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

Optimal. Leaf size=572 \[ \frac{2 B i^2 n (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac{2 B^2 i^2 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b^3 g}-\frac{B^2 i^2 n^2 (b c-a d)^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac{2 B^2 i^2 n^2 (b c-a d)^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac{d i^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g}-\frac{B d i^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac{2 B i^2 n (b c-a d)^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}-\frac{i^2 (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g}+\frac{B i^2 n (b c-a d)^2 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac{i^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b g}+\frac{B^2 i^2 n^2 (b c-a d)^2 \log (c+d x)}{b^3 g} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 5.08149, antiderivative size = 1790, normalized size of antiderivative = 3.13, number of steps used = 82, number of rules used = 27, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2500, 2440, 2434, 2433, 2375, 2317, 2374, 6589, 2499, 2302, 30, 2396, 2525, 2486, 31} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2523

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

Rule 12

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

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 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 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 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 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 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]

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 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 2486

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(172 c+172 d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac{29584 d (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{172 d (172 c+172 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 (a g+b g x)}\right ) \, dx\\ &=\frac{\left (29584 (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx}{b^2}+\frac{(172 d) \int (172 c+172 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}+\frac{(29584 d (b c-a d)) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g}\\ &=\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{(B n) \int \frac{29584 (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b g}-\frac{(59168 B d (b c-a d) n) \int \frac{(b c-a d) 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}{b^2 g}-\frac{\left (59168 B (b c-a d)^2 n\right ) \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 (a g+b g x)}{a+b x} \, dx}{b^3 g}\\ &=\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{(29584 B (b c-a d) n) \int \frac{(c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b g}-\frac{\left (59168 B (b c-a d)^2 n\right ) \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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b^3 g}-\frac{\left (59168 B d (b c-a d)^2 n\right ) \int \frac{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}{b^2 g}\\ &=\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{(29584 B (b c-a d) n) \int \left (\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b (a+b x)}\right ) \, dx}{b g}-\frac{\left (59168 B d (b c-a d)^2 n\right ) \int \left (-\frac{a \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)}+\frac{c \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)}\right ) \, dx}{b^2 g}-\frac{\left (59168 B (b c-a d)^3 n\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b^3 g}\\ &=\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{(29584 B d (b c-a d) n) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}+\frac{(59168 a B d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g}-\frac{(59168 B c d (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 g}-\frac{\left (29584 B (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 B (b c-a d)^3 n\right ) \int \left (\frac{d \left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{\left (29584 B^2 d (b c-a d) n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g}-\frac{\left (59168 B (b c-a d)^2 n\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 B d (b c-a d)^2 n\right ) \int \frac{\left (-A-B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{c+d x} \, dx}{b^3 g}+\frac{\left (59168 B^2 c (b c-a d) 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}{b^2 g}-\frac{\left (59168 a B^2 d (b c-a d) 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}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^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}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{\left (59168 B (b c-a d)^2 n\right ) \int \left (\frac{A \log (a g+b g x)}{a+b x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{b^2 g}-\frac{\left (59168 B d (b c-a d)^2 n\right ) \int \left (\frac{A \log (a g+b g x)}{-c-d x}+\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b^3 g}+\frac{\left (59168 B^2 c (b c-a d) 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}{b^2 g}-\frac{\left (59168 a B^2 d (b c-a d) 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}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^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}{b^3 g}+\frac{\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac{\left (59168 A B (b c-a d)^2 n\right ) \int \frac{\log (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 B^2 (b c-a d)^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 A B d (b c-a d)^2 n\right ) \int \frac{\log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac{\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}+\frac{\left (59168 B^2 c (b c-a d) n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b g}-\frac{\left (59168 a B^2 d (b c-a d) n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 B^2 c d (b c-a d) n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 g}+\frac{\left (59168 a B^2 d^2 (b c-a d) n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^2 g}-\frac{\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{\left (59168 A B (b c-a d)^2 n\right ) \int \frac{\log \left (\frac{b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx}{b^2}-\frac{\left (59168 A B (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}-\frac{\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac{\log \left ((a+b x)^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac{\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac{\log \left ((c+d x)^{-n}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac{\left (59168 B^2 c (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g}-\frac{\left (59168 a B^2 d (b c-a d) n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g}-\frac{\left (59168 a B^2 d (b c-a d) n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g}-\frac{\left (59168 B^2 c d (b c-a d) n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log ^2(a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac{\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac{\log ^2(a g+b g x)}{c+d x} \, dx}{b^3 g}-\frac{\left (59168 B^2 d (b c-a d)^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 (a g+b g x)}{-c-d x} \, dx}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{\left (59168 A B (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac{\left (59168 A B (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^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{-b c g+a d g}{d}-\frac{b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (\frac{-b c g+a d g}{d}+\frac{b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (\frac{b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx}{b^2}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}-\frac{\left (59168 B^2 c (b c-a d) 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 )}{b^2 g}-\frac{\left (59168 a B^2 d (b c-a d) 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 )}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^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 )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^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{b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx}{b^2}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{\left (29584 B^2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (x^{-n}\right )}{\frac{-b c g+a d g}{d}+\frac{b g x}{d}} \, dx,x,c+d x\right )}{b^2 d}+\frac{\left (59168 B^2 (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\left (\frac{-b c+a d}{d}-\frac{b x}{d}\right )^n\right )}{\frac{-b c+a d}{d}-\frac{b x}{d}} \, dx,x,-c-d x\right )}{b^2 d g}+\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c g+a d g}{d}-\frac{b g x}{d}\right )}{\frac{-b c g+a d g}{d}-\frac{b g x}{d}} \, dx,x,-c-d x\right )}{b^2 d}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b g \left (\frac{b c g-a d g}{b g}+\frac{d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^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}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^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 (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{\left (59168 B^2 (b c-a d)^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 )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (x^{-n}\right ) \log \left (1+\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}+\frac{\left (29584 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{-b c g+a d g}{b g}-\frac{d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^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 (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{\left (29584 B^2 d (b c-a d)^2\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 )}{b^4 g}-\frac{\left (29584 B^2 d (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c g+a d g}{b g}-\frac{d x}{b g}} \, dx,x,a g+b g x\right )}{b^4 g^2}-\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^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 (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^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 )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (1-\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^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 (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^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 )}{b^3 g}-\frac{\left (59168 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac{29584 A B d (b c-a d) n x}{b^2 g}-\frac{29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac{14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac{29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac{29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac{59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac{29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac{29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac{14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac{59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac{29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac{59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac{29584 B^2 (b c-a d)^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac{29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left (\frac{b (c+d x)}{b c-a d}\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 (a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac{59168 A B (b c-a d)^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 a B^2 d (b c-a d) n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{29584 B^2 (b c-a d)^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 (b c-a d)^2 n \log \left ((a+b x)^n\right ) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^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 (a+b x)}{b c-a d}\right )}{b^3 g}+\frac{59168 B^2 c (b c-a d) n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac{59168 B^2 (b c-a d)^2 n^2 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{b^3 g}\\ \end{align*}

Mathematica [B]  time = 3.0083, size = 1654, normalized size = 2.89 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(i^2*(6*b*d*(2*b*c - a*d)*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 3*b^2*d^
2*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 6*(b*c - a*d)^2*Log[a + b*x]*(
A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 - 12*b*B*c*n*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + Log[a + b*x]) + 2*(-
(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + (-(b*d*x) + a*d*
Log[a + b*x])*Log[(a + b*x)/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 6*b^2*B*c^2*n*(A + B*
Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x]
- Log[c/d + x] - Log[(a + b*x)/(c + d*x)]) - 2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b
*(c + d*x))/(b*c - a*d)])) + 3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(-4*a
*d^2*(a + b*x)*(-1 + Log[a/b + x]) + 2*a^2*d^2*Log[a/b + x]^2 + 4*a*b*d*(c + d*x)*(-1 + Log[c/d + x]) + d^2*(b
*x*(2*a - b*x) + 2*b^2*x^2*Log[a/b + x] - 2*a^2*Log[a + b*x]) - 2*d^2*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x])*
(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)]) + b^2*(d*x*(-2*c + d*x) - 2*d^2*x^2*Log[c/d + x] + 2*
c^2*Log[c + d*x]) - 4*a^2*d^2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])) + 4*b*B^2*c*n^2*(Log[(a + b*x)/(c + d*x)]*(-(a*d*Log[(a + b*x)/(c + d*x)]^2) + 6*(b*c - a*d)*Log[(b*
c - a*d)/(b*c + b*d*x)] + 3*d*Log[(a + b*x)/(c + d*x)]*(a + b*x + a*Log[(b*c - a*d)/(b*c + b*d*x)])) + 6*(b*c
- a*d + a*d*Log[(a + b*x)/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 6*a*d*PolyLog[3, (d*(a + b*x))
/(b*(c + d*x))]) - B^2*n^2*(6*b^2*c^2*Log[(b*c - a*d)/(c + d*x)] - 12*a*b*c*d*Log[(b*c - a*d)/(c + d*x)] + 6*a
^2*d^2*Log[(b*c - a*d)/(c + d*x)] + 6*a*b*c*d*Log[(a + b*x)/(c + d*x)] - 6*a^2*d^2*Log[(a + b*x)/(c + d*x)] +
6*b^2*c*d*x*Log[(a + b*x)/(c + d*x)] - 6*a*b*d^2*x*Log[(a + b*x)/(c + d*x)] + 9*a^2*d^2*Log[(a + b*x)/(c + d*x
)]^2 + 6*a*b*d^2*x*Log[(a + b*x)/(c + d*x)]^2 - 3*b^2*d^2*x^2*Log[(a + b*x)/(c + d*x)]^2 - 2*a^2*d^2*Log[(a +
b*x)/(c + d*x)]^3 + 6*b^2*c^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + 12*a*b*c*d*Log[(a + b*
x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] - 18*a^2*d^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*
x)] + 6*a^2*d^2*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*(b^2*c^2 + 2*a*b*c*d - 3*a^2*d^2
+ 2*a^2*d^2*Log[(a + b*x)/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 12*a^2*d^2*PolyLog[3, (d*(a +
b*x))/(b*(c + d*x))]) - 6*b^2*B^2*c^2*n^2*(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c + d*x)]^2 - 2*L
og[(a + b*x)/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])))
/(6*b^3*g)

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Maple [F]  time = 0.687, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{2}}{bgx+ag} \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((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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