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

Optimal. Leaf size=183 \[ -\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^2 i (b c-a d)^2}-\frac{b (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i (a+b x) (b c-a d)^2}-\frac{2 b B (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac{2 b B^2 (c+d x)}{g^2 i (a+b x) (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 6.33136, antiderivative size = 1684, normalized size of antiderivative = 9.2, number of steps used = 87, 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)^2*(c*i + d*i*x)),x]

[Out]

(-2*B^2)/((b*c - a*d)*g^2*i*(a + b*x)) - (2*B^2*d*Log[a + b*x])/((b*c - a*d)^2*g^2*i) + (A*B*d*Log[a + b*x]^2)
/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[a + b*x]^2)/((b*c - a*d)^2*g^2*i) - (B^2*d*Log[a + b*x]*Log[(c + d*x)^(-1)
]^2)/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/((b*c - a*d)^2*g^
2*i) + (B^2*d*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^2*g^2*i) + (B^2*d
*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^2*g^2*i) - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]
))/((b*c - a*d)*g^2*i*(a + b*x)) - (2*B*d*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^2*g^
2*i) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((b*c - a*d)*g^2*i*(a + b*x)) - (d*Log[a + b*x]*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])^2)/((b*c - a*d)^2*g^2*i) + (2*B^2*d*Log[c + d*x])/((b*c - a*d)^2*g^2*i) - (B^2*d*Log[a +
b*x]^2*Log[c + d*x])/((b*c - a*d)^2*g^2*i) - (2*A*B*d*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c -
a*d)^2*g^2*i) - (2*B^2*d*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*Log[
a + b*x]*Log[(c + d*x)^(-1)]*Log[c + d*x])/((b*c - a*d)^2*g^2*i) + (2*B^2*d*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)^2*g^2*i) + (2*B
*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^2*g^2*i) + (d*(A + B*Log[(e*(a + b*x))/(c +
d*x)])^2*Log[c + d*x])/((b*c - a*d)^2*g^2*i) + (A*B*d*Log[c + d*x]^2)/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[c +
d*x]^2)/((b*c - a*d)^2*g^2*i) - (B^2*d*Log[a + b*x]*Log[c + d*x]^2)/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[(e*(a +
b*x))/(c + d*x)]*Log[c + d*x]^2)/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[c + d*x]^3)/(3*(b*c - a*d)^2*g^2*i) - (2*
A*B*d*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*Log[a + b*x]*Log[(b*(c + d
*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) + (B^2*d*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^
2*g^2*i) - (2*A*B*d*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*PolyLog[2, -((d
*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^2*g^2*i) + (2*B^2*d*Log[a + b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d
))])/((b*c - a*d)^2*g^2*i) - (2*A*B*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*
PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c +
d*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) + (2*B^2*d*(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)^2*g^2*i) - (2*B^2*d*Log[(e*(a + b*x))/(c + d*x
)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*PolyLog[3, -((d*(a + b*x))/(b*c
- a*d))])/((b*c - a*d)^2*g^2*i) - (2*B^2*d*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^2*g^2*i) - (2*
B^2*d*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^2*g^2*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}{(89 c+89 d x) (a g+b g x)^2} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{(b d) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{d^2 \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{b \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{89 (b c-a d) g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{(2 B d) \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}{89 (b c-a d)^2 g^2}-\frac{(2 B d) \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}{89 (b c-a d)^2 g^2}+\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)^2 (c+d x)} \, dx}{89 (b c-a d) g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{(2 B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{89 g^2}+\frac{(2 B d) \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}{89 (b c-a d)^2 e g^2}-\frac{(2 B d) \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}{89 (b c-a d)^2 e g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\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)^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}{89 g^2}+\frac{(2 B d) \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}{89 (b c-a d)^2 e g^2}-\frac{(2 B d) \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}{89 (b c-a d)^2 e g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{(2 b B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{(2 b B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{89 (b c-a d) g^2}+\frac{(2 B d) \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}{89 (b c-a d) g^2}-\frac{(2 B d) \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}{89 (b c-a d) g^2}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{89 (b c-a d) g^2}+\frac{(2 B d) \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}{89 (b c-a d) g^2}-\frac{(2 B d) \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}{89 (b c-a d) g^2}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{89 g^2}-\frac{(2 b B d) \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}{89 (b c-a d)^2 g^2}+\frac{\left (2 B d^2\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}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{89 (b c-a d) g^2}+\frac{\left (2 B^2 d\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}{89 (b c-a d) g^2}+\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 e g^2}-\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 e g^2}\\ &=\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^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}{89 g^2}-\frac{(2 b B d) \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}{89 (b c-a d)^2 g^2}-\frac{\left (b B^2 d\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B d^2\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}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \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 )}{89 b (b c-a d) g^2}+\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 e g^2}-\frac{\left (2 B^2 d\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}{89 (b c-a d)^2 e g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}-\frac{(2 A b B d) \int \frac{\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 b B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 b B^2 d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 b B^2 d\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 A B d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 A B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{89 b (b c-a d)^2 g^2}-\frac{\left (2 B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2\right ) \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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}{89 (b c-a d) g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac{\left (b B^2 d\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 b B^2 d\right ) \int \frac{\log (a+b x) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}-\frac{\left (2 b B^2 d\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 b B^2 d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 A B d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (B^2 d^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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}{89 (b c-a d) g^2}-\frac{\left (2 b B^2 d \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac{(2 A B d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2 \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac{\left (B^2 d\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}+\frac{\left (B^2 d^2\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 )}{89 b (b c-a d)^2 g^2}+\frac{\left (2 B^2 d^2\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 )}{89 b (b c-a d)^2 g^2}-\frac{\left (2 B^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-b c+a d}{b}+\frac{d x}{b}\right )}{-\frac{-b c+a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{89 b (b c-a d)^2 g^2}+\frac{\left (2 B^2 d \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac{\left (b B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (\frac{1}{x}\right )}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}+\frac{\left (b B^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{-b c+a d}{d}+\frac{b x}{d}} \, dx,x,c+d x\right )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}+\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}-2 \frac{\left (2 B^2 d\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 )}{89 (b c-a d)^2 g^2}\\ &=-\frac{2 B^2}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d) g^2 (a+b x)}-\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{89 (b c-a d)^2 g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d) g^2 (a+b x)}-\frac{d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log ^2(a+b x) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 A B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \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)}{89 (b c-a d)^2 g^2}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{89 (b c-a d)^2 g^2}+\frac{A B d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(c+d x)}{89 (b c-a d)^2 g^2}-\frac{B^2 d \log (a+b x) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^3(c+d x)}{267 (b c-a d)^2 g^2}-\frac{2 A B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 A B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \log \left (\frac{1}{c+d x}\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}+\frac{2 B^2 d \left (\log (a+b x)+\log \left (\frac{1}{c+d x}\right )-\log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \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 )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{89 (b c-a d)^2 g^2}-\frac{2 B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{89 (b c-a d)^2 g^2}\\ \end{align*}

Mathematica [A]  time = 0.663086, size = 186, normalized size = 1.02 \[ -\frac{3 \left (A^2+2 A B+2 B^2\right ) (-d (a+b x) \log (c+d x)-a d+b c)+3 d \left (A^2+2 A B+2 B^2\right ) (a+b x) \log (a+b x)+3 B (a A d+A b d x+b B (c+d x)) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )+6 B (A+B) (b c-a d) \log \left (\frac{e (a+b x)}{c+d x}\right )+B^2 d (a+b x) \log ^3\left (\frac{e (a+b x)}{c+d x}\right )}{3 g^2 i (a+b x) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.054, size = 1201, normalized size = 6.6 \begin{align*} \text{result 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)^2/(d*i*x+c*i),x)

[Out]

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

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Maxima [B]  time = 1.53342, size = 1361, normalized size = 7.44 \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)^2/(d*i*x+c*i),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.521856, size = 512, normalized size = 2.8 \begin{align*} -\frac{{\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{3} + 3 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b c - 3 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a d + 3 \,{\left (B^{2} b c + A B a d +{\left (A B + B^{2}\right )} b d x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 3 \,{\left (A^{2} a d +{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b d x + 2 \,{\left (A B + B^{2}\right )} b c\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{3 \,{\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} i x +{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2} i\right )}} \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)^2/(d*i*x+c*i),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.5508, size = 541, normalized size = 2.96 \begin{align*} - \frac{B^{2} d \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{3}}{3 a^{2} d^{2} g^{2} i - 6 a b c d g^{2} i + 3 b^{2} c^{2} g^{2} i} + \frac{\left (2 A B + 2 B^{2}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A^{2} + 2 A B + 2 B^{2}\right ) \left (\frac{d \log{\left (x + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac{d \log{\left (x + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac{1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) + \frac{\left (- A B a d - A B b d x - B^{2} b c - B^{2} b d x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{3} d^{2} g^{2} i - 2 a^{2} b c d g^{2} i + a^{2} b d^{2} g^{2} i x + a b^{2} c^{2} g^{2} i - 2 a b^{2} c d g^{2} i x + b^{3} c^{2} g^{2} i x} \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)**2/(d*i*x+c*i),x)

[Out]

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

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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 )}^{2}{\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)^2/(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)^2*(d*i*x + c*i)), x)