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

Optimal. Leaf size=375 \[ \frac {b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g i^3 (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {B d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g i^3 (c+d x) (b c-a d)^3}+\frac {4 A b B d (a+b x)}{g i^3 (c+d x) (b c-a d)^3}+\frac {B^2 d^2 (a+b x)^2}{4 g i^3 (c+d x)^2 (b c-a d)^3}+\frac {4 b B^2 d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {4 b B^2 d (a+b x)}{g i^3 (c+d x) (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 7.37, antiderivative size = 1899, normalized size of antiderivative = 5.06, number of steps used = 117, number of rules used = 31, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.738, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 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)])^2/((a*g + b*g*x)*(c*i + d*i*x)^3),x]

[Out]

B^2/(4*(b*c - a*d)*g*i^3*(c + d*x)^2) + (7*b*B^2)/(2*(b*c - a*d)^2*g*i^3*(c + d*x)) + (7*b^2*B^2*Log[a + b*x])
/(2*(b*c - a*d)^3*g*i^3) - (A*b^2*B*Log[a + b*x]^2)/((b*c - a*d)^3*g*i^3) + (3*b^2*B^2*Log[a + b*x]^2)/(2*(b*c
- a*d)^3*g*i^3) + (b^2*B^2*Log[a + b*x]*Log[(c + d*x)^(-1)]^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)^(-1)]^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)]^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)]^2)
/((b*c - a*d)^3*g*i^3) - (B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)*g*i^3*(c + d*x)^2) - (3*b*B*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g*i^3*(c + d*x)) - (3*b^2*B*Log[a + b*x]*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/((b*c - a*d)^3*g*i^3) + (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(2*(b*c - a*d)*g*i^3*(c +
d*x)^2) + (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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)])^2)/((b*c - a*d)^3*g*i^3) - (7*b^2*B^2*Log[c + d*x])/(2*(b*c - a*d)^3*g*i^3) +
(b^2*B^2*Log[a + b*x]^2*Log[c + d*x])/((b*c - a*d)^3*g*i^3) + (2*A*b^2*B*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*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^3*g
*i^3) + (2*b^2*B^2*Log[a + b*x]*Log[(c + d*x)^(-1)]*Log[c + d*x])/((b*c - a*d)^3*g*i^3) - (2*b^2*B^2*Log[-((d*
(a + b*x))/(b*c - a*d))]*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b
*c - a*d)^3*g*i^3) + (3*b^2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^3*g*i^3) - (b^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[c + d*x])/((b*c - a*d)^3*g*i^3) - (A*b^2*B*Log[c + d*x]^2)/((b*c -
a*d)^3*g*i^3) + (3*b^2*B^2*Log[c + d*x]^2)/(2*(b*c - a*d)^3*g*i^3) + (b^2*B^2*Log[a + b*x]*Log[c + d*x]^2)/((b
*c - a*d)^3*g*i^3) - (b^2*B^2*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x]^2)/((b*c - a*d)^3*g*i^3) - (b^2*B^2*Lo
g[c + d*x]^3)/(3*(b*c - a*d)^3*g*i^3) + (2*A*b^2*B*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*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
]^2*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^3*g*i^3) + (2*A*b^2*B*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))
])/((b*c - a*d)^3*g*i^3) - (3*b^2*B^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*g*i^3) - (2*b^2
*B^2*Log[a + b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*g*i^3) + (2*A*b^2*B*PolyLog[2, (b*(
c + d*x))/(b*c - a*d)])/((b*c - a*d)^3*g*i^3) - (3*b^2*B^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)
^3*g*i^3) + (2*b^2*B^2*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^3*g*i^3) - (2*b
^2*B^2*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(b*c - a*d
)])/((b*c - a*d)^3*g*i^3) + (2*b^2*B^2*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])
/((b*c - a*d)^3*g*i^3) + (2*b^2*B^2*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^3*g*i^3) + (2*b^2*B
^2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^3*g*i^3) + (2*b^2*B^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 (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(104 c+104 d x)^3 (a g+b g x)} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d) g (c+d x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\frac {b^3 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 d\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{1124864 (b c-a d)^3 g}-\frac {(b d) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{1124864 (b c-a d)^2 g}-\frac {d \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3} \, dx}{1124864 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{562432 (b c-a d)^3 g}-\frac {(b B) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{562432 (b c-a d)^2 g}-\frac {B \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^3} \, dx}{1124864 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {B \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^3} \, dx}{1124864 g}-\frac {(b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{562432 (b c-a d) g}-\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{562432 (b c-a d)^3 e g}+\frac {\left (b^2 B\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {B \int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{1124864 g}-\frac {(b B) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{562432 (b c-a d) g}-\frac {\left (b^2 B\right ) \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^3 e g}+\frac {\left (b^2 B\right ) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^3 e g}\\ &=\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {\left (b^3 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^3 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^2 g}+\frac {\left (b^2 B\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^2 g}+\frac {(b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1124864 (b c-a d)^2 g}+\frac {(b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{562432 (b c-a d)^2 g}+\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1124864 (b c-a d) g}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{562432 (b c-a d)^2 g}+\frac {\left (b^2 B\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{562432 (b c-a d)^2 g}+\frac {\left (b B^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1124864 (b c-a d)^2 g}+\frac {\left (b B^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{562432 (b c-a d)^2 g}+\frac {B^2 \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{2249728 (b c-a d) g}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {B^2 \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{2249728 g}+\frac {\left (b^3 B\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B d\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^2 g}-\frac {\left (b^2 B^2\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^2 g}+\frac {\left (b B^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1124864 (b c-a d) g}+\frac {\left (b B^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{562432 (b c-a d) g}+\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1124864 (b c-a d)^3 e g}-\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{1124864 (b c-a d)^3 e g}+\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{562432 (b c-a d)^3 e g}-\frac {\left (b^2 B^2\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 e g}\\ &=-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {B^2 \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}{2249728 g}+\frac {\left (b^3 B\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B d\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{562432 (b c-a d)^3 g}-\frac {(A b B) \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 )}{562432 (b c-a d)^2 g}+\frac {\left (b B^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}{1124864 (b c-a d) g}+\frac {\left (b B^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}{562432 (b c-a d) g}+\frac {\left (b^2 B^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{1124864 (b c-a d)^3 e g}-\frac {\left (b^2 B^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{1124864 (b c-a d)^3 e g}+\frac {\left (b^2 B^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{562432 (b c-a d)^3 e g}-\frac {\left (b^2 B^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{562432 (b c-a d)^3 e g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {\left (A b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}+\frac {\left (A b^3 B\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^3 B^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^3 B^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {(A b B d) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{562432 (b c-a d)^2 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^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}{562432 (b c-a d)^2 g}+\frac {\left (b^3 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (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(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {\left (A b^2 B\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{1124864 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (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(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{1124864 (b c-a d)^3 g}-\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}-\frac {\left (b B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{1124864 (b c-a d)^3 g}-\frac {\left (b B^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{-\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 )}{562432 (b c-a d)^3 g}+\frac {\left (b B^2 d\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 )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\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 )}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (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(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^3(c+d x)}{3374592 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2(c+d x)}{1124864 (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(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^3(c+d x)}{3374592 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}-\frac {\left (b^3 B^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{1124864 d (b c-a d)^3 g}-\frac {\left (b^3 B^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 )}{1124864 d (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^3(c+d x)}{3374592 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}-\frac {\left (b^2 B^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{562432 (b c-a d)^3 g}+\frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^3(c+d x)}{3374592 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+2 \frac {\left (b^2 B^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 )}{562432 (b c-a d)^3 g}\\ &=\frac {B^2}{4499456 (b c-a d) g (c+d x)^2}+\frac {7 b B^2}{2249728 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 \log (a+b x)}{2249728 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(a+b x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(a+b x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {1}{c+d x}\right )}{1124864 (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 (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1124864 (b c-a d)^3 g}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2249728 (b c-a d) g (c+d x)^2}-\frac {3 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1124864 (b c-a d)^3 g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2249728 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1124864 (b c-a d)^3 g}-\frac {7 b^2 B^2 \log (c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2(a+b x) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{562432 (b c-a d)^3 g}+\frac {3 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1124864 (b c-a d)^3 g}-\frac {A b^2 B \log ^2(c+d x)}{1124864 (b c-a d)^3 g}+\frac {3 b^2 B^2 \log ^2(c+d x)}{2249728 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^3(c+d x)}{3374592 (b c-a d)^3 g}+\frac {A b^2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {A b^2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {3 b^2 B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1124864 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}-\frac {b^2 B^2 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{562432 (b c-a d)^3 g}+\frac {b^2 B^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{562432 (b c-a d)^3 g}\\ \end {align*}

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Mathematica [A]  time = 1.26, size = 290, normalized size = 0.77 \[ \frac {6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (a+b x)+\frac {6 b \left (2 A^2-6 A B+7 B^2\right ) (b c-a d)}{c+d x}+\frac {3 \left (2 A^2-2 A B+B^2\right ) (b c-a d)^2}{(c+d x)^2}+\frac {6 B \left (B d (a+b x) (a d-4 b c-3 b d x)+2 A b^2 (c+d x)^2\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+\frac {6 B (b c-a d) (A (-2 a d+6 b c+4 b d x)+B (a d-7 b c-6 b d x)) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2}+4 b^2 B^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b^2 \left (2 A^2-6 A B+7 B^2\right ) \log (c+d x)}{12 g i^3 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 545, normalized size = 1.45 \[ \frac {3 \, {\left (6 \, A^{2} - 14 \, A B + 15 \, B^{2}\right )} b^{2} c^{2} - 24 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b c d + 3 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2} + 4 \, {\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + B^{2} b^{2} c^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 6 \, {\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A B b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2} - 2 \, {\left (B^{2} a b d^{2} - 2 \, {\left (A B - B^{2}\right )} b^{2} c d\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} c d - {\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} a b d^{2}\right )} x + 6 \, {\left ({\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} d^{2} x^{2} + 2 \, A^{2} b^{2} c^{2} - 8 \, {\left (A B - B^{2}\right )} a b c d + {\left (2 \, A B - B^{2}\right )} a^{2} d^{2} + 2 \, {\left (2 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b^{2} c d - {\left (2 \, A B - 3 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*(6*A^2 - 14*A*B + 15*B^2)*b^2*c^2 - 24*(A^2 - 2*A*B + 2*B^2)*a*b*c*d + 3*(2*A^2 - 2*A*B + B^2)*a^2*d^2
+ 4*(B^2*b^2*d^2*x^2 + 2*B^2*b^2*c*d*x + B^2*b^2*c^2)*log((b*e*x + a*e)/(d*x + c))^3 + 6*((2*A*B - 3*B^2)*b^2
*d^2*x^2 + 2*A*B*b^2*c^2 - 4*B^2*a*b*c*d + B^2*a^2*d^2 - 2*(B^2*a*b*d^2 - 2*(A*B - B^2)*b^2*c*d)*x)*log((b*e*x
+ a*e)/(d*x + c))^2 + 6*((2*A^2 - 6*A*B + 7*B^2)*b^2*c*d - (2*A^2 - 6*A*B + 7*B^2)*a*b*d^2)*x + 6*((2*A^2 - 6
*A*B + 7*B^2)*b^2*d^2*x^2 + 2*A^2*b^2*c^2 - 8*(A*B - B^2)*a*b*c*d + (2*A*B - B^2)*a^2*d^2 + 2*(2*(A^2 - 2*A*B
+ 2*B^2)*b^2*c*d - (2*A*B - 3*B^2)*a*b*d^2)*x)*log((b*e*x + a*e)/(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 = 1.56, size = 628, normalized size = 1.67 \[ \frac {{\left (4 \, B^{2} b^{2} i e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{3} + 12 \, A B b^{2} i e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {24 \, {\left (b x e + a e\right )} B^{2} b d i e \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + 12 \, A^{2} b^{2} i e^{2} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {48 \, {\left (b x e + a e\right )} A B b d i e \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {48 \, {\left (b x e + a e\right )} B^{2} b d i e \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {6 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} - \frac {24 \, {\left (b x e + a e\right )} A^{2} b d i e}{d x + c} + \frac {48 \, {\left (b x e + a e\right )} A B b d i e}{d x + c} - \frac {48 \, {\left (b x e + a e\right )} B^{2} b d i e}{d x + c} + \frac {12 \, {\left (b x e + a e\right )}^{2} A B d^{2} i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {6 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {6 \, {\left (b x e + a e\right )}^{2} A^{2} d^{2} i}{{\left (d x + c\right )}^{2}} - \frac {6 \, {\left (b x e + a e\right )}^{2} A B d^{2} i}{{\left (d x + c\right )}^{2}} + \frac {3 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} i}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{12 \, {\left (b^{2} c^{2} g e - 2 \, a b c d g e + a^{2} d^{2} g e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*B^2*b^2*i*e^2*log((b*x*e + a*e)/(d*x + c))^3 + 12*A*B*b^2*i*e^2*log((b*x*e + a*e)/(d*x + c))^2 - 24*(b
*x*e + a*e)*B^2*b*d*i*e*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c) + 12*A^2*b^2*i*e^2*log((b*x*e + a*e)/(d*x + c
)) - 48*(b*x*e + a*e)*A*B*b*d*i*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 48*(b*x*e + a*e)*B^2*b*d*i*e*log((b
*x*e + a*e)/(d*x + c))/(d*x + c) + 6*(b*x*e + a*e)^2*B^2*d^2*i*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c)^2 - 24
*(b*x*e + a*e)*A^2*b*d*i*e/(d*x + c) + 48*(b*x*e + a*e)*A*B*b*d*i*e/(d*x + c) - 48*(b*x*e + a*e)*B^2*b*d*i*e/(
d*x + c) + 12*(b*x*e + a*e)^2*A*B*d^2*i*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 6*(b*x*e + a*e)^2*B^2*d^2*i
*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 6*(b*x*e + a*e)^2*A^2*d^2*i/(d*x + c)^2 - 6*(b*x*e + a*e)^2*A*B*d^
2*i/(d*x + c)^2 + 3*(b*x*e + a*e)^2*B^2*d^2*i/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e -
a*d*e)*(b*c - a*d)))/(b^2*c^2*g*e - 2*a*b*c*d*g*e + a^2*d^2*g*e)

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maple [B]  time = 0.05, size = 2842, normalized size = 7.58 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b^2/(d*x+c)*a*c+1/i^3/(a*d-b*c)^4/g*A^2*b^3*ln(b/
d*e+(a*d-b*c)/(d*x+c)/d*e)*c+7/2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^3*c+1/3/i^3/(a*d-b*c)
^4/g*B^2*b^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^3*c-3/2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b
^3*c-1/4*d^3/i^3/(a*d-b*c)^4/g*B^2/(d*x+c)^2*a^3-1/2*d^3/i^3/(a*d-b*c)^4/g*A^2/(d*x+c)^2*a^3+7/2/i^3/(a*d-b*c)
^4/g*B^2*b^3/(d*x+c)*c^2-3/2/i^3/(a*d-b*c)^4/g*A^2*b^3*c+1/4/i^3/(a*d-b*c)^4/g*B^2/(d*x+c)^2*b^3*c^3+1/i^3/(a*
d-b*c)^4/g*A^2*b^3/(d*x+c)*c^2+1/2/i^3/(a*d-b*c)^4/g*A^2/(d*x+c)^2*b^3*c^3-3/i^3/(a*d-b*c)^4/g*A*B*b^3/(d*x+c)
*c^2-1/2/i^3/(a*d-b*c)^4/g*A*B/(d*x+c)^2*b^3*c^3-3/2*d^2/i^3/(a*d-b*c)^4/g*A*B/(d*x+c)^2*a^2*b*c+3/2*d/i^3/(a*
d-b*c)^4/g*A*B/(d*x+c)^2*a*b^2*c^2-3/2*d^2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)^2*a^2
*b*c+15/4*d/i^3/(a*d-b*c)^4/g*B^2*b^2*a-15/4/i^3/(a*d-b*c)^4/g*B^2*b^3*c+3/2*d/i^3/(a*d-b*c)^4/g*A^2*b^2*a+1/2
*d^3/i^3/(a*d-b*c)^4/g*A*B/(d*x+c)^2*a^3+d^2/i^3/(a*d-b*c)^4/g*A^2*b/(d*x+c)*a^2-1/3*d/i^3/(a*d-b*c)^4/g*B^2*b
^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^3*a+7/2*d^2/i^3/(a*d-b*c)^4/g*B^2*b/(d*x+c)*a^2+1/2*d^3/i^3/(a*d-b*c)^4/g*B
^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)^2*a^3-7/2*d/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b
^2*a-1/2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)^2*b^3*c^3+1/i^3/(a*d-b*c)^4/g*B^2*ln(b/
d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b^3/(d*x+c)*c^2+1/2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+
c)^2*b^3*c^3+3/2*d/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b^2*a-1/2*d^3/i^3/(a*d-b*c)^4/g*B^2
*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+c)^2*a^3-d/i^3/(a*d-b*c)^4/g*A^2*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a
+1/i^3/(a*d-b*c)^4/g*A*B*b^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*c-3/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d
*x+c)/d*e)*b^3*c-3/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^3/(d*x+c)*c^2+3*d^2/i^3/(a*d-b*c)^4
/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)^2*a^2*b*c-3*d/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/
d*e)/(d*x+c)^2*a*b^2*c^2-4*d/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2/(d*x+c)*a*c+3/2*d/i^3/(
a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)^2*a*b^2*c^2+1/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c
)/(d*x+c)/d*e)/(d*x+c)^2*b^3*c^3+d^2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b/(d*x+c)*a^2+6*d
/i^3/(a*d-b*c)^4/g*A*B*b^2/(d*x+c)*a*c-7/2*d/i^3/(a*d-b*c)^4/g*A*B*b^2*a+7/2/i^3/(a*d-b*c)^4/g*A*B*b^3*c-3*d^2
/i^3/(a*d-b*c)^4/g*A*B*b/(d*x+c)*a^2+3/2*d^2/i^3/(a*d-b*c)^4/g*A^2/(d*x+c)^2*a^2*b*c-3*d^2/i^3/(a*d-b*c)^4/g*B
^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b/(d*x+c)*a^2-2*d/i^3/(a*d-b*c)^4/g*A^2*b^2/(d*x+c)*a*c-7*d/i^3/(a*d-b*c)^4
/g*B^2*b^2/(d*x+c)*a*c-3/4*d/i^3/(a*d-b*c)^4/g*B^2/(d*x+c)^2*a*b^2*c^2+3/4*d^2/i^3/(a*d-b*c)^4/g*B^2/(d*x+c)^2
*a^2*b*c-3/2*d/i^3/(a*d-b*c)^4/g*A^2/(d*x+c)^2*b^2*c^2*a-d^3/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/
d*e)/(d*x+c)^2*a^3+3*d/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2*a+6*d/i^3/(a*d-b*c)^4/g*B^2*l
n(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2/(d*x+c)*a*c-d/i^3/(a*d-b*c)^4/g*A*B*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a
+3/2*d^2/i^3/(a*d-b*c)^4/g*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+c)^2*a^2*b*c-3/2*d/i^3/(a*d-b*c)^4/g*B^2
*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+c)^2*a*b^2*c^2+2*d^2/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/
d*e)*b/(d*x+c)*a^2+2/i^3/(a*d-b*c)^4/g*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^3/(d*x+c)*c^2

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maxima [B]  time = 3.06, size = 2116, normalized size = 5.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^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(b*e*x/(d*x + c) + a*e/(d*x + c))^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(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/12*B
^2*(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))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(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) - (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))/(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*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)) - 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/(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)

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mupad [B]  time = 8.10, size = 984, normalized size = 2.62 \[ -{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {\frac {B^2\,b^2\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B^2\,x\,\left (a\,d-b\,c\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}+\frac {B\,b^2\,\left (2\,A-3\,B\right )}{2\,g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\frac {2\,A^2\,a\,d-6\,A^2\,b\,c+B^2\,a\,d-15\,B^2\,b\,c-2\,A\,B\,a\,d+14\,A\,B\,b\,c}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (2\,b\,d\,A^2-6\,b\,d\,A\,B+7\,b\,d\,B^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}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B^2}{b\,d\,g\,i^3\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,\left (\frac {a^2\,d^2-3\,a\,b\,c\,d+2\,b^2\,c^2}{2\,b^3\,d}-\frac {c\,\left (a\,d-b\,c\right )}{2\,b^2\,d}\right )\,\left (2\,A-3\,B\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B\,x\,\left (2\,A-3\,B\right )\,\left (a\,d-b\,c\right )}{g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {d\,x^2}{b}+\frac {c^2}{b\,d}+\frac {2\,c\,x}{b}}-\frac {B^2\,b^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^3}{3\,g\,i^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (A^2-3\,A\,B+\frac {7\,B^2}{2}\right )\,\left (2\,g\,a^3\,d^3\,i^3-2\,g\,a^2\,b\,c\,d^2\,i^3-2\,g\,a\,b^2\,c^2\,d\,i^3+2\,g\,b^3\,c^3\,i^3\right )\,1{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2+7\,B^2\,b^2\right )}+\frac {b^3\,d\,x\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,\left (A^2-3\,A\,B+\frac {7\,B^2}{2}\right )\,4{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2+7\,B^2\,b^2\right )}\right )\,\left (A^2-3\,A\,B+\frac {7\,B^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)))^2/((a*g + b*g*x)*(c*i + d*i*x)^3),x)

[Out]

(b^2*atan((b^2*(A^2 + (7*B^2)/2 - 3*A*B)*(2*a^3*d^3*g*i^3 + 2*b^3*c^3*g*i^3 - 2*a*b^2*c^2*d*g*i^3 - 2*a^2*b*c*
d^2*g*i^3)*1i)/(g*i^3*(a*d - b*c)^3*(2*A^2*b^2 + 7*B^2*b^2 - 6*A*B*b^2)) + (b^3*d*x*(a^2*d^2*g*i^3 + b^2*c^2*g
*i^3 - 2*a*b*c*d*g*i^3)*(A^2 + (7*B^2)/2 - 3*A*B)*4i)/(g*i^3*(a*d - b*c)^3*(2*A^2*b^2 + 7*B^2*b^2 - 6*A*B*b^2)
))*(A^2 + (7*B^2)/2 - 3*A*B)*2i)/(g*i^3*(a*d - b*c)^3) - ((2*A^2*a*d - 6*A^2*b*c + B^2*a*d - 15*B^2*b*c - 2*A*
B*a*d + 14*A*B*b*c)/(2*(a*d - b*c)) - (x*(2*A^2*b*d + 7*B^2*b*d - 6*A*B*b*d))/(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))*(B^2/(b*d*g*i^3*(a*d - b*c)) + (B*b^2*((a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d)/(2*b^3*d) - (c*(a*d
- b*c))/(2*b^2*d))*(2*A - 3*B))/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B*x*(2*A - 3*B)
*(a*d - b*c))/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((d*x^2)/b + c^2/(b*d) + (2*c*x)/b
) - log((e*(a + b*x))/(c + d*x))^2*(((B^2*b^2*((a^2*d^2 + 2*b^2*c^2 - 3*a*b*c*d)/(2*b^3*d) - (c*(a*d - b*c))/(
2*b^2*d)))/(g*i^3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*x*(a*d - b*c))/(g*i^3*(a^3*d^3 -
b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/((d*x^2)/b + c^2/(b*d) + (2*c*x)/b) + (B*b^2*(2*A - 3*B))/(2*g*i^3
*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (B^2*b^2*log((e*(a + b*x))/(c + d*x))^3)/(3*g*i^3*(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 [B]  time = 14.45, size = 1488, normalized size = 3.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B**2*b**2*log(e*(a + b*x)/(c + d*x))**3/(3*a**3*d**3*g*i**3 - 9*a**2*b*c*d**2*g*i**3 + 9*a*b**2*c**2*d*g*i**3
- 3*b**3*c**3*g*i**3) + b**2*(2*A**2 - 6*A*B + 7*B**2)*log(x + (2*A**2*a*b**2*d + 2*A**2*b**3*c - 6*A*B*a*b**
2*d - 6*A*B*b**3*c + 7*B**2*a*b**2*d + 7*B**2*b**3*c - a**4*b**2*d**4*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3
+ 4*a**3*b**3*c*d**3*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 - 6*a**2*b**4*c**2*d**2*(2*A**2 - 6*A*B + 7*B**
2)/(a*d - b*c)**3 + 4*a*b**5*c**3*d*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 - b**6*c**4*(2*A**2 - 6*A*B + 7*B
**2)/(a*d - b*c)**3)/(4*A**2*b**3*d - 12*A*B*b**3*d + 14*B**2*b**3*d))/(2*g*i**3*(a*d - b*c)**3) - b**2*(2*A**
2 - 6*A*B + 7*B**2)*log(x + (2*A**2*a*b**2*d + 2*A**2*b**3*c - 6*A*B*a*b**2*d - 6*A*B*b**3*c + 7*B**2*a*b**2*d
+ 7*B**2*b**3*c + a**4*b**2*d**4*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a**3*b**3*c*d**3*(2*A**2 - 6*A*
B + 7*B**2)/(a*d - b*c)**3 + 6*a**2*b**4*c**2*d**2*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 - 4*a*b**5*c**3*d*
(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3 + b**6*c**4*(2*A**2 - 6*A*B + 7*B**2)/(a*d - b*c)**3)/(4*A**2*b**3*d
- 12*A*B*b**3*d + 14*B**2*b**3*d))/(2*g*i**3*(a*d - b*c)**3) + (-2*A*B*a*d + 6*A*B*b*c + 4*A*B*b*d*x + B**2*a*
d - 7*B**2*b*c - 6*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**2*c**2*d**2*g*i**3 + 4*a**2*c*d**3*g*i**3*x +
2*a**2*d**4*g*i**3*x**2 - 4*a*b*c**3*d*g*i**3 - 8*a*b*c**2*d**2*g*i**3*x - 4*a*b*c*d**3*g*i**3*x**2 + 2*b**2*c
**4*g*i**3 + 4*b**2*c**3*d*g*i**3*x + 2*b**2*c**2*d**2*g*i**3*x**2) + (-2*A*B*b**2*c**2 - 4*A*B*b**2*c*d*x - 2
*A*B*b**2*d**2*x**2 - B**2*a**2*d**2 + 4*B**2*a*b*c*d + 2*B**2*a*b*d**2*x + 4*B**2*b**2*c*d*x + 3*B**2*b**2*d*
*2*x**2)*log(e*(a + b*x)/(c + d*x))**2/(2*a**3*c**2*d**3*g*i**3 + 4*a**3*c*d**4*g*i**3*x + 2*a**3*d**5*g*i**3*
x**2 - 6*a**2*b*c**3*d**2*g*i**3 - 12*a**2*b*c**2*d**3*g*i**3*x - 6*a**2*b*c*d**4*g*i**3*x**2 + 6*a*b**2*c**4*
d*g*i**3 + 12*a*b**2*c**3*d**2*g*i**3*x + 6*a*b**2*c**2*d**3*g*i**3*x**2 - 2*b**3*c**5*g*i**3 - 4*b**3*c**4*d*
g*i**3*x - 2*b**3*c**3*d**2*g*i**3*x**2) + (-2*A**2*a*d + 6*A**2*b*c + 2*A*B*a*d - 14*A*B*b*c - B**2*a*d + 15*
B**2*b*c + x*(4*A**2*b*d - 12*A*B*b*d + 14*B**2*b*d))/(4*a**2*c**2*d**2*g*i**3 - 8*a*b*c**3*d*g*i**3 + 4*b**2*
c**4*g*i**3 + x**2*(4*a**2*d**4*g*i**3 - 8*a*b*c*d**3*g*i**3 + 4*b**2*c**2*d**2*g*i**3) + x*(8*a**2*c*d**3*g*i
**3 - 16*a*b*c**2*d**2*g*i**3 + 8*b**2*c**3*d*g*i**3))

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