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

Optimal. Leaf size=231 \[ \frac{b \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i^2 n (b c-a d)^2}-\frac{d (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g i^2 (c+d x) (b c-a d)^2}+\frac{2 A B d n (a+b x)}{g i^2 (c+d x) (b c-a d)^2}+\frac{2 B^2 d n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g i^2 (c+d x) (b c-a d)^2}-\frac{2 B^2 d n^2 (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 6.11183, antiderivative size = 1803, normalized size of antiderivative = 7.81, number of steps used = 83, number of rules used = 31, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.689, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2525, 44, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

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

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(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 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 44

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] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2500

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2375

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

Rule 2374

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

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]

Rule 2434

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.)
))/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, In
t[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/
(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]

Rule 2499

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.))^(m_.))/((j_.) + (k_.)*(x_)), x_Symbol] :> Simp[((s + t*Log[i*(g + h*x)^n])^(m + 1)
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(k*n*t*(m + 1)), x] + (-Dist[(b*p*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*
(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)
/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] &
& EqQ[h*j - g*k, 0] && IGtQ[m, 0]

Rule 2396

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(198 c+198 d x)^2 (a g+b g x)} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g (c+d x)}\right ) \, dx\\ &=\frac{b^2 \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{39204 (b c-a d)^2 g}-\frac{(b d) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{39204 (b c-a d)^2 g}-\frac{d \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{39204 (b c-a d) g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{(b B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{(b B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{(B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{(B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{19602 g}-\frac{(b B n) \int \frac{(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d)^2 g}+\frac{(b B n) \int \frac{(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)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{(B n) \int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19602 g}-\frac{(b B n) \int \frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac{(b B n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{\left (b^2 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{(b B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac{(b B n) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{19602 (b c-a d) g}+\frac{(b B n) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{19602 (b c-a d) g}+\frac{(B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{\left (b^2 B n\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{(b B d n) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac{(A b B n) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}-\frac{\left (b B^2 n\right ) \int \frac{\log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac{\left (b B^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{19602 (b c-a d) g}\\ &=-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{\left (b^2 B^2\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{39204 (b c-a d)^2 g}+\frac{\left (b^2 B n\right ) \int \left (\frac{A \log (c+d x)}{a+b x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac{(b B d n) \int \left (\frac{A \log (c+d x)}{c+d x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac{(A B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{19602 (b c-a d) g}+\frac{\left (B^2 n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{19602 g}+\frac{\left (b B^2 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{19602 (b c-a d)^2 g}\\ &=-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{(A b B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (A b^2 B n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (b^2 B^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{(A B d n) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac{(A b B d n) \int \frac{\log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 d n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac{\left (B^2 n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19602 g}+\frac{\left (b^2 B^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b^2 B^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 d n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 d n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{(A b B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac{(A b B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b^2 B^2 n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (b^2 B^2 n\right ) \int \frac{\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{(A b B d n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b^2 B^2 n^2\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{39204 (b c-a d)^2 g}+\frac{\left (b^2 B^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 d n^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{39204 (b c-a d)^2 g}+\frac{\left (b B^2 d n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{19602 (b c-a d) g}+\frac{\left (b^2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{(A b B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right ) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{39204 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 d n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{\left (B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^n\right )}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{39204 (b c-a d)^2 g}-\frac{\left (B^2 d n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )^{-n}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{39204 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{d \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac{\left (B^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^n\right ) \log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{\left (b^2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{39204 d (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19602 (b c-a d)^2 g}-\frac{\left (b^2 B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{39204 d (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}-\frac{\left (b B^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}+\frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+2 \frac{\left (b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19602 (b c-a d)^2 g}\\ &=\frac{B^2 n^2}{19602 (b c-a d) g (c+d x)}+\frac{b B^2 n^2 \log (a+b x)}{19602 (b c-a d)^2 g}-\frac{A b B n \log ^2(a+b x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(a+b x)}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{39204 (b c-a d)^2 g}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d) g (c+d x)}-\frac{b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{19602 (b c-a d)^2 g}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d) g (c+d x)}+\frac{b \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{A b B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19602 (b c-a d)^2 g}+\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39204 (b c-a d)^2 g}+\frac{b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19602 (b c-a d)^2 g}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39204 (b c-a d)^2 g}-\frac{A b B n \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log ^2(c+d x)}{39204 (b c-a d)^2 g}+\frac{b B^2 n^2 \log (a+b x) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39204 (b c-a d)^2 g}-\frac{b B^2 n^2 \log ^3(c+d x)}{117612 (b c-a d)^2 g}+\frac{A b B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{39204 (b c-a d)^2 g}+\frac{b B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39204 (b c-a d)^2 g}-\frac{b B^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{A b B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}-\frac{b B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{19602 (b c-a d)^2 g}+\frac{b B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{19602 (b c-a d)^2 g}\\ \end{align*}

Mathematica [B]  time = 0.950208, size = 789, normalized size = 3.42 \[ \frac{b \log (a+b x) \left (2 A B \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2\right )}{g i^2 (b c-a d)^2}+\frac{2 A B \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2}{g i^2 (c+d x) (b c-a d)}-\frac{b \log (c+d x) \left (2 A B \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )^2-2 B^2 n \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+A^2-2 A B n+2 B^2 n^2\right )}{g i^2 (b c-a d)^2}+\frac{\log ^2\left (\frac{a+b x}{c+d x}\right ) \left (b B^2 c n \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )+b B^2 d n x \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )-a B^2 d n^2+A b B c n+A b B d n x-b B^2 d n^2 x\right )}{g i^2 (c+d x) (b c-a d)^2}-\frac{2 B n \log \left (\frac{a+b x}{c+d x}\right ) \left (-B \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\frac{a+b x}{c+d x}\right )\right )-A+B n\right )}{g i^2 (c+d x) (b c-a d)}+\frac{b B^2 n^2 \log ^3\left (\frac{a+b x}{c+d x}\right )}{3 g i^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.689, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.52465, size = 1369, normalized size = 5.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.517245, size = 954, normalized size = 4.13 \begin{align*} \frac{3 \, A^{2} b c - 3 \, A^{2} a d +{\left (B^{2} b d n^{2} x + B^{2} b c n^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )^{3} + 6 \,{\left (B^{2} b c - B^{2} a d\right )} n^{2} + 3 \,{\left (B^{2} b c - B^{2} a d +{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right )^{2} - 3 \,{\left (B^{2} a d n^{2} - A B b c n +{\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} - 6 \,{\left (A B b c - A B a d\right )} n + 3 \,{\left (2 \, A B b c - 2 \, A B a d +{\left (B^{2} b d n x + B^{2} b c n\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\left (B^{2} b c - B^{2} a d\right )} n - 2 \,{\left (B^{2} a d n - A B b c +{\left (B^{2} b d n - A B b d\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 3 \,{\left (2 \, B^{2} a d n^{2} - 2 \, A B a d n + A^{2} b c +{\left (2 \, B^{2} b d n^{2} - 2 \, A B b d n + A^{2} b d\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{3 \,{\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}{\left (d i x + c i\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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