\(\int \frac {(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{c i+d i x} \, dx\) [187]

Optimal result

Integrand size = 45, antiderivative size = 573 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx=-\frac {B (b c-a d) g^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2 i}-\frac {2 (b c-a d) g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 i}+\frac {b^2 g^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^3 i}-\frac {4 B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i}+\frac {B^2 (b c-a d)^2 g^2 n^2 \log (c+d x)}{d^3 i}+\frac {B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}-\frac {4 B^2 (b c-a d)^2 g^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {2 B (b c-a d)^2 g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {B^2 (b c-a d)^2 g^2 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}+\frac {2 B^2 (b c-a d)^2 g^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i} \] Output:

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

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.33 \[ \int \frac {(a g+b g x)^2 \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^2 \left (-2 b d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+2 A^2 (b c-a d)^2 \log (c+d x)-4 A B (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-2 B^2 (b c-a d)^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 A B (b c-a d)^2 n \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) n \left (2 a d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b c \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-a B d 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 )+b B c 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 )-B (b c-a d) n \left (2 A b d x+2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B (b c-a d) n \log (c+d x)-2 (b c-a d) \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) 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 )+4 B^2 (b c-a d)^2 n \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )+n \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{2 d^3 i} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 507, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

g**2*(Integral(A**2*a**2/(c + d*x), x) + Integral(A**2*b**2*x**2/(c + d*x) 
, x) + Integral(B**2*a**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + 
d*x), x) + Integral(2*A*B*a**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c 
+ d*x), x) + Integral(2*A**2*a*b*x/(c + d*x), x) + Integral(B**2*b**2*x**2 
*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + Integral(2*A*B 
*b**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) + Integra 
l(2*B**2*a*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c + d*x), x) + 
Integral(4*A*B*a*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c + d*x), x) 
)/i
 

Maxima [F]

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

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

Output:

2*A^2*a*b*g^2*(x/(d*i) - c*log(d*x + c)/(d^2*i)) + 1/2*A^2*b^2*g^2*(2*c^2* 
log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A^2*a^2*g^2*log(d*i*x + 
c*i)/(d*i) + 1/2*(B^2*b^2*d^2*g^2*x^2 - 2*(b^2*c*d*g^2 - 2*a*b*d^2*g^2)*B^ 
2*x + 2*(b^2*c^2*g^2 - 2*a*b*c*d*g^2 + a^2*d^2*g^2)*B^2*log(d*x + c))*log( 
(d*x + c)^n)^2/(d^3*i) - integrate(-(B^2*a^2*d^2*g^2*log(e)^2 + 2*A*B*a^2* 
d^2*g^2*log(e) + (B^2*b^2*d^2*g^2*log(e)^2 + 2*A*B*b^2*d^2*g^2*log(e))*x^2 
 + (B^2*b^2*d^2*g^2*x^2 + 2*B^2*a*b*d^2*g^2*x + B^2*a^2*d^2*g^2)*log((b*x 
+ a)^n)^2 + 2*(B^2*a*b*d^2*g^2*log(e)^2 + 2*A*B*a*b*d^2*g^2*log(e))*x + 2* 
(B^2*a^2*d^2*g^2*log(e) + A*B*a^2*d^2*g^2 + (B^2*b^2*d^2*g^2*log(e) + A*B* 
b^2*d^2*g^2)*x^2 + 2*(B^2*a*b*d^2*g^2*log(e) + A*B*a*b*d^2*g^2)*x)*log((b* 
x + a)^n) - (2*B^2*a^2*d^2*g^2*log(e) + 2*A*B*a^2*d^2*g^2 + 2*(b^2*c^2*g^2 
*n - 2*a*b*c*d*g^2*n + a^2*d^2*g^2*n)*B^2*log(d*x + c) + (2*A*B*b^2*d^2*g^ 
2 + (g^2*n + 2*g^2*log(e))*B^2*b^2*d^2)*x^2 + 2*(2*A*B*a*b*d^2*g^2 - (b^2* 
c*d*g^2*n - 2*(g^2*n + g^2*log(e))*a*b*d^2)*B^2)*x + 2*(B^2*b^2*d^2*g^2*x^ 
2 + 2*B^2*a*b*d^2*g^2*x + B^2*a^2*d^2*g^2)*log((b*x + a)^n))*log((d*x + c) 
^n))/(d^3*i*x + c*d^2*i), x)
 

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c i+d i x} \, dx=\frac {g^{2} i \left (-2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2}}{d x +c}d x \right ) a^{2} b^{2} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{d x +c}d x \right ) a^{3} b \,d^{3}-2 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x^{2}}{d x +c}d x \right ) b^{4} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2} x}{d x +c}d x \right ) a \,b^{3} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x^{2}}{d x +c}d x \right ) a \,b^{3} d^{3}-8 \left (\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{d x +c}d x \right ) a^{2} b^{2} d^{3}-2 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{2}+4 \,\mathrm {log}\left (d x +c \right ) a^{3} b c d -2 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2}-4 a^{3} b \,d^{2} x +2 a^{2} b^{2} c d x -a^{2} b^{2} d^{2} x^{2}\right )}{2 d^{3}} \] Input:

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

Output:

(g**2*i*( - 2*int(log(((a + b*x)**n*e)/(c + d*x)**n)**2/(c + d*x),x)*a**2* 
b**2*d**3 - 4*int(log(((a + b*x)**n*e)/(c + d*x)**n)/(c + d*x),x)*a**3*b*d 
**3 - 2*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c + d*x),x)*b**4 
*d**3 - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x)/(c + d*x),x)*a*b** 
3*d**3 - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c + d*x),x)*a*b* 
*3*d**3 - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(c + d*x),x)*a**2*b 
**2*d**3 - 2*log(c + d*x)*a**4*d**2 + 4*log(c + d*x)*a**3*b*c*d - 2*log(c 
+ d*x)*a**2*b**2*c**2 - 4*a**3*b*d**2*x + 2*a**2*b**2*c*d*x - a**2*b**2*d* 
*2*x**2))/(2*d**3)