2 + log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3*c**2*d + 6*log(((a + b*x) 
**n*e)/(c + d*x)**n)**2*a*b**3*c*d**2*x + 3*log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*a*b**3*d**3*x**2 + 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**4*c* 
d**2*x**2 + 2*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**4*d**3*x**3 + 6*log 
(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*c**2*d + 2*log(((a + b*x)**n*e)/ 
(c + d*x)**n)*a**2*b**2*c*d**2*n + 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a 
**2*b**2*c*d**2*x + 2*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*d**3*n* 
x + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*d**3*x**2 - 2*log(((a + 
 b*x)**n*e)/(c + d*x)**n)*a*b**3*c**3 - 2*log(((a + b*x)**n*e)/(c + d*x)** 
n)*a*b**3*c**2*d*n + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*c*d**2*x* 
*2 + 2*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*d**3*n*x**2 + 4*log(((a + 
 b*x)**n*e)/(c + d*x)**n)*a*b**3*d**3*x**3 - 2*log(((a + b*x)**n*e)/(c ...