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

Optimal. Leaf size=297 \[ \frac{2 B n \text{PolyLog}\left (2,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac{2 B n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{g}+\frac{2 B^2 n^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{g}+\frac{\log \left (1-\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 5.18411, antiderivative size = 2233, normalized size of antiderivative = 7.52, number of steps used = 43, number of rules used = 21, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.656, Rules used = {2524, 2528, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2437, 2435, 2315} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 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 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 2437

Int[(Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)])/(x_), x_Symbol] :> Dist[m, In
t[(Log[i + j*x]*Log[c*(d + e*x)^n])/x, x], x] - Dist[m*Log[i + j*x] - Log[h*(i + j*x)^m], Int[Log[c*(d + e*x)^
n]/x, x], x] /; FreeQ[{c, d, e, h, i, j, m, n}, x] && NeQ[e*i - d*j, 0] && NeQ[i + j*x, h*(i + j*x)^m]

Rule 2435

Int[(Log[(a_) + (b_.)*(x_)]*Log[(c_) + (d_.)*(x_)])/(x_), x_Symbol] :> Simp[Log[-((b*x)/a)]*Log[a + b*x]*Log[c
+ d*x], x] + (Simp[(1*(Log[-((b*x)/a)] - Log[-(((b*c - a*d)*x)/(a*(c + d*x)))] + Log[(b*c - a*d)/(b*(c + d*x)
)])*Log[(a*(c + d*x))/(c*(a + b*x))]^2)/2, x] - Simp[(1*(Log[-((b*x)/a)] - Log[-((d*x)/c)])*(Log[a + b*x] + Lo
g[(a*(c + d*x))/(c*(a + b*x))])^2)/2, x] + Simp[(Log[c + d*x] - Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1
+ (b*x)/a], x] + Simp[(Log[a + b*x] + Log[(a*(c + d*x))/(c*(a + b*x))])*PolyLog[2, 1 + (d*x)/c], x] + Simp[Lo
g[(a*(c + d*x))/(c*(a + b*x))]*PolyLog[2, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[Log[(a*(c + d*x))/(c*(a + b*
x))]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))], x] - Simp[PolyLog[3, 1 + (b*x)/a], x] - Simp[PolyLog[3, 1 + (d*x
)/c], x] + Simp[PolyLog[3, (c*(a + b*x))/(a*(c + d*x))], x] - Simp[PolyLog[3, (d*(a + b*x))/(b*(c + d*x))], x]
) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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

Mathematica [B]  time = 0.473154, size = 1441, normalized size = 4.85 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.528, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{gx+f} \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/(g*x+f),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{A^{2} \log \left (g x + f\right )}{g} + \int \frac{B^{2} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{2} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + 2 \,{\left (B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{2} \log \left (e\right ) + A B\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{g x + f}\,{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/(g*x+f),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{g x + f}, x\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/(g*x+f),x, algorithm="fricas")

[Out]

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

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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/(g*x+f),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}}{g x + f}\,{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/(g*x+f),x, algorithm="giac")

[Out]

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