\(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{c i+d i x} \, dx\) [186]

Optimal result

Integrand size = 45, antiderivative size = 768 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx=\frac {b B^2 (b c-a d)^2 g^3 n^2 x}{3 d^3 i}+\frac {7 B (b c-a d)^2 g^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3 i}-\frac {b^2 B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^4 i}+\frac {3 (b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i}-\frac {3 b^2 (b c-a d) g^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^4 i}+\frac {b^3 g^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d^4 i}+\frac {6 B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i}+\frac {B^2 (b c-a d)^3 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 d^4 i}-\frac {2 B^2 (b c-a d)^3 g^3 n^2 \log (c+d x)}{d^4 i}-\frac {7 B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4 i}+\frac {6 B^2 (b c-a d)^3 g^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {2 B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i}+\frac {7 B^2 (b c-a d)^3 g^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4 i}-\frac {2 B^2 (b c-a d)^3 g^3 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \] Output:

1/3*b*B^2*(-a*d+b*c)^2*g^3*n^2*x/d^3/i+7/3*B*(-a*d+b*c)^2*g^3*n*(b*x+a)*(A 
+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3/i-1/3*b^2*B*(-a*d+b*c)*g^3*n*(d*x+c)^2*( 
A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4/i+3*(-a*d+b*c)^2*g^3*(b*x+a)*(A+B*ln(e* 
((b*x+a)/(d*x+c))^n))^2/d^3/i-3/2*b^2*(-a*d+b*c)*g^3*(d*x+c)^2*(A+B*ln(e*( 
(b*x+a)/(d*x+c))^n))^2/d^4/i+1/3*b^3*g^3*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x 
+c))^n))^2/d^4/i+6*B*(-a*d+b*c)^3*g^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln 
((-a*d+b*c)/b/(d*x+c))/d^4/i+(-a*d+b*c)^3*g^3*(A+B*ln(e*((b*x+a)/(d*x+c))^ 
n))^2*ln((-a*d+b*c)/b/(d*x+c))/d^4/i+1/3*B^2*(-a*d+b*c)^3*g^3*n^2*ln((b*x+ 
a)/(d*x+c))/d^4/i-2*B^2*(-a*d+b*c)^3*g^3*n^2*ln(d*x+c)/d^4/i-7/3*B*(-a*d+b 
*c)^3*g^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d/(b*x+a))/d^4/ 
i+6*B^2*(-a*d+b*c)^3*g^3*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/d^4/i+2*B*(-a* 
d+b*c)^3*g^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*polylog(2,d*(b*x+a)/b/(d*x+ 
c))/d^4/i+7/3*B^2*(-a*d+b*c)^3*g^3*n^2*polylog(2,b*(d*x+c)/d/(b*x+a))/d^4/ 
i-2*B^2*(-a*d+b*c)^3*g^3*n^2*polylog(3,d*(b*x+a)/b/(d*x+c))/d^4/i
 

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 1072, normalized size of antiderivative = 1.40 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx =\text {Too large to display} \] Input:

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

Output:

(g^3*(6*b*d*(b*c - a*d)^2*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 3*d 
^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 2 
*d^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 6*A^2*(b*c - a 
*d)^3*Log[c + d*x] + 12*A*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n]*L 
og[(b*c - a*d)/(b*c + b*d*x)] + 6*B^2*(b*c - a*d)^3*Log[e*((a + b*x)/(c + 
d*x))^n]^2*Log[(b*c - a*d)/(b*c + b*d*x)] - 6*A*B*(b*c - a*d)^3*n*(Log[(b* 
c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + Log[(b*c - 
a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 6*B*(b*c 
 - a*d)^2*n*(2*a*d*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2 
*b*c*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - a*B*d*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)]) + b*B*c*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*(b*c - a*d)^2*n*(2*A*b*d*x + 2*B*d*(a + b*x)*Log[e*((a + b*x)/(c + 
 d*x))^n] - 2*B*(b*c - a*d)*n*Log[c + d*x] - 2*(b*c - a*d)*(A + B*Log[e*(( 
a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*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)])) + 2*B*(b*c - a*d)*n*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c 
- a*d)*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - d^2*(a + b*x)^2*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n]) - 2*B*(b*c - a*d)^2*n*Log[c + d*x] - 2*(...
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 679, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx=\frac {g^{3} \left (\int \frac {A^{2} a^{3}}{c + d x}\, dx + \int \frac {A^{2} b^{3} x^{3}}{c + d x}\, dx + \int \frac {B^{2} a^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B a^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {3 A^{2} a b^{2} x^{2}}{c + d x}\, dx + \int \frac {3 A^{2} a^{2} b x}{c + d x}\, dx + \int \frac {B^{2} b^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {2 A B b^{3} x^{3} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {3 B^{2} a b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {3 B^{2} a^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c + d x}\, dx + \int \frac {6 A B a b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {6 A B a^{2} b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx\right )}{i} \] Input:

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

Output:

g**3*(Integral(A**2*a**3/(c + d*x), x) + Integral(A**2*b**3*x**3/(c + d*x) 
, x) + Integral(B**2*a**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + 
d*x), x) + Integral(2*A*B*a**3*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c 
+ d*x), x) + Integral(3*A**2*a*b**2*x**2/(c + d*x), x) + Integral(3*A**2*a 
**2*b*x/(c + d*x), x) + Integral(B**2*b**3*x**3*log(e*(a/(c + d*x) + b*x/( 
c + d*x))**n)**2/(c + d*x), x) + Integral(2*A*B*b**3*x**3*log(e*(a/(c + d* 
x) + b*x/(c + d*x))**n)/(c + d*x), x) + Integral(3*B**2*a*b**2*x**2*log(e* 
(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + Integral(3*B**2*a**2* 
b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + Integral(6* 
A*B*a*b**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) + In 
tegral(6*A*B*a**2*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x 
))/i
 

Maxima [F]

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

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

Output:

3*A^2*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A^2*b^3*g^3*(6*c^ 
3*log(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2* 
A^2*a*b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A^2 
*a^3*g^3*log(d*i*x + c*i)/(d*i) + 1/6*(2*B^2*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^ 
2*g^3 - 3*a*b^2*d^3*g^3)*B^2*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 
3*a^2*b*d^3*g^3)*B^2*x - 6*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^ 
2*g^3 - a^3*d^3*g^3)*B^2*log(d*x + c))*log((d*x + c)^n)^2/(d^4*i) - integr 
ate(-1/3*(3*B^2*a^3*d^3*g^3*log(e)^2 + 6*A*B*a^3*d^3*g^3*log(e) + 3*(B^2*b 
^3*d^3*g^3*log(e)^2 + 2*A*B*b^3*d^3*g^3*log(e))*x^3 + 9*(B^2*a*b^2*d^3*g^3 
*log(e)^2 + 2*A*B*a*b^2*d^3*g^3*log(e))*x^2 + 3*(B^2*b^3*d^3*g^3*x^3 + 3*B 
^2*a*b^2*d^3*g^3*x^2 + 3*B^2*a^2*b*d^3*g^3*x + B^2*a^3*d^3*g^3)*log((b*x + 
 a)^n)^2 + 9*(B^2*a^2*b*d^3*g^3*log(e)^2 + 2*A*B*a^2*b*d^3*g^3*log(e))*x + 
 6*(B^2*a^3*d^3*g^3*log(e) + A*B*a^3*d^3*g^3 + (B^2*b^3*d^3*g^3*log(e) + A 
*B*b^3*d^3*g^3)*x^3 + 3*(B^2*a*b^2*d^3*g^3*log(e) + A*B*a*b^2*d^3*g^3)*x^2 
 + 3*(B^2*a^2*b*d^3*g^3*log(e) + A*B*a^2*b*d^3*g^3)*x)*log((b*x + a)^n) - 
(6*B^2*a^3*d^3*g^3*log(e) + 6*A*B*a^3*d^3*g^3 + 2*(3*A*B*b^3*d^3*g^3 + (g^ 
3*n + 3*g^3*log(e))*B^2*b^3*d^3)*x^3 - 6*(b^3*c^3*g^3*n - 3*a*b^2*c^2*d*g^ 
3*n + 3*a^2*b*c*d^2*g^3*n - a^3*d^3*g^3*n)*B^2*log(d*x + c) + 3*(6*A*B*a*b 
^2*d^3*g^3 - (b^3*c*d^2*g^3*n - 3*(g^3*n + 2*g^3*log(e))*a*b^2*d^3)*B^2)*x 
^2 + 6*(3*A*B*a^2*b*d^3*g^3 + (b^3*c^2*d*g^3*n - 3*a*b^2*c*d^2*g^3*n + ...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx=\frac {g^{3} i \left (-6 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2}}{d x +c}d x \right ) a^{3} b^{2} d^{4}-12 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{d x +c}d x \right ) a^{4} b \,d^{4}-6 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x^{3}}{d x +c}d x \right ) b^{5} d^{4}-18 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x^{2}}{d x +c}d x \right ) a \,b^{4} d^{4}-18 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x}{d x +c}d x \right ) a^{2} b^{3} d^{4}-12 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x^{3}}{d x +c}d x \right ) a \,b^{4} d^{4}-36 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x^{2}}{d x +c}d x \right ) a^{2} b^{3} d^{4}-36 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{d x +c}d x \right ) a^{3} b^{2} d^{4}-6 \,\mathrm {log}\left (d x +c \right ) a^{5} d^{3}+18 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{2}-18 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d +6 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3}-18 a^{4} b \,d^{3} x +18 a^{3} b^{2} c \,d^{2} x -9 a^{3} b^{2} d^{3} x^{2}-6 a^{2} b^{3} c^{2} d x +3 a^{2} b^{3} c \,d^{2} x^{2}-2 a^{2} b^{3} d^{3} x^{3}\right )}{6 d^{4}} \] Input:

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

Output:

(g**3*i*( - 6*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(c + d*x),x)*a**3* 
b**2*d**4 - 12*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(c + d*x),x)*a**4*b* 
d**4 - 6*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**3)/(c + d*x),x)*b** 
5*d**4 - 18*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c + d*x),x)* 
a*b**4*d**4 - 18*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(c + d*x),x 
)*a**2*b**3*d**4 - 12*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**3)/(c + d 
*x),x)*a*b**4*d**4 - 36*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c + 
 d*x),x)*a**2*b**3*d**4 - 36*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(c 
 + d*x),x)*a**3*b**2*d**4 - 6*log(c + d*x)*a**5*d**3 + 18*log(c + d*x)*a** 
4*b*c*d**2 - 18*log(c + d*x)*a**3*b**2*c**2*d + 6*log(c + d*x)*a**2*b**3*c 
**3 - 18*a**4*b*d**3*x + 18*a**3*b**2*c*d**2*x - 9*a**3*b**2*d**3*x**2 - 6 
*a**2*b**3*c**2*d*x + 3*a**2*b**3*c*d**2*x**2 - 2*a**2*b**3*d**3*x**3))/(6 
*d**4)