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

Optimal. Leaf size=402 \[ \frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i^3 n (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {B d^2 n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g i^3 (c+d x) (b c-a d)^3}+\frac {4 A b B d n (a+b x)}{g i^3 (c+d x) (b c-a d)^3}+\frac {B^2 d^2 n^2 (a+b x)^2}{4 g i^3 (c+d x)^2 (b c-a d)^3}+\frac {4 b B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {4 b B^2 d n^2 (a+b x)}{g i^3 (c+d x) (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 7.04, antiderivative size = 2025, normalized size of antiderivative = 5.04, number of steps used = 111, number of rules used = 31, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.689, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 12, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2525, 44, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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}{(206 c+206 d x)^3 (a g+b g x)} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d) g (c+d x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\frac {b^3 \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}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 d\right ) \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}{8741816 (b c-a d)^3 g}-\frac {(b d) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8741816 (b c-a d)^2 g}-\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3} \, dx}{8741816 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}-\frac {(b B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}-\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^3} \, dx}{8741816 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{8741816 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}-\frac {(b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {(B n) \int \left (\frac {b^3 \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 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^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 (c+d x)^2}-\frac {b^2 d \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}{8741816 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}+\frac {\left (b^2 B n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}-\frac {(b B n) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4370908 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (b^3 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^3 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}+\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}+\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8741816 (b c-a d)^2 g}+\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}+\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8741816 (b c-a d) g}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B d 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}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}-\frac {\left (b^2 B^2 n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8741816 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{17483632 (b c-a d) g}\\ &=-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B n\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B d 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}{4370908 (b c-a d)^3 g}-\frac {(A b B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{17483632 g}+\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8741816 (b c-a d) g}+\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d) g}\\ &=-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (A b^3 B n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {(A b B d n) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B d n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{17483632 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{8741816 (b c-a d) g}+\frac {\left (b B^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4370908 (b c-a d) g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B d n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}+\frac {\left (b^3 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{8741816 (b c-a d)^3 g}-\frac {\left (b B^2 d n\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^3 B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8741816 d (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^3 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{8741816 d (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+2 \frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}\\ \end {align*}

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Mathematica [B]  time = 1.37, size = 971, normalized size = 2.42 \[ \frac {4 b^2 B^2 n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )-\frac {6 B n \left (-2 A c^2 b^2-2 A d^2 x^2 b^2+3 B d^2 n x^2 b^2-4 A c d x b^2+4 B c d n x b^2-2 B (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) b^2+2 B n (c+d x)^2 \log \left (\frac {a+b x}{c+d x}\right ) b^2+4 a B c d n b+2 a B d^2 n x b-a^2 B d^2 n\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}-\frac {6 B (b c-a d) n \left (-6 A b c+7 b B n c+2 a A d-a B d n-4 A b d x+6 b B d n x+2 B (-3 b c+a d-2 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (3 b c-a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}+\frac {3 (b c-a d)^2 \left (2 A^2-2 B n A+B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B n (B n-2 A) \log \left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{(c+d x)^2}+6 b^2 \log (a+b x) \left (2 A^2-6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B n (3 B n-2 A) \log \left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+\frac {6 b (b c-a d) \left (2 A^2-6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B n (3 B n-2 A) \log \left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{c+d x}-6 b^2 \left (2 A^2-6 B n A+7 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B n (3 B n-2 A) \log \left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-2 A+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{12 (b c-a d)^3 g i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.09, size = 1076, normalized size = 2.68 \[ \frac {18 \, A^{2} b^{2} c^{2} - 24 \, A^{2} a b c d + 6 \, A^{2} a^{2} d^{2} + 4 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} b^{2} c d n^{2} x + B^{2} b^{2} c^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, {\left (15 \, B^{2} b^{2} c^{2} - 16 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n^{2} + 6 \, {\left (3 \, B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + B^{2} b^{2} c^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e)^{2} + 6 \, {\left (2 \, A B b^{2} c^{2} n - {\left (4 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2} - {\left (3 \, B^{2} b^{2} d^{2} n^{2} - 2 \, A B b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (2 \, A B b^{2} c d n - {\left (2 \, B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (7 \, A B b^{2} c^{2} - 8 \, A B a b c d + A B a^{2} d^{2}\right )} n + 6 \, {\left (2 \, A^{2} b^{2} c d - 2 \, A^{2} a b d^{2} + 7 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} - 6 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 6 \, {\left (6 \, A B b^{2} c^{2} - 8 \, A B a b c d + 2 \, A B a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} b^{2} c d n x + B^{2} b^{2} c^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (7 \, B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A B b^{2} c d - 2 \, A B a b d^{2} - 3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n\right )} x + 2 \, {\left (2 \, A B b^{2} c^{2} - {\left (3 \, B^{2} b^{2} d^{2} n - 2 \, A B b^{2} d^{2}\right )} x^{2} - {\left (4 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A B b^{2} c d - {\left (2 \, B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 6 \, {\left (2 \, A^{2} b^{2} c^{2} + {\left (8 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2} + {\left (7 \, B^{2} b^{2} d^{2} n^{2} - 6 \, A B b^{2} d^{2} n + 2 \, A^{2} b^{2} d^{2}\right )} x^{2} - 2 \, {\left (4 \, A B a b c d - A B a^{2} d^{2}\right )} n + 2 \, {\left (2 \, A^{2} b^{2} c d + {\left (4 \, B^{2} b^{2} c d + 3 \, B^{2} a b d^{2}\right )} n^{2} - 2 \, {\left (2 \, A B b^{2} c d + A B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 8.67, size = 664, normalized size = 1.65 \[ \frac {1}{12} \, {\left (\frac {4 \, B^{2} b^{2} i n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3}}{b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g} - 6 \, {\left (\frac {4 \, {\left (b x + a\right )} B^{2} b d i n^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}} - \frac {{\left (b x + a\right )}^{2} B^{2} d^{2} i n^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} - \frac {2 \, {\left (A B b^{2} i n + B^{2} b^{2} i n\right )}}{b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (\frac {{\left (B^{2} d^{2} i n^{2} - 2 \, A B d^{2} i n - 2 \, B^{2} d^{2} i n\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (B^{2} b d i n^{2} - A B b d i n - B^{2} b d i n\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) - \frac {12 \, {\left (A^{2} b^{2} + 2 \, A B b^{2} + B^{2} b^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} c^{2} g i - 2 \, a b c d g i + a^{2} d^{2} g i} + \frac {3 \, {\left (B^{2} d^{2} i n^{2} - 2 \, A B d^{2} i n - 2 \, B^{2} d^{2} i n + 2 \, A^{2} d^{2} i + 4 \, A B d^{2} i + 2 \, B^{2} d^{2} i\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} - \frac {24 \, {\left (2 \, B^{2} b d i n^{2} - 2 \, A B b d i n - 2 \, B^{2} b d i n + A^{2} b d i + 2 \, A B b d i + B^{2} b d i\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right ) \left (d i x +c i \right )^{3}}\, dx \]

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)/(d*i*x+c*i)^3,x)

[Out]

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

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maxima [B]  time = 2.75, size = 2126, normalized size = 5.29 \[ \text {result too large to display} \]

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)/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.73, size = 1007, normalized size = 2.50 \[ {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )}{2\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,b^2\,\left (\frac {c\,g\,i^3\,n\,\left (a\,d-b\,c\right )}{2\,b}-\frac {g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,\left (a\,d-b\,c\right )}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (g\,c^2\,i^3+2\,g\,c\,d\,i^3\,x+g\,d^2\,i^3\,x^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-6\,A^2\,b\,c+B^2\,a\,d\,n^2-15\,B^2\,b\,c\,n^2-2\,A\,B\,a\,d\,n+14\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (2\,b\,d\,A^2-6\,b\,d\,A\,B\,n+7\,b\,d\,B^2\,n^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,a\,d^3\,g\,i^3-2\,b\,c\,d^2\,g\,i^3\right )+x\,\left (4\,a\,c\,d^2\,g\,i^3-4\,b\,c^2\,d\,g\,i^3\right )-2\,b\,c^3\,g\,i^3+2\,a\,c^2\,d\,g\,i^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B^2\,n}{x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3}+\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )\,\left (\frac {c\,g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2}{2\,b}-\frac {g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,{\left (a\,d-b\,c\right )}^2}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3\right )}\right )-\frac {B^2\,b^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (\frac {g\,a^3\,d^3\,i^3-g\,a^2\,b\,c\,d^2\,i^3-g\,a\,b^2\,c^2\,d\,i^3+g\,b^3\,c^3\,i^3}{g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3}+2\,b\,d\,x\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2\,n+7\,B^2\,b^2\,n^2\right )}\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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