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

Optimal. Leaf size=369 \[ -\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 i (a+b x)^2 (b c-a d)^3}-\frac {b^2 B n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g^3 i n (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^3 i (a+b x) (b c-a d)^3}+\frac {4 b B d n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {b^2 B^2 n^2 (c+d x)^2}{4 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {4 b B^2 d n^2 (c+d x)}{g^3 i (a+b x) (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 7.20, antiderivative size = 2025, normalized size of antiderivative = 5.49, number of steps used = 111, number of rules used = 31, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.689, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 30

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 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 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 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 2344

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

Rule 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 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 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 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 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 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 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 2411

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

Rule 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 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 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 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 2488

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

Rule 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 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 2506

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

Rule 2507

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

Rule 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 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 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 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 6610

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

Rule 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]]

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}{(192 c+192 d x) (a g+b g x)^3} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d) g^3 (a+b x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b 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+b x} \, dx}{192 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{192 (b c-a d)^3 g^3}-\frac {(b d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{192 (b c-a d)^2 g^3}+\frac {b \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{192 (b c-a d) g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {\left (B 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 ) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (B 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 (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{96 (b c-a d)^2 g^3}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{192 (b c-a d) g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}+\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{192 g^3}-\frac {\left (B d^2 n\right ) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (B 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 (c+d x)}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^3 g^3}-\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{96 (b c-a d) g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}+\frac {(B n) \int \left (\frac {b \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)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{192 g^3}-\frac {\left (B d^2 n\right ) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}+\frac {\left (B 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 (c+d x)}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}-\frac {(B d n) \int \left (\frac {b \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)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{96 (b c-a d) g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}+\frac {\left (b B 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}{192 (b c-a d)^3 g^3}+\frac {\left (b B 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}{96 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{192 (b c-a d)^2 g^3}-\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{96 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{96 (b c-a d)^2 g^3}+\frac {\left (B d^2 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{96 (b c-a d)^2 g^3}+\frac {(b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{192 (b c-a d) g^3}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}+\frac {\left (b B 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 (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B 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 (c+d x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}-\frac {\left (B^2 d^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}-\frac {\left (B^2 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}{192 (b c-a d)^3 g^3}+\frac {\left (B^2 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 (c+d x)}{a+b x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 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}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 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 (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{192 (b c-a d)^2 g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{96 (b c-a d)^2 g^3}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{384 (b c-a d) g^3}\\ &=-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{192 (b c-a d)^3 g^3}+\frac {\left (b B d^2 n\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B d^3 n\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{96 b (b c-a d)^2 g^3}+\frac {\left (B^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{384 g^3}-\frac {\left (B^2 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}{192 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 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}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{192 (b c-a d) g^3}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{96 (b c-a d) g^3}\\ &=-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (A b B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (A B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{96 b (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}+\frac {\left (B^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{384 g^3}-\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{192 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{192 (b c-a d)^3 g^3}+\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{192 (b c-a d) g^3}-\frac {\left (B^2 d n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{96 (b c-a d) g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (b B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{96 (b c-a d)^2 g^3}+\frac {\left (b B^2 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 (c+d x)}{a+b x} \, dx}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (A B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 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 )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 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 )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 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 )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^3\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 )}{192 b (b c-a d)^3 g^3}-\frac {\left (B^2 d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{96 b (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{192 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{96 b (b c-a d)^3 g^3}-\frac {\left (B^2 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 {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^3(c+d x)}{576 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 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 {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 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 )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^3(c+d x)}{576 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 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 {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (b B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{192 (b c-a d)^3 g^3}-\frac {\left (b B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{192 (b c-a d)^3 g^3}+\frac {\left (B^2 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 )}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^3(c+d x)}{576 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 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 {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}-\frac {\left (B^2 d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}+\frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^3(c+d x)}{576 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 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 {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+2 \frac {\left (B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{96 (b c-a d)^3 g^3}\\ &=-\frac {B^2 n^2}{768 (b c-a d) g^3 (a+b x)^2}+\frac {7 B^2 d n^2}{384 (b c-a d)^2 g^3 (a+b x)}+\frac {7 B^2 d^2 n^2 \log (a+b x)}{384 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(a+b x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(a+b x)}{128 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{192 (b c-a d)^3 g^3}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{384 (b c-a d) g^3 (a+b x)^2}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 (b c-a d)^2 g^3 (a+b x)}+\frac {B 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 )}{64 (b c-a d)^3 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{384 (b c-a d) g^3 (a+b x)^2}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^2 g^3 (a+b x)}+\frac {d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{192 (b c-a d)^3 g^3}-\frac {7 B^2 d^2 n^2 \log (c+d x)}{384 (b c-a d)^3 g^3}+\frac {A B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{64 (b c-a d)^3 g^3}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{192 (b c-a d)^3 g^3}-\frac {A B d^2 n \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^2(c+d x)}{128 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n^2 \log ^3(c+d x)}{576 (b c-a d)^3 g^3}+\frac {A B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{192 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{192 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}-\frac {B^2 d^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {A B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{64 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}-\frac {B^2 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 {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{96 (b c-a d)^3 g^3}+\frac {B^2 d^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{96 (b c-a d)^3 g^3}\\ \end {align*}

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Mathematica [B]  time = 1.40, size = 975, normalized size = 2.64 \[ \frac {4 B^2 d^2 n^2 (a+b x)^2 \log ^3\left (\frac {a+b x}{c+d x}\right )+6 B n \left (2 A d^2 x^2 b^2+3 B d^2 n x^2 b^2-B c^2 n b^2+2 B c d n x b^2+4 a B c d n b+4 a A d^2 x b+4 a B d^2 n x b+2 a^2 A d^2+2 B d^2 (a+b x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B d^2 n (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right )\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B (b c-a d) n \left (2 A b c+b B n c-6 a A d-7 a B d n-4 A b d x-6 b B d n x+2 B (b (c-2 d x)-3 a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-b c+3 a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )-3 (b c-a d)^2 \left (2 A^2+2 B n A+B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B n (2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (2 A+B n-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+6 d (b c-a d) (a+b x) \left (2 A^2+6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B n (2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (2 A+3 B n-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+6 d^2 (a+b x)^2 \log (a+b x) \left (2 A^2+6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B n (2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (2 A+3 B n-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 d^2 (a+b x)^2 \left (2 A^2+6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B n (2 A+3 B n) \log \left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (2 A+3 B n-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{12 (b c-a d)^3 g^3 i (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*B^2*d^2*n^2*(a + b*x)^2*Log[(a + b*x)/(c + d*x)]^3 + 6*B*n*Log[(a + b*x)/(c + d*x)]^2*(2*a^2*A*d^2 - b^2*B*
c^2*n + 4*a*b*B*c*d*n + 4*a*A*b*d^2*x + 2*b^2*B*c*d*n*x + 4*a*b*B*d^2*n*x + 2*A*b^2*d^2*x^2 + 3*b^2*B*d^2*n*x^
2 + 2*B*d^2*(a + b*x)^2*Log[e*((a + b*x)/(c + d*x))^n] - 2*B*d^2*n*(a + b*x)^2*Log[(a + b*x)/(c + d*x)]) - 6*B
*(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(2*A*b*c - 6*a*A*d + b*B*c*n - 7*a*B*d*n - 4*A*b*d*x - 6*b*B*d*n*x + 2
*B*(-3*a*d + b*(c - 2*d*x))*Log[e*((a + b*x)/(c + d*x))^n] + 2*B*n*(-(b*c) + 3*a*d + 2*b*d*x)*Log[(a + b*x)/(c
+ d*x)]) - 3*(b*c - a*d)^2*(2*A^2 + 2*A*B*n + B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(2*A +
B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(2*
A + B*n - 2*B*n*Log[(a + b*x)/(c + d*x)])) + 6*d*(b*c - a*d)*(a + b*x)*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B^2*Lo
g[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(2*A + 3*B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d
*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(2*A + 3*B*n - 2*B*n*Log[(a + b*x)/(c + d*x)])) + 6*d^2*(a + b*x)^
2*Log[a + b*x]*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(2*A + 3*B*n)*Log
[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(2*A + 3*B*n
- 2*B*n*Log[(a + b*x)/(c + d*x)])) - 6*d^2*(a + b*x)^2*(2*A^2 + 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/
(c + d*x))^n]^2 - 2*B*n*(2*A + 3*B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Lo
g[e*((a + b*x)/(c + d*x))^n]*(2*A + 3*B*n - 2*B*n*Log[(a + b*x)/(c + d*x)]))*Log[c + d*x])/(12*(b*c - a*d)^3*g
^3*i*(a + b*x)^2)

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fricas [B]  time = 1.00, size = 1068, normalized size = 2.89 \[ -\frac {6 \, A^{2} b^{2} c^{2} - 24 \, A^{2} a b c d + 18 \, A^{2} a^{2} d^{2} - 4 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} a b d^{2} n^{2} x + B^{2} a^{2} d^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, {\left (B^{2} b^{2} c^{2} - 16 \, B^{2} a b c d + 15 \, B^{2} a^{2} d^{2}\right )} n^{2} + 6 \, {\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + 3 \, B^{2} a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} x - 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} a b d^{2} x + B^{2} a^{2} d^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e)^{2} - 6 \, {\left (2 \, A B a^{2} d^{2} n - {\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d\right )} n^{2} + {\left (3 \, B^{2} b^{2} d^{2} n^{2} + 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (2 \, A B a b d^{2} n + {\left (B^{2} b^{2} c d + 2 \, B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, {\left (A B b^{2} c^{2} - 8 \, A B a b c d + 7 \, A B a^{2} d^{2}\right )} n - 6 \, {\left (2 \, A^{2} b^{2} c d - 2 \, A^{2} a b d^{2} + 7 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + 6 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 6 \, {\left (2 \, A B b^{2} c^{2} - 8 \, A B a b c d + 6 \, A B a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} a b d^{2} n x + B^{2} a^{2} d^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + 7 \, B^{2} a^{2} d^{2}\right )} n - 2 \, {\left (2 \, A B b^{2} c d - 2 \, A B a b d^{2} + 3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n\right )} x - 2 \, {\left (2 \, A B a^{2} d^{2} + {\left (3 \, B^{2} b^{2} d^{2} n + 2 \, A B b^{2} d^{2}\right )} x^{2} - {\left (B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d\right )} n + 2 \, {\left (2 \, A B a b d^{2} + {\left (B^{2} b^{2} c d + 2 \, B^{2} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) - 6 \, {\left (2 \, A^{2} a^{2} d^{2} - {\left (B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d\right )} n^{2} + {\left (7 \, B^{2} b^{2} d^{2} n^{2} + 6 \, A B b^{2} d^{2} n + 2 \, A^{2} b^{2} d^{2}\right )} x^{2} - 2 \, {\left (A B b^{2} c^{2} - 4 \, A B a b c d\right )} n + 2 \, {\left (2 \, A^{2} a b d^{2} + {\left (3 \, B^{2} b^{2} c d + 4 \, B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left (A B b^{2} c d + 2 \, A B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{12 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*A^2*b^2*c^2 - 24*A^2*a*b*c*d + 18*A^2*a^2*d^2 - 4*(B^2*b^2*d^2*n^2*x^2 + 2*B^2*a*b*d^2*n^2*x + B^2*a^
2*d^2*n^2)*log((b*x + a)/(d*x + c))^3 + 3*(B^2*b^2*c^2 - 16*B^2*a*b*c*d + 15*B^2*a^2*d^2)*n^2 + 6*(B^2*b^2*c^2
- 4*B^2*a*b*c*d + 3*B^2*a^2*d^2 - 2*(B^2*b^2*c*d - B^2*a*b*d^2)*x - 2*(B^2*b^2*d^2*x^2 + 2*B^2*a*b*d^2*x + B^
2*a^2*d^2)*log((b*x + a)/(d*x + c)))*log(e)^2 - 6*(2*A*B*a^2*d^2*n - (B^2*b^2*c^2 - 4*B^2*a*b*c*d)*n^2 + (3*B^
2*b^2*d^2*n^2 + 2*A*B*b^2*d^2*n)*x^2 + 2*(2*A*B*a*b*d^2*n + (B^2*b^2*c*d + 2*B^2*a*b*d^2)*n^2)*x)*log((b*x + a
)/(d*x + c))^2 + 6*(A*B*b^2*c^2 - 8*A*B*a*b*c*d + 7*A*B*a^2*d^2)*n - 6*(2*A^2*b^2*c*d - 2*A^2*a*b*d^2 + 7*(B^2
*b^2*c*d - B^2*a*b*d^2)*n^2 + 6*(A*B*b^2*c*d - A*B*a*b*d^2)*n)*x + 6*(2*A*B*b^2*c^2 - 8*A*B*a*b*c*d + 6*A*B*a^
2*d^2 - 2*(B^2*b^2*d^2*n*x^2 + 2*B^2*a*b*d^2*n*x + B^2*a^2*d^2*n)*log((b*x + a)/(d*x + c))^2 + (B^2*b^2*c^2 -
8*B^2*a*b*c*d + 7*B^2*a^2*d^2)*n - 2*(2*A*B*b^2*c*d - 2*A*B*a*b*d^2 + 3*(B^2*b^2*c*d - B^2*a*b*d^2)*n)*x - 2*(
2*A*B*a^2*d^2 + (3*B^2*b^2*d^2*n + 2*A*B*b^2*d^2)*x^2 - (B^2*b^2*c^2 - 4*B^2*a*b*c*d)*n + 2*(2*A*B*a*b*d^2 + (
B^2*b^2*c*d + 2*B^2*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) - 6*(2*A^2*a^2*d^2 - (B^2*b^2*c^2 - 8*B^2*
a*b*c*d)*n^2 + (7*B^2*b^2*d^2*n^2 + 6*A*B*b^2*d^2*n + 2*A^2*b^2*d^2)*x^2 - 2*(A*B*b^2*c^2 - 4*A*B*a*b*c*d)*n +
2*(2*A^2*a*b*d^2 + (3*B^2*b^2*c*d + 4*B^2*a*b*d^2)*n^2 + 2*(A*B*b^2*c*d + 2*A*B*a*b*d^2)*n)*x)*log((b*x + a)/
(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^
2*d + 3*a^3*b^2*c*d^2 - a^4*b*d^3)*g^3*i*x + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*g^3*i)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 2.12, size = 2126, normalized size = 5.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c
*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^
2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)*g^3*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2 + A*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d
+ a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^
4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x
+ c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/
12*((3*b^2*c^2 - 48*a*b*c*d + 45*a^2*d^2 - 4*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^3 + 4*(b^2*d^2
*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x + c)^3 + 18*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 6*(3*
b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c)^2 -
42*(b^2*c*d - a*b*d^2)*x - 42*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 6*(7*b^2*d^2*x^2 + 14*a*b*
d^2*x + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^
2*d^2)*log(b*x + a))*log(d*x + c))*n^2/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*
d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c
^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)*x) + 6*(b^2*c^2 - 8*a*b*c*d + 7*a^
2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d
*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2
+ 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))*n*log(e*(b*x/(
d*x + c) + a/(d*x + c))^n)/(a^2*b^3*c^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i +
(b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3
*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g^3*i - a^4*b*d^3*g^3*i)*x))*B^2 - 1/2*(b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2
+ 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(d*x +
c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a
*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))*A*B*n/(a^2*b^3*c^3*
g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3
*a^2*b^3*c*d^2*g^3*i - a^3*b^2*d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c*d^2*g
^3*i - a^4*b*d^3*g^3*i)*x) + 1/2*A^2*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2
+ 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*lo
g(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^
2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))

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mupad [B]  time = 8.49, size = 1011, normalized size = 2.74 \[ \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B^2\,n}{x^2\,\left (b^3\,c\,g^3\,i-a\,b^2\,d\,g^3\,i\right )+x\,\left (2\,a\,b^2\,c\,g^3\,i-2\,a^2\,b\,d\,g^3\,i\right )-a^3\,d\,g^3\,i+a^2\,b\,c\,g^3\,i}-\frac {d^2\,\left (3\,n\,B^2+2\,A\,B\right )\,\left (\frac {a\,g^3\,i\,n\,{\left (a\,d-b\,c\right )}^2}{2\,d}+\frac {g^3\,i\,n\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {b\,g^3\,i\,n\,x\,{\left (a\,d-b\,c\right )}^2}{d}\right )}{g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (x^2\,\left (b^3\,c\,g^3\,i-a\,b^2\,d\,g^3\,i\right )+x\,\left (2\,a\,b^2\,c\,g^3\,i-2\,a^2\,b\,d\,g^3\,i\right )-a^3\,d\,g^3\,i+a^2\,b\,c\,g^3\,i\right )}\right )-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {d^2\,\left (3\,n\,B^2+2\,A\,B\right )}{2\,g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {B^2\,d^2\,\left (\frac {g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,g^3\,i\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b\,g^3\,i\,n\,x\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (i\,a^2\,g^3+2\,i\,a\,b\,g^3\,x+i\,b^2\,g^3\,x^2\right )}\right )-\frac {\frac {6\,A^2\,a\,d-2\,A^2\,b\,c+15\,B^2\,a\,d\,n^2-B^2\,b\,c\,n^2+14\,A\,B\,a\,d\,n-2\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (2\,b\,d\,A^2+6\,b\,d\,A\,B\,n+7\,b\,d\,B^2\,n^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}-\frac {B^2\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (\frac {i\,a^3\,d^3\,g^3-i\,a^2\,b\,c\,d^2\,g^3-i\,a\,b^2\,c^2\,d\,g^3+i\,b^3\,c^3\,g^3}{i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3}+2\,b\,d\,x\right )\,\left (A^2+3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,d^2+6\,A\,B\,d^2\,n+7\,B^2\,d^2\,n^2\right )}\right )\,\left (A^2+3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*((a + b*x)/(c + d*x))^n)*((B^2*n)/(x^2*(b^3*c*g^3*i - a*b^2*d*g^3*i) + x*(2*a*b^2*c*g^3*i - 2*a^2*b*d*g^
3*i) - a^3*d*g^3*i + a^2*b*c*g^3*i) - (d^2*(3*B^2*n + 2*A*B)*((a*g^3*i*n*(a*d - b*c)^2)/(2*d) + (g^3*i*n*(a*d
- b*c)^2*(2*a*d - b*c))/(2*d^2) + (b*g^3*i*n*x*(a*d - b*c)^2)/d))/(g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*
a*b*c*d)*(x^2*(b^3*c*g^3*i - a*b^2*d*g^3*i) + x*(2*a*b^2*c*g^3*i - 2*a^2*b*d*g^3*i) - a^3*d*g^3*i + a^2*b*c*g^
3*i))) - log(e*((a + b*x)/(c + d*x))^n)^2*((d^2*(3*B^2*n + 2*A*B))/(2*g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 -
2*a*b*c*d)) - (B^2*d^2*((g^3*i*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2) + (a*g^3*i*n*(a*d - b*c))/(2*d) + (b*g^3*
i*n*x*(a*d - b*c))/d))/(g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*g^3*i + b^2*g^3*i*x^2 + 2*a*b
*g^3*i*x))) - ((6*A^2*a*d - 2*A^2*b*c + 15*B^2*a*d*n^2 - B^2*b*c*n^2 + 14*A*B*a*d*n - 2*A*B*b*c*n)/(2*(a*d - b
*c)) + (x*(2*A^2*b*d + 7*B^2*b*d*n^2 + 6*A*B*b*d*n))/(a*d - b*c))/(x^2*(2*b^3*c*g^3*i - 2*a*b^2*d*g^3*i) + x*(
4*a*b^2*c*g^3*i - 4*a^2*b*d*g^3*i) - 2*a^3*d*g^3*i + 2*a^2*b*c*g^3*i) + (d^2*atan((d^2*((a^3*d^3*g^3*i + b^3*c
^3*g^3*i - a*b^2*c^2*d*g^3*i - a^2*b*c*d^2*g^3*i)/(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2*a*b*c*d*g^3*i) + 2*b*d*x)
*(A^2 + (7*B^2*n^2)/2 + 3*A*B*n)*(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2*a*b*c*d*g^3*i)*2i)/(g^3*i*(a*d - b*c)^3*(2
*A^2*d^2 + 7*B^2*d^2*n^2 + 6*A*B*d^2*n)))*(A^2 + (7*B^2*n^2)/2 + 3*A*B*n)*2i)/(g^3*i*(a*d - b*c)^3) - (B^2*d^2
*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g^3*i*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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