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

Optimal. Leaf size=507 \[ -\frac{b^3 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{2 b^3 B (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac{3 b^2 d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^4 i (a+b x)^2 (b c-a d)^4}+\frac{3 b^2 B d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{3 b d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 i (a+b x) (b c-a d)^4}-\frac{6 b B d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac{d^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^4 i (b c-a d)^4}-\frac{2 b^3 B^2 (c+d x)^3}{27 g^4 i (a+b x)^3 (b c-a d)^4}+\frac{3 b^2 B^2 d (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{6 b B^2 d^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \]

[Out]

(-6*b*B^2*d^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B^2*d*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a
+ b*x)^2) - (2*b^3*B^2*(c + d*x)^3)/(27*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (6*b*B*d^2*(c + d*x)*(A + B*Log[(e
*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (2*b^3*B*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*
(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g
^4*i*(a + b*x)) + (3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^4*i*(a + b*x
)^2) - (b^3*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])^3)/(3*B*(b*c - a*d)^4*g^4*i)

________________________________________________________________________________________

Rubi [C]  time = 8.43056, antiderivative size = 2044, normalized size of antiderivative = 4.03, number of steps used = 151, number of rules used = 31, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.738, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2)/(27*(b*c - a*d)*g^4*i*(a + b*x)^3) + (19*B^2*d)/(36*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (85*B^2*d^2)/(
18*(b*c - a*d)^3*g^4*i*(a + b*x)) - (85*B^2*d^3*Log[a + b*x])/(18*(b*c - a*d)^4*g^4*i) + (A*B*d^3*Log[a + b*x]
^2)/((b*c - a*d)^4*g^4*i) + (11*B^2*d^3*Log[a + b*x]^2)/(6*(b*c - a*d)^4*g^4*i) - (B^2*d^3*Log[a + b*x]*Log[(c
+ d*x)^(-1)]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/((b
*c - a*d)^4*g^4*i) + (B^2*d^3*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^4
*g^4*i) + (B^2*d^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^4*g^4*i) - (2*B*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(9*(b*c - a*d)*g^4*i*(a + b*x)^3) + (5*B*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*(b*c
- a*d)^2*g^4*i*(a + b*x)^2) - (11*B*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^3*g^4*i*(a + b*x)
) - (11*B*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^4*g^4*i) - (A + B*Log[(e*(a +
b*x))/(c + d*x)])^2/(3*(b*c - a*d)*g^4*i*(a + b*x)^3) + (d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a
*d)^2*g^4*i*(a + b*x)^2) - (d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*i*(a + b*x)) - (d^3
*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^4*i) + (85*B^2*d^3*Log[c + d*x])/(18*(b
*c - a*d)^4*g^4*i) - (B^2*d^3*Log[a + b*x]^2*Log[c + d*x])/((b*c - a*d)^4*g^4*i) - (2*A*B*d^3*Log[-((d*(a + b*
x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c +
d*x])/(3*(b*c - a*d)^4*g^4*i) - (2*B^2*d^3*Log[a + b*x]*Log[(c + d*x)^(-1)]*Log[c + d*x])/((b*c - a*d)^4*g^4*i
) + (2*B^2*d^3*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)^4*g^4*i) + (11*B*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(3*(
b*c - a*d)^4*g^4*i) + (d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[c + d*x])/((b*c - a*d)^4*g^4*i) + (A*B*d
^3*Log[c + d*x]^2)/((b*c - a*d)^4*g^4*i) + (11*B^2*d^3*Log[c + d*x]^2)/(6*(b*c - a*d)^4*g^4*i) - (B^2*d^3*Log[
a + b*x]*Log[c + d*x]^2)/((b*c - a*d)^4*g^4*i) + (B^2*d^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x]^2)/((b*c -
a*d)^4*g^4*i) + (B^2*d^3*Log[c + d*x]^3)/(3*(b*c - a*d)^4*g^4*i) - (2*A*B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/
(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*(b*c - a*d)^
4*g^4*i) + (B^2*d^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (2*A*B*d^3*PolyLog[
2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))]
)/(3*(b*c - a*d)^4*g^4*i) + (2*B^2*d^3*Log[a + b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g
^4*i) - (2*A*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (11*B^2*d^3*PolyLog[2, (b*(c
+ d*x))/(b*c - a*d)])/(3*(b*c - a*d)^4*g^4*i) - (2*B^2*d^3*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])/((b*c - a*d)^4*g^4*i) + (2*B^2*d^3*(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)^4*g^4*i) - (2*B^2*d^3*Log[(e*(a + b*x))/(c + d*x)]*PolyL
og[2, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^4*g^4*i) - (2*B^2*d^3*PolyLog[3, -((d*(a + b*x))/(b*c - a*d
))])/((b*c - a*d)^4*g^4*i) - (2*B^2*d^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (2*B^2*
d^3*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^4*g^4*i)

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 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 12

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

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 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 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 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 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 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 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 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 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 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 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2375

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]

Rule 2434

Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.)
))/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, In
t[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/
(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]

Rule 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 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 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

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

Rubi steps

\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(91 c+91 d x) (a g+b g x)^4} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d) g^4 (a+b x)^4}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b d^3\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{d^4 \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (b d^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}-\frac{(b d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^3} \, dx}{91 (b c-a d)^2 g^4}+\frac{b \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{91 (b c-a d) g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^3\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}{91 (b c-a d)^4 g^4}-\frac{\left (2 B d^3\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^2\right ) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}-\frac{(B d) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}+\frac{(2 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)^4 (c+d x)} \, dx}{273 (b c-a d) g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{(2 B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{273 g^4}+\frac{\left (2 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}-\frac{(B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{91 (b c-a d) g^4}+\frac{\left (2 B d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (2 B d^3\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}{91 (b c-a d)^4 e g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{(2 B) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^4}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{273 g^4}+\frac{\left (2 B d^2\right ) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b 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 (a+b x)}+\frac{d^2 \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}{91 (b c-a d)^2 g^4}-\frac{(B d) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b 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 (a+b x)^2}+\frac{b d^2 \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^3 \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}{91 (b c-a d) g^4}+\frac{\left (2 B d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (2 B d^3\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}{91 (b c-a d)^4 e g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac{\left (b B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac{\left (B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 b B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{273 (b c-a d)^3 g^4}+\frac{\left (b B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}+\frac{\left (2 b B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{91 (b c-a d)^3 g^4}+\frac{\left (2 B d^3\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}{91 (b c-a d)^3 g^4}-\frac{\left (2 B d^3\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}{91 (b c-a d)^3 g^4}-\frac{(2 b B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{273 (b c-a d)^2 g^4}-\frac{(b B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{91 (b c-a d)^2 g^4}+\frac{(2 b B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{273 (b c-a d) g^4}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 g^4}-\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{273 (b c-a d)^3 g^4}+\frac{\left (B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac{\left (2 B d^3\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}{91 (b c-a d)^3 g^4}-\frac{\left (2 B d^3\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}{91 (b c-a d)^3 g^4}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{273 (b c-a d)^2 g^4}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{182 (b c-a d)^2 g^4}+\frac{\left (2 B^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{819 (b c-a d) g^4}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{819 g^4}-\frac{\left (2 b B d^3\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^4\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{91 (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{273 (b c-a d)^2 g^4}+\frac{\left (B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{91 (b c-a d)^2 g^4}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{273 (b c-a d) g^4}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{182 (b c-a d) g^4}+\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}-\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}+\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}+\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}\\ &=\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{819 g^4}-\frac{\left (2 b B d^3\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}{91 (b c-a d)^4 g^4}-\frac{\left (b B^2 d^3\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B d^4\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}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{91 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{273 (b c-a d)^2 g^4}+\frac{\left (B^2 d^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{91 (b c-a d)^2 g^4}+\frac{\left (2 B^2 d^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{91 (b c-a d)^2 g^4}-\frac{\left (B^2 d\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{273 (b c-a d) g^4}-\frac{\left (B^2 d\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{182 (b c-a d) g^4}+\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}-\frac{\left (2 B^2 d^3\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}{273 (b c-a d)^4 e g^4}+\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (B^2 d^3\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}{91 (b c-a d)^4 e g^4}+\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}-\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^4 e g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}-\frac{\left (2 A b B d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 b B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac{\left (2 b B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{273 (b c-a d)^4 g^4}+\frac{\left (b B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (b B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 b B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B^2 d^3\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 A B d^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{91 b (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{273 (b c-a d)^4 g^4}-\frac{\left (B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{273 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{273 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 b B^2 d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{273 (b c-a d)^4 g^4}-\frac{\left (b B^2 d^3\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (b B^2 d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B^2 d^3\right ) \int \frac{\log (a+b x) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}-\frac{\left (2 b B^2 d^3\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 b B^2 d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{273 (b c-a d)^4 g^4}+\frac{\left (B^2 d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^4\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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}{91 (b c-a d)^3 g^4}-\frac{\left (2 b B^2 d^3 \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}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 A B d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{273 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{273 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4 \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}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac{\left (B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^4\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 )}{91 b (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac{\left (b B^2 d^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 )}{91 (b c-a d)^4 g^4}+\frac{\left (b B^2 d^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 )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}+\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}-\frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}-2 \frac{\left (2 B^2 d^3\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 )}{91 (b c-a d)^4 g^4}\\ &=-\frac{2 B^2}{2457 (b c-a d) g^4 (a+b x)^3}+\frac{19 B^2 d}{3276 (b c-a d)^2 g^4 (a+b x)^2}-\frac{85 B^2 d^2}{1638 (b c-a d)^3 g^4 (a+b x)}-\frac{85 B^2 d^3 \log (a+b x)}{1638 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(a+b x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(a+b x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{819 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{546 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{273 (b c-a d)^4 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{273 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{182 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{91 (b c-a d)^4 g^4}+\frac{85 B^2 d^3 \log (c+d x)}{1638 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log ^2(a+b x) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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)}{91 (b c-a d)^4 g^4}+\frac{11 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{273 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{91 (b c-a d)^4 g^4}+\frac{A B d^3 \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{11 B^2 d^3 \log ^2(c+d x)}{546 (b c-a d)^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{91 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^3(c+d x)}{273 (b c-a d)^4 g^4}-\frac{2 A B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{B^2 d^3 \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 A B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{11 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{273 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}+\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \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 )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{91 (b c-a d)^4 g^4}-\frac{2 B^2 d^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{91 (b c-a d)^4 g^4}\\ \end{align*}

Mathematica [A]  time = 1.46497, size = 442, normalized size = 0.87 \[ -\frac{18 B \left (18 a^2 b d^2 (A d x+B (c+d x))+6 a^3 A d^3+9 a b^2 d \left (2 A d^2 x^2+B \left (-c^2+2 c d x+3 d^2 x^2\right )\right )+b^3 \left (6 A d^3 x^3+B \left (-3 c^2 d x+2 c^3+6 c d^2 x^2+11 d^3 x^3\right )\right )\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )-6 d^2 \left (18 A^2+66 A B+85 B^2\right ) (a+b x)^2 (a d-b c)-6 d^3 \left (18 A^2+66 A B+85 B^2\right ) (a+b x)^3 \log (c+d x)-3 d \left (18 A^2+30 A B+19 B^2\right ) (a+b x) (b c-a d)^2+4 \left (9 A^2+6 A B+2 B^2\right ) (b c-a d)^3+6 d^3 \left (18 A^2+66 A B+85 B^2\right ) (a+b x)^3 \log (a+b x)+6 B (b c-a d) \left (3 d (6 A+5 B) (a+b x) (a d-b c)+4 (3 A+B) (b c-a d)^2+6 d^2 (6 A+11 B) (a+b x)^2\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )+36 B^2 d^3 (a+b x)^3 \log ^3\left (\frac{e (a+b x)}{c+d x}\right )}{108 g^4 i (a+b x)^3 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.059, size = 3093, normalized size = 6.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i),x)

[Out]

1/3*e^3/i/(a*d-b*c)^5/g^4*A^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-6*e*d^3/i/(a*d-b*c)^5/g^4*A*B*b/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+6*e*d^2/i/(a*d-b*c)^5/g^4*A*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c-2/9*
e^3*d/i/(a*d-b*c)^5/g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3/2*e^
2*d^2/i/(a*d-b*c)^5/g^4*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-3/2*
e^2*d/i/(a*d-b*c)^5/g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c+3/2*
e^2*d^2/i/(a*d-b*c)^5/g^4*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-3/2*
e^2*d/i/(a*d-b*c)^5/g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-1/3*e^
3*d/i/(a*d-b*c)^5/g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-3*e*d^
3/i/(a*d-b*c)^5/g^4*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a+3*e*d^2/i/(a
*d-b*c)^5/g^4*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c-6*e*d^3/i/(a*d-b
*c)^5/g^4*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+6*e*d^2/i/(a*d-b*c)^5/g^
4*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+2/3*e^3/i/(a*d-b*c)^5/g^4*A*B*
b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+3/2*e^2*d^2/i/(a*d-b*c)^5/g^4*A*B*
b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-3/2*e^2*d/i/(a*d-b*c)^5/g^4*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)^2*c-2/9*e^3*d/i/(a*d-b*c)^5/g^4*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a-3/4*e^2*d/i/(a*d-b*c)^5/
g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c-2/27*e^3*d/i/(a*d-b*c)^5/g^4*B^2*b^3/(b*e/d+e/(d*x+c)*a-e/
d/(d*x+c)*b*c)^3*a+3/2*e^2*d^2/i/(a*d-b*c)^5/g^4*A^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-3/2*e^2*d/i/(
a*d-b*c)^5/g^4*A^2*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c-1/3*e^3*d/i/(a*d-b*c)^5/g^4*A^2*b^3/(b*e/d+e/(d
*x+c)*a-e/d/(d*x+c)*b*c)^3*a+2/9*e^3/i/(a*d-b*c)^5/g^4*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+
(a*d-b*c)*e/d/(d*x+c))*c+d^3/i/(a*d-b*c)^5/g^4*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b*c+1/3*e^3/i/(a*d-b*c)^5
/g^4*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c-1/3*d^4/i/(a*d-b*c)^5/g
^4*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*a-d^4/i/(a*d-b*c)^5/g^4*A^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-3*e*d^3
/i/(a*d-b*c)^5/g^4*A^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+3*e*d^2/i/(a*d-b*c)^5/g^4*A^2*b^2/(b*e/d+e/(d*x
+c)*a-e/d/(d*x+c)*b*c)*c-6*e*d^3/i/(a*d-b*c)^5/g^4*B^2*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+2/9*e^3/i/(a*d-
b*c)^5/g^4*A*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c+6*e*d^2/i/(a*d-b*c)^5/g^4*B^2*b^2/(b*e/d+e/(d*x+c)*
a-e/d/(d*x+c)*b*c)*c+3/4*e^2*d^2/i/(a*d-b*c)^5/g^4*B^2*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+2/27*e^3/i/
(a*d-b*c)^5/g^4*B^2*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-d^4/i/(a*d-b*c)^5/g^4*A*B*ln(b*e/d+(a*d-b*c)*e
/d/(d*x+c))^2*a+1/3*d^3/i/(a*d-b*c)^5/g^4*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*b*c-2/3*e^3*d/i/(a*d-b*c)^5/g^
4*A*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-3*e^2*d/i/(a*d-b*c)^5/g^4*A*
B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+3*e^2*d^2/i/(a*d-b*c)^5/g^4*A*B*
b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-6*e*d^3/i/(a*d-b*c)^5/g^4*A*B*b/(b
*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+6*e*d^2/i/(a*d-b*c)^5/g^4*A*B*b^2/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+d^3/i/(a*d-b*c)^5/g^4*A^2*ln(b*e/d+(a*d-b*c)*e/d
/(d*x+c))*b*c

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Maxima [B]  time = 2.77982, size = 4636, normalized size = 9.14 \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

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

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Fricas [A]  time = 0.602219, size = 1991, normalized size = 3.93 \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

-1/108*(4*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 81*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + 324*(A^2 + 2*A*B + 2*B^2)*a
^2*b*c*d^2 - (198*A^2 + 510*A*B + 575*B^2)*a^3*d^3 + 36*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d
^3*x + B^2*a^3*d^3)*log((b*e*x + a*e)/(d*x + c))^3 + 6*((18*A^2 + 66*A*B + 85*B^2)*b^3*c*d^2 - (18*A^2 + 66*A*
B + 85*B^2)*a*b^2*d^3)*x^2 + 18*((6*A*B + 11*B^2)*b^3*d^3*x^3 + 2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2
*b*c*d^2 + 6*A*B*a^3*d^3 + 3*(2*B^2*b^3*c*d^2 + 3*(2*A*B + 3*B^2)*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2*d - 6*B^2*a*
b^2*c*d^2 - 6*(A*B + B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((18*A^2 + 30*A*B + 19*B^2)*b^3*c^2
*d - 54*(2*A^2 + 6*A*B + 7*B^2)*a*b^2*c*d^2 + (90*A^2 + 294*A*B + 359*B^2)*a^2*b*d^3)*x + 6*((18*A^2 + 66*A*B
+ 85*B^2)*b^3*d^3*x^3 + 18*A^2*a^3*d^3 + 4*(3*A*B + B^2)*b^3*c^3 - 27*(2*A*B + B^2)*a*b^2*c^2*d + 108*(A*B + B
^2)*a^2*b*c*d^2 + 3*(2*(6*A*B + 11*B^2)*b^3*c*d^2 + 9*(2*A^2 + 6*A*B + 7*B^2)*a*b^2*d^3)*x^2 - 3*((6*A*B + 5*B
^2)*b^3*c^2*d - 18*(2*A*B + 3*B^2)*a*b^2*c*d^2 - 18*(A^2 + 2*A*B + 2*B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x
+ c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^4*i*x^3 + 3*(a*b^6*c^
4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^
4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^
2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4*i)

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Sympy [B]  time = 50.194, size = 2388, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

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

[Out]

-B**2*d**3*log(e*(a + b*x)/(c + d*x))**3/(3*a**4*d**4*g**4*i - 12*a**3*b*c*d**3*g**4*i + 18*a**2*b**2*c**2*d**
2*g**4*i - 12*a*b**3*c**3*d*g**4*i + 3*b**4*c**4*g**4*i) + d**3*(18*A**2 + 66*A*B + 85*B**2)*log(x + (18*A**2*
a*d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A*B*b*c*d**3 + 85*B**2*a*d**4 + 85*B**2*b*c*d**3 - a**5*d**8*(1
8*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 5*a**4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 10*a
**3*b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5*(18*A**2 + 66*A*B + 85
*B**2)/(a*d - b*c)**4 - 5*a*b**4*c**4*d**4*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + b**5*c**5*d**3*(18*A*
*2 + 66*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*b*d**4 + 132*A*B*b*d**4 + 170*B**2*b*d**4))/(18*g**4*i*(a*d -
b*c)**4) - d**3*(18*A**2 + 66*A*B + 85*B**2)*log(x + (18*A**2*a*d**4 + 18*A**2*b*c*d**3 + 66*A*B*a*d**4 + 66*A
*B*b*c*d**3 + 85*B**2*a*d**4 + 85*B**2*b*c*d**3 + a**5*d**8*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - 5*a*
*4*b*c*d**7*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 10*a**3*b**2*c**2*d**6*(18*A**2 + 66*A*B + 85*B**2)/
(a*d - b*c)**4 - 10*a**2*b**3*c**3*d**5*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 + 5*a*b**4*c**4*d**4*(18*A
**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4 - b**5*c**5*d**3*(18*A**2 + 66*A*B + 85*B**2)/(a*d - b*c)**4)/(36*A**2*
b*d**4 + 132*A*B*b*d**4 + 170*B**2*b*d**4))/(18*g**4*i*(a*d - b*c)**4) + (66*A*B*a**2*d**2 - 42*A*B*a*b*c*d +
90*A*B*a*b*d**2*x + 12*A*B*b**2*c**2 - 18*A*B*b**2*c*d*x + 36*A*B*b**2*d**2*x**2 + 85*B**2*a**2*d**2 - 23*B**2
*a*b*c*d + 147*B**2*a*b*d**2*x + 4*B**2*b**2*c**2 - 15*B**2*b**2*c*d*x + 66*B**2*b**2*d**2*x**2)*log(e*(a + b*
x)/(c + d*x))/(18*a**6*d**3*g**4*i - 54*a**5*b*c*d**2*g**4*i + 54*a**5*b*d**3*g**4*i*x + 54*a**4*b**2*c**2*d*g
**4*i - 162*a**4*b**2*c*d**2*g**4*i*x + 54*a**4*b**2*d**3*g**4*i*x**2 - 18*a**3*b**3*c**3*g**4*i + 162*a**3*b*
*3*c**2*d*g**4*i*x - 162*a**3*b**3*c*d**2*g**4*i*x**2 + 18*a**3*b**3*d**3*g**4*i*x**3 - 54*a**2*b**4*c**3*g**4
*i*x + 162*a**2*b**4*c**2*d*g**4*i*x**2 - 54*a**2*b**4*c*d**2*g**4*i*x**3 - 54*a*b**5*c**3*g**4*i*x**2 + 54*a*
b**5*c**2*d*g**4*i*x**3 - 18*b**6*c**3*g**4*i*x**3) + (-6*A*B*a**3*d**3 - 18*A*B*a**2*b*d**3*x - 18*A*B*a*b**2
*d**3*x**2 - 6*A*B*b**3*d**3*x**3 - 18*B**2*a**2*b*c*d**2 - 18*B**2*a**2*b*d**3*x + 9*B**2*a*b**2*c**2*d - 18*
B**2*a*b**2*c*d**2*x - 27*B**2*a*b**2*d**3*x**2 - 2*B**2*b**3*c**3 + 3*B**2*b**3*c**2*d*x - 6*B**2*b**3*c*d**2
*x**2 - 11*B**2*b**3*d**3*x**3)*log(e*(a + b*x)/(c + d*x))**2/(6*a**7*d**4*g**4*i - 24*a**6*b*c*d**3*g**4*i +
18*a**6*b*d**4*g**4*i*x + 36*a**5*b**2*c**2*d**2*g**4*i - 72*a**5*b**2*c*d**3*g**4*i*x + 18*a**5*b**2*d**4*g**
4*i*x**2 - 24*a**4*b**3*c**3*d*g**4*i + 108*a**4*b**3*c**2*d**2*g**4*i*x - 72*a**4*b**3*c*d**3*g**4*i*x**2 + 6
*a**4*b**3*d**4*g**4*i*x**3 + 6*a**3*b**4*c**4*g**4*i - 72*a**3*b**4*c**3*d*g**4*i*x + 108*a**3*b**4*c**2*d**2
*g**4*i*x**2 - 24*a**3*b**4*c*d**3*g**4*i*x**3 + 18*a**2*b**5*c**4*g**4*i*x - 72*a**2*b**5*c**3*d*g**4*i*x**2
+ 36*a**2*b**5*c**2*d**2*g**4*i*x**3 + 18*a*b**6*c**4*g**4*i*x**2 - 24*a*b**6*c**3*d*g**4*i*x**3 + 6*b**7*c**4
*g**4*i*x**3) + (198*A**2*a**2*d**2 - 126*A**2*a*b*c*d + 36*A**2*b**2*c**2 + 510*A*B*a**2*d**2 - 138*A*B*a*b*c
*d + 24*A*B*b**2*c**2 + 575*B**2*a**2*d**2 - 73*B**2*a*b*c*d + 8*B**2*b**2*c**2 + x**2*(108*A**2*b**2*d**2 + 3
96*A*B*b**2*d**2 + 510*B**2*b**2*d**2) + x*(270*A**2*a*b*d**2 - 54*A**2*b**2*c*d + 882*A*B*a*b*d**2 - 90*A*B*b
**2*c*d + 1077*B**2*a*b*d**2 - 57*B**2*b**2*c*d))/(108*a**6*d**3*g**4*i - 324*a**5*b*c*d**2*g**4*i + 324*a**4*
b**2*c**2*d*g**4*i - 108*a**3*b**3*c**3*g**4*i + x**3*(108*a**3*b**3*d**3*g**4*i - 324*a**2*b**4*c*d**2*g**4*i
+ 324*a*b**5*c**2*d*g**4*i - 108*b**6*c**3*g**4*i) + x**2*(324*a**4*b**2*d**3*g**4*i - 972*a**3*b**3*c*d**2*g
**4*i + 972*a**2*b**4*c**2*d*g**4*i - 324*a*b**5*c**3*g**4*i) + x*(324*a**5*b*d**3*g**4*i - 972*a**4*b**2*c*d*
*2*g**4*i + 972*a**3*b**3*c**2*d*g**4*i - 324*a**2*b**4*c**3*g**4*i))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{4}{\left (d i x + c i\right )}}\,{d x} \end{align*}

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)^4/(d*i*x+c*i),x, algorithm="giac")

[Out]

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