3.193 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4 (c i+d i x)} \, dx\)

Optimal. Leaf size=543 \[ -\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac {2 b^3 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {3 b^2 B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {d^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g^4 i n (b c-a d)^4}-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^4 i (a+b x) (b c-a d)^4}-\frac {6 b B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac {2 b^3 B^2 n^2 (c+d x)^3}{27 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B^2 d n^2 (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {6 b B^2 d^2 n^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \]

[Out]

-6*b*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/4*b^2*B^2*d*n^2*(d*x+c)^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2
/27*b^3*B^2*n^2*(d*x+c)^3/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-6*b*B*d^2*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a
*d+b*c)^4/g^4/i/(b*x+a)+3/2*b^2*B*d*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2
/9*b^3*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-3*b*d^2*(d*x+c)*(A+B*ln(e*((
b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/2*b^2*d*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b
*c)^4/g^4/i/(b*x+a)^2-1/3*b^3*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-1/3*d^3
*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^4/g^4/i/n

________________________________________________________________________________________

Rubi [C]  time = 8.35, antiderivative size = 2180, normalized size of antiderivative = 4.01, number of steps used = 143, 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)^4*(c*i + d*i*x)),x]

[Out]

(-2*B^2*n^2)/(27*(b*c - a*d)*g^4*i*(a + b*x)^3) + (19*B^2*d*n^2)/(36*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (85*B^
2*d^2*n^2)/(18*(b*c - a*d)^3*g^4*i*(a + b*x)) - (85*B^2*d^3*n^2*Log[a + b*x])/(18*(b*c - a*d)^4*g^4*i) + (A*B*
d^3*n*Log[a + b*x]^2)/((b*c - a*d)^4*g^4*i) + (11*B^2*d^3*n^2*Log[a + b*x]^2)/(6*(b*c - a*d)^4*g^4*i) + (B^2*d
^3*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*((a + b*x)/(c + d*x))^n]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*Log[a
+ b*x]*Log[e*((a + b*x)/(c + d*x))^n]^2)/((b*c - a*d)^4*g^4*i) - (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
)/(9*(b*c - a*d)*g^4*i*(a + b*x)^3) + (5*B*d*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*(b*c - a*d)^2*g^4*i*
(a + b*x)^2) - (11*B*d^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3*g^4*i*(a + b*x)) - (11*B*d
^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^4*g^4*i) - (A + B*Log[e*((a + b*x)/(c
+ d*x))^n])^2/(3*(b*c - a*d)*g^4*i*(a + b*x)^3) + (d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)
^2*g^4*i*(a + b*x)^2) - (d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g^4*i*(a + b*x)) - (d^3*
Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^4*i) + (85*B^2*d^3*n^2*Log[c + d*x])/(
18*(b*c - a*d)^4*g^4*i) - (2*A*B*d^3*n*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^4*g^4*i) -
(11*B^2*d^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*(b*c - a*d)^4*g^4*i) - (B^2*d^3*Log[(a + b
*x)^n]^2*Log[c + d*x])/((b*c - a*d)^4*g^4*i) + (11*B*d^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
)/(3*(b*c - a*d)^4*g^4*i) + (d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[c + d*x])/((b*c - a*d)^4*g^4*i)
+ (A*B*d^3*n*Log[c + d*x]^2)/((b*c - a*d)^4*g^4*i) + (11*B^2*d^3*n^2*Log[c + d*x]^2)/(6*(b*c - a*d)^4*g^4*i) -
(B^2*d^3*n^2*Log[a + b*x]*Log[c + d*x]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*n*Log[e*((a + b*x)/(c + d*x))^n]*L
og[c + d*x]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*n^2*Log[c + d*x]^3)/(3*(b*c - a*d)^4*g^4*i) - (2*A*B*d^3*n*Log
[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*n^2*Log[a + b*x]*Log[(b*(c + d*x
))/(b*c - a*d)])/(3*(b*c - a*d)^4*g^4*i) + (B^2*d^3*Log[(a + b*x)^n]^2*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c -
a*d)^4*g^4*i) - (2*B^2*d^3*n*Log[a + b*x]*Log[c + d*x]*Log[(c + d*x)^(-n)])/((b*c - a*d)^4*g^4*i) - (B^2*d^3*
Log[a + b*x]*Log[(c + d*x)^(-n)]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c
+ d*x)^(-n)]^2)/((b*c - a*d)^4*g^4*i) + (2*B^2*d^3*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)^4*g^4*i) - (2*A*B*d^3*n*PolyLog[
2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*
d))])/(3*(b*c - a*d)^4*g^4*i) + (2*B^2*d^3*n*Log[(a + b*x)^n]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c
- a*d)^4*g^4*i) - (2*A*B*d^3*n*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*n^2*
PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*(b*c - a*d)^4*g^4*i) - (2*B^2*d^3*n*Log[(c + d*x)^(-n)]*PolyLog[2, (
b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) + (2*B^2*d^3*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)^4*g^4*i) - (2*B^2*d^3*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)^4*g^4*i) - (2*B^2*d^3*n^2*Pol
yLog[3, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^4*i) - (2*B^2*d^3*n^2*PolyLog[3, (b*(c + d*x))/(b*c -
a*d)])/((b*c - a*d)^4*g^4*i) - (2*B^2*d^3*n^2*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^4*g^4*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}{(193 c+193 d x) (a g+b g x)^4} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d) g^4 (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b d^3\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}{193 (b c-a d)^4 g^4}+\frac {d^4 \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}{193 (b c-a d)^4 g^4}+\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)^2} \, dx}{193 (b c-a d)^3 g^4}-\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)^3} \, dx}{193 (b c-a d)^2 g^4}+\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)^4} \, dx}{193 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {\left (2 B d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (2 B d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 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 )}{(a+b x)^2 (c+d x)} \, dx}{193 (b c-a d)^3 g^4}-\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)^3 (c+d x)} \, dx}{193 (b c-a d)^2 g^4}+\frac {(2 B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{579 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {(2 B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{579 g^4}+\frac {\left (2 B d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (2 B d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 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)^2 (c+d x)} \, dx}{193 (b c-a d)^2 g^4}-\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)^3 (c+d x)} \, dx}{193 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {(2 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)^4}-\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)^3}+\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)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{579 g^4}+\frac {\left (2 B d^3 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}{193 (b c-a d)^3 g^4}-\frac {\left (2 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)}{(a+b x) (c+d x)} \, dx}{193 (b c-a d)^3 g^4}+\frac {\left (2 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 )}{(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}{193 (b c-a d)^2 g^4}-\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)^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}{193 (b c-a d) g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{579 (b c-a d)^4 g^4}-\frac {\left (b B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{579 (b c-a d)^4 g^4}+\frac {\left (B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 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)^2} \, dx}{579 (b c-a d)^3 g^4}+\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)^2} \, dx}{193 (b c-a d)^3 g^4}+\frac {\left (2 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)^2} \, dx}{193 (b c-a d)^3 g^4}+\frac {\left (2 B d^3 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}{193 (b c-a d)^3 g^4}-\frac {\left (2 B d^3 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}{193 (b c-a d)^3 g^4}-\frac {(2 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)^3} \, dx}{579 (b c-a d)^2 g^4}-\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)^3} \, dx}{193 (b c-a d)^2 g^4}+\frac {(2 b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{579 (b c-a d) g^4}\\ &=-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {\left (2 b 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)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B d^4 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{193 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3 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}{579 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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}{579 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (B^2 d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{579 (b c-a d)^3 g^4}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{193 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{193 (b c-a d)^3 g^4}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{579 (b c-a d)^2 g^4}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{386 (b c-a d)^2 g^4}+\frac {\left (2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{1737 (b c-a d) g^4}\\ &=\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B d^4 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 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 )}{193 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{1737 g^4}+\frac {\left (2 B^2 d^3 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}{579 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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}{579 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (B^2 d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{579 (b c-a d)^2 g^4}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{193 (b c-a d)^2 g^4}+\frac {\left (2 B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{193 (b c-a d)^2 g^4}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{579 (b c-a d) g^4}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{386 (b c-a d) g^4}\\ &=\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{193 (b c-a d)^4 g^4}-\frac {\left (2 A b B d^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 b 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)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^4 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 A B d^4 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 )}{193 b (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 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}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^3 g^4}+\frac {\left (2 B^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{1737 g^4}+\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{579 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{579 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{579 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{579 (b c-a d)^4 g^4}-\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^2 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}{579 (b c-a d)^2 g^4}+\frac {\left (B^2 d^2 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}{193 (b c-a d)^2 g^4}+\frac {\left (2 B^2 d^2 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}{193 (b c-a d)^2 g^4}-\frac {\left (B^2 d 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}{579 (b c-a d) g^4}-\frac {\left (B^2 d 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}{386 (b c-a d) g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}-\frac {\left (2 b B^2 d^3 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^4 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{579 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{579 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{579 (b c-a d)^4 g^4}-\frac {\left (b B^2 d^3 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{579 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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}{193 (b c-a d)^3 g^4}-\frac {\left (2 b 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 (c+d x)}{a+b x} \, dx}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 A B d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{579 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{579 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 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}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 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}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}+\frac {\left (B^2 d^4\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 )}{193 b (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^4 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 )}{193 b (b c-a d)^4 g^4}+\frac {\left (B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^4 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 )}{193 b (b c-a d)^4 g^4}+\frac {\left (2 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 ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n^2 \log ^3(c+d x)}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log (a+b x) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n^2 \log ^3(c+d x)}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}+\frac {\left (b B^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log (a+b x) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n^2 \log ^3(c+d x)}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}+\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log (a+b x) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n^2 \log ^3(c+d x)}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}-2 \frac {\left (2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}\\ &=-\frac {2 B^2 n^2}{5211 (b c-a d) g^4 (a+b x)^3}+\frac {19 B^2 d n^2}{6948 (b c-a d)^2 g^4 (a+b x)^2}-\frac {85 B^2 d^2 n^2}{3474 (b c-a d)^3 g^4 (a+b x)}-\frac {85 B^2 d^3 n^2 \log (a+b x)}{3474 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(a+b x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(a+b x)}{1158 (b c-a d)^4 g^4}+\frac {B^2 d^3 \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 )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{193 (b c-a d)^4 g^4}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1737 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1158 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{579 (b c-a d)^4 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{579 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{386 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{193 (b c-a d)^4 g^4}+\frac {85 B^2 d^3 n^2 \log (c+d x)}{3474 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{579 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{193 (b c-a d)^4 g^4}+\frac {A B d^3 n \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {11 B^2 d^3 n^2 \log ^2(c+d x)}{1158 (b c-a d)^4 g^4}-\frac {B^2 d^3 n^2 \log (a+b x) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 n^2 \log ^3(c+d x)}{579 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}-\frac {B^2 d^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{193 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}+\frac {2 B^2 d^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 A B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {11 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{579 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 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 )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{193 (b c-a d)^4 g^4}-\frac {2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{193 (b c-a d)^4 g^4}\\ \end {align*}

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Mathematica [B]  time = 1.92, size = 1295, normalized size = 2.38 \[ -\frac {4 \left (9 A^2+6 B n A+2 B^2 n^2+9 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+9 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B n (3 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (3 A+B n-3 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) (b c-a d)^3-3 d (a+b x) \left (18 A^2+30 B n A+19 B^2 n^2+18 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+18 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B n (6 A+5 B n) \log \left (\frac {a+b x}{c+d x}\right )+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (6 A+5 B n-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) (b c-a d)^2+6 d^2 (a+b x)^2 \left (18 A^2+66 B n A+85 B^2 n^2+18 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+18 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B n (6 A+11 B n) \log \left (\frac {a+b x}{c+d x}\right )+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (6 A+11 B n-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) (b c-a d)+6 B n \log \left (\frac {a+b x}{c+d x}\right ) \left (4 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 B n \log \left (\frac {a+b x}{c+d x}\right )\right ) (b c-a d)^2+3 d (a d-b c) (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )+6 d^2 (a+b x)^2 \left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) (b c-a d)+36 B^2 d^3 n^2 (a+b x)^3 \log ^3\left (\frac {a+b x}{c+d x}\right )+18 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (6 A d^3 x^3 b^3+11 B d^3 n x^3 b^3+6 B c d^2 n x^2 b^3+2 B c^3 n b^3-3 B c^2 d n x b^3+18 a A d^3 x^2 b^2+27 a B d^3 n x^2 b^2-9 a B c^2 d n b^2+18 a B c d^2 n x b^2+18 a^2 B c d^2 n b+18 a^2 A d^3 x b+18 a^2 B d^3 n x b+6 a^3 A d^3+6 B d^3 (a+b x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-6 B d^3 n (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right )\right )+6 d^3 (a+b x)^3 \log (a+b x) \left (18 A^2+66 B n A+85 B^2 n^2+18 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+18 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B n (6 A+11 B n) \log \left (\frac {a+b x}{c+d x}\right )+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (6 A+11 B n-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 d^3 (a+b x)^3 \left (18 A^2+66 B n A+85 B^2 n^2+18 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+18 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-6 B n (6 A+11 B n) \log \left (\frac {a+b x}{c+d x}\right )+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (6 A+11 B n-6 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{108 (b c-a d)^4 g^4 i (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(36*B^2*d^3*n^2*(a + b*x)^3*Log[(a + b*x)/(c + d*x)]^3 + 18*B*n*Log[(a + b*x)/(c + d*x)]^2*(6*a^3*A*d^3
+ 2*b^3*B*c^3*n - 9*a*b^2*B*c^2*d*n + 18*a^2*b*B*c*d^2*n + 18*a^2*A*b*d^3*x - 3*b^3*B*c^2*d*n*x + 18*a*b^2*B*
c*d^2*n*x + 18*a^2*b*B*d^3*n*x + 18*a*A*b^2*d^3*x^2 + 6*b^3*B*c*d^2*n*x^2 + 27*a*b^2*B*d^3*n*x^2 + 6*A*b^3*d^3
*x^3 + 11*b^3*B*d^3*n*x^3 + 6*B*d^3*(a + b*x)^3*Log[e*((a + b*x)/(c + d*x))^n] - 6*B*d^3*n*(a + b*x)^3*Log[(a
+ b*x)/(c + d*x)]) - 3*d*(b*c - a*d)^2*(a + b*x)*(18*A^2 + 30*A*B*n + 19*B^2*n^2 + 18*B^2*Log[e*((a + b*x)/(c
+ d*x))^n]^2 - 6*B*n*(6*A + 5*B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[
e*((a + b*x)/(c + d*x))^n]*(6*A + 5*B*n - 6*B*n*Log[(a + b*x)/(c + d*x)])) + 6*d^2*(b*c - a*d)*(a + b*x)^2*(18
*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A + 11*B*n)*Log[(a + b*x)/(c
+ d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 11*B*n - 6*B*n*Lo
g[(a + b*x)/(c + d*x)])) + 6*d^3*(a + b*x)^3*Log[a + b*x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*Log[e*((a +
b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A + 11*B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2
+ 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 11*B*n - 6*B*n*Log[(a + b*x)/(c + d*x)])) + 4*(b*c - a*d)^3*(9*A^2
+ 6*A*B*n + 2*B^2*n^2 + 9*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(3*A + B*n)*Log[(a + b*x)/(c + d*x)] +
9*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(3*A + B*n - 3*B*n*Log[(a + b*x)/(c
+ d*x)])) + 6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(3*d*(-(b*c) + a*d)*(a + b*x)*(6*A + 5*B*n + 6*B*Log[e
*((a + b*x)/(c + d*x))^n] - 6*B*n*Log[(a + b*x)/(c + d*x)]) + 6*d^2*(a + b*x)^2*(6*A + 11*B*n + 6*B*Log[e*((a
+ b*x)/(c + d*x))^n] - 6*B*n*Log[(a + b*x)/(c + d*x)]) + 4*(b*c - a*d)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c
+ d*x))^n] - 3*B*n*Log[(a + b*x)/(c + d*x)])) - 6*d^3*(a + b*x)^3*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*Log
[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A + 11*B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c +
d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 11*B*n - 6*B*n*Log[(a + b*x)/(c + d*x)]))*Log[c + d*x])/((
b*c - a*d)^4*g^4*i*(a + b*x)^3)

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fricas [B]  time = 0.88, size = 1910, normalized size = 3.52 \[ \text {result too large to display} \]

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)^4/(d*i*x+c*i),x, algorithm="fricas")

[Out]

-1/108*(36*A^2*b^3*c^3 - 162*A^2*a*b^2*c^2*d + 324*A^2*a^2*b*c*d^2 - 198*A^2*a^3*d^3 + 36*(B^2*b^3*d^3*n^2*x^3
+ 3*B^2*a*b^2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + B^2*a^3*d^3*n^2)*log((b*x + a)/(d*x + c))^3 + (8*B^2*b^3*
c^3 - 81*B^2*a*b^2*c^2*d + 648*B^2*a^2*b*c*d^2 - 575*B^2*a^3*d^3)*n^2 + 6*(18*A^2*b^3*c*d^2 - 18*A^2*a*b^2*d^3
+ 85*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 + 66*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(2*B^2*b^3*c^3 - 9*
B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2 - 11*B^2*a^3*d^3 + 6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2
*d - 6*B^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^3)*x + 6*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B
^2*a^3*d^3)*log((b*x + a)/(d*x + c)))*log(e)^2 + 18*(6*A*B*a^3*d^3*n + (11*B^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*
x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 +
9*B^2*a*b^2*d^3)*n^2)*x^2 + 3*(6*A*B*a^2*b*d^3*n - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n^2)
*x)*log((b*x + a)/(d*x + c))^2 + 6*(4*A*B*b^3*c^3 - 27*A*B*a*b^2*c^2*d + 108*A*B*a^2*b*c*d^2 - 85*A*B*a^3*d^3)
*n - 3*(18*A^2*b^3*c^2*d - 108*A^2*a*b^2*c*d^2 + 90*A^2*a^2*b*d^3 + (19*B^2*b^3*c^2*d - 378*B^2*a*b^2*c*d^2 +
359*B^2*a^2*b*d^3)*n^2 + 6*(5*A*B*b^3*c^2*d - 54*A*B*a*b^2*c*d^2 + 49*A*B*a^2*b*d^3)*n)*x + 6*(12*A*B*b^3*c^3
- 54*A*B*a*b^2*c^2*d + 108*A*B*a^2*b*c*d^2 - 66*A*B*a^3*d^3 + 6*(6*A*B*b^3*c*d^2 - 6*A*B*a*b^2*d^3 + 11*(B^2*b
^3*c*d^2 - B^2*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*d^3*n*x^3 + 3*B^2*a*b^2*d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + B^2*a
^3*d^3*n)*log((b*x + a)/(d*x + c))^2 + (4*B^2*b^3*c^3 - 27*B^2*a*b^2*c^2*d + 108*B^2*a^2*b*c*d^2 - 85*B^2*a^3*
d^3)*n - 3*(6*A*B*b^3*c^2*d - 36*A*B*a*b^2*c*d^2 + 30*A*B*a^2*b*d^3 + (5*B^2*b^3*c^2*d - 54*B^2*a*b^2*c*d^2 +
49*B^2*a^2*b*d^3)*n)*x + 6*(6*A*B*a^3*d^3 + (11*B^2*b^3*d^3*n + 6*A*B*b^3*d^3)*x^3 + 3*(6*A*B*a*b^2*d^3 + (2*B
^2*b^3*c*d^2 + 9*B^2*a*b^2*d^3)*n)*x^2 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2)*n + 3*(6*A*B
*a^2*b*d^3 - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6*B^2*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) + 6*
(18*A^2*a^3*d^3 + (85*B^2*b^3*d^3*n^2 + 66*A*B*b^3*d^3*n + 18*A^2*b^3*d^3)*x^3 + (4*B^2*b^3*c^3 - 27*B^2*a*b^2
*c^2*d + 108*B^2*a^2*b*c*d^2)*n^2 + 3*(18*A^2*a*b^2*d^3 + (22*B^2*b^3*c*d^2 + 63*B^2*a*b^2*d^3)*n^2 + 6*(2*A*B
*b^3*c*d^2 + 9*A*B*a*b^2*d^3)*n)*x^2 + 6*(2*A*B*b^3*c^3 - 9*A*B*a*b^2*c^2*d + 18*A*B*a^2*b*c*d^2)*n + 3*(18*A^
2*a^2*b*d^3 - (5*B^2*b^3*c^2*d - 54*B^2*a*b^2*c*d^2 - 36*B^2*a^2*b*d^3)*n^2 - 6*(A*B*b^3*c^2*d - 6*A*B*a*b^2*c
*d^2 - 6*A*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*
b^4*c*d^3 + a^4*b^3*d^4)*g^4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^
5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4
*i*x + (a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4*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)^4/(d*i*x+c*i),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{4} \left (d i x +c i \right )}\, dx \]

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)^4/(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)^4/(d*i*x+c*i),x)

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maxima [B]  time = 2.98, size = 3445, normalized size = 6.34 \[ \text {result too large to display} \]

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)^4/(d*i*x+c*i),x, algorithm="maxima")

[Out]

-1/6*B^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5
*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2
*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a
^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*
d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i) - 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a
^3*b*c*d^3 + a^4*d^4)*g^4*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2 - 1/3*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 -
7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3
)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a
^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3
)*g^4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i)
- 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i))*log(e*(b
*x/(d*x + c) + a/(d*x + c))^n) - 1/108*((8*b^3*c^3 - 81*a*b^2*c^2*d + 648*a^2*b*c*d^2 - 575*a^3*d^3 + 36*(b^3*
d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^3 - 36*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^
2*b*d^3*x + a^3*d^3)*log(d*x + c)^3 + 510*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 198*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3
*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 -
6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c)^2 - 3*(19*b^3*c^2*d -
378*a*b^2*c*d^2 + 359*a^2*b*d^3)*x + 510*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a
) - 6*(85*b^3*d^3*x^3 + 255*a*b^2*d^3*x^2 + 255*a^2*b*d^3*x + 85*a^3*d^3 + 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 +
3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*
x + a))*log(d*x + c))*n^2/(a^3*b^4*c^4*g^4*i - 4*a^4*b^3*c^3*d*g^4*i + 6*a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3
*g^4*i + a^7*d^4*g^4*i + (b^7*c^4*g^4*i - 4*a*b^6*c^3*d*g^4*i + 6*a^2*b^5*c^2*d^2*g^4*i - 4*a^3*b^4*c*d^3*g^4*
i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6*c^4*g^4*i - 4*a^2*b^5*c^3*d*g^4*i + 6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*
c*d^3*g^4*i + a^5*b^2*d^4*g^4*i)*x^2 + 3*(a^2*b^5*c^4*g^4*i - 4*a^3*b^4*c^3*d*g^4*i + 6*a^4*b^3*c^2*d^2*g^4*i
- 4*a^5*b^2*c*d^3*g^4*i + a^6*b*d^4*g^4*i)*x) + 6*(4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 +
66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2
- 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^
2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^
3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^
3*d^3)*log(b*x + a))*log(d*x + c))*n*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(a^3*b^4*c^4*g^4*i - 4*a^4*b^3*c^3
*d*g^4*i + 6*a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3*g^4*i + a^7*d^4*g^4*i + (b^7*c^4*g^4*i - 4*a*b^6*c^3*d*g^4*
i + 6*a^2*b^5*c^2*d^2*g^4*i - 4*a^3*b^4*c*d^3*g^4*i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6*c^4*g^4*i - 4*a^2*b^5*
c^3*d*g^4*i + 6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*c*d^3*g^4*i + a^5*b^2*d^4*g^4*i)*x^2 + 3*(a^2*b^5*c^4*g^4*i
- 4*a^3*b^4*c^3*d*g^4*i + 6*a^4*b^3*c^2*d^2*g^4*i - 4*a^5*b^2*c*d^3*g^4*i + a^6*b*d^4*g^4*i)*x))*B^2 - 1/18*(4
*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 +
3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x
+ a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*
x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d
^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*A*B*n/(a^3*b^4*c^
4*g^4*i - 4*a^4*b^3*c^3*d*g^4*i + 6*a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3*g^4*i + a^7*d^4*g^4*i + (b^7*c^4*g^4
*i - 4*a*b^6*c^3*d*g^4*i + 6*a^2*b^5*c^2*d^2*g^4*i - 4*a^3*b^4*c*d^3*g^4*i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6
*c^4*g^4*i - 4*a^2*b^5*c^3*d*g^4*i + 6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*c*d^3*g^4*i + a^5*b^2*d^4*g^4*i)*x^2
+ 3*(a^2*b^5*c^4*g^4*i - 4*a^3*b^4*c^3*d*g^4*i + 6*a^4*b^3*c^2*d^2*g^4*i - 4*a^5*b^2*c*d^3*g^4*i + a^6*b*d^4*g
^4*i)*x) - 1/6*A^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3
- 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2
- a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3
*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^
2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i) - 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2
*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i))

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mupad [B]  time = 10.34, size = 1921, normalized size = 3.54 \[ \text {result too large to display} \]

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)^4*(c*i + d*i*x)),x)

[Out]

((198*A^2*a^2*d^2 + 36*A^2*b^2*c^2 + 575*B^2*a^2*d^2*n^2 + 8*B^2*b^2*c^2*n^2 - 126*A^2*a*b*c*d + 510*A*B*a^2*d
^2*n + 24*A*B*b^2*c^2*n - 73*B^2*a*b*c*d*n^2 - 138*A*B*a*b*c*d*n)/(6*(a*d - b*c)) + (x^2*(18*A^2*b^2*d^2 + 85*
B^2*b^2*d^2*n^2 + 66*A*B*b^2*d^2*n))/(a*d - b*c) + (x*(90*A^2*a*b*d^2 - 18*A^2*b^2*c*d + 359*B^2*a*b*d^2*n^2 -
19*B^2*b^2*c*d*n^2 + 294*A*B*a*b*d^2*n - 30*A*B*b^2*c*d*n))/(2*(a*d - b*c)))/(x*(54*a^4*b*d^2*g^4*i + 54*a^2*
b^3*c^2*g^4*i - 108*a^3*b^2*c*d*g^4*i) + x^2*(54*a*b^4*c^2*g^4*i + 54*a^3*b^2*d^2*g^4*i - 108*a^2*b^3*c*d*g^4*
i) + x^3*(18*b^5*c^2*g^4*i + 18*a^2*b^3*d^2*g^4*i - 36*a*b^4*c*d*g^4*i) + 18*a^5*d^2*g^4*i + 18*a^3*b^2*c^2*g^
4*i - 36*a^4*b*c*d*g^4*i) - log(e*((a + b*x)/(c + d*x))^n)^2*((d^3*(11*B^2*n + 6*A*B))/(6*g^4*i*n*(a*d - b*c)*
(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*d^3*(x*(b*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*
d^2) + (a*g^4*i*n*(a*d - b*c))/(3*d)) + (2*a*b*g^4*i*n*(a*d - b*c))/(3*d) + (b*g^4*i*n*(a*d - b*c)*(3*a*d - b*
c))/(3*d^2)) + a*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*g^4*i*n*(a*d - b*c))/(3*d)) + (g^4*i*n*(a*d
- b*c)*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*d^3) + (b^2*g^4*i*n*x^2*(a*d - b*c))/d))/(g^4*i*n*(a*d - b*c)*(a
^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*g^4*i + b^3*g^4*i*x^3 + 3*a^2*b*g^4*i*x + 3*a*b^2*g^4*i
*x^2))) - log(e*((a + b*x)/(c + d*x))^n)*((6*B^2*a*d*n - 3*B^2*b*c*n + 3*B^2*b*d*n*x)/(x*(9*a^4*b*d^2*g^4*i +
9*a^2*b^3*c^2*g^4*i - 18*a^3*b^2*c*d*g^4*i) + x^2*(9*a*b^4*c^2*g^4*i + 9*a^3*b^2*d^2*g^4*i - 18*a^2*b^3*c*d*g^
4*i) + x^3*(3*b^5*c^2*g^4*i + 3*a^2*b^3*d^2*g^4*i - 6*a*b^4*c*d*g^4*i) + 3*a^5*d^2*g^4*i + 3*a^3*b^2*c^2*g^4*i
- 6*a^4*b*c*d*g^4*i) - (d^3*(11*B^2*n + 6*A*B)*(x*(b*((a*g^4*i*n*(a*d - b*c)^3)/d + (g^4*i*n*(a*d - b*c)^3*(3
*a*d - b*c))/(2*d^2)) + (2*a*b*g^4*i*n*(a*d - b*c)^3)/d + (b*g^4*i*n*(a*d - b*c)^3*(3*a*d - b*c))/d^2) + a*((a
*g^4*i*n*(a*d - b*c)^3)/d + (g^4*i*n*(a*d - b*c)^3*(3*a*d - b*c))/(2*d^2)) + (g^4*i*n*(a*d - b*c)^3*(3*a^2*d^2
+ b^2*c^2 - 3*a*b*c*d))/d^3 + (3*b^2*g^4*i*n*x^2*(a*d - b*c)^3)/d))/(3*g^4*i*n*(a*d - b*c)*(a^3*d^3 - b^3*c^3
+ 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(x*(9*a^4*b*d^2*g^4*i + 9*a^2*b^3*c^2*g^4*i - 18*a^3*b^2*c*d*g^4*i) + x^2*(9
*a*b^4*c^2*g^4*i + 9*a^3*b^2*d^2*g^4*i - 18*a^2*b^3*c*d*g^4*i) + x^3*(3*b^5*c^2*g^4*i + 3*a^2*b^3*d^2*g^4*i -
6*a*b^4*c*d*g^4*i) + 3*a^5*d^2*g^4*i + 3*a^3*b^2*c^2*g^4*i - 6*a^4*b*c*d*g^4*i))) + (d^3*atan((d^3*(A^2 + (85*
B^2*n^2)/18 + (11*A*B*n)/3)*(18*a^4*d^4*g^4*i - 18*b^4*c^4*g^4*i + 36*a*b^3*c^3*d*g^4*i - 36*a^3*b*c*d^3*g^4*i
)*1i)/(g^4*i*(a*d - b*c)^4*(18*A^2*d^3 + 85*B^2*d^3*n^2 + 66*A*B*d^3*n)) + (b*d^4*x*(A^2 + (85*B^2*n^2)/18 + (
11*A*B*n)/3)*(a^3*d^3*g^4*i - b^3*c^3*g^4*i + 3*a*b^2*c^2*d*g^4*i - 3*a^2*b*c*d^2*g^4*i)*36i)/(g^4*i*(a*d - b*
c)^4*(18*A^2*d^3 + 85*B^2*d^3*n^2 + 66*A*B*d^3*n)))*(A^2 + (85*B^2*n^2)/18 + (11*A*B*n)/3)*2i)/(g^4*i*(a*d - b
*c)^4) - (B^2*d^3*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g^4*i*n*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2))

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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)**4/(d*i*x+c*i),x)

[Out]

Timed out

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