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

Optimal. Leaf size=365 \[ -\frac{b^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac{2 b^2 B (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^3}+\frac{d^2 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac{2 A B d^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac{2 b d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^2 i^2 (b c-a d)^3}-\frac{2 b^2 B^2 (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac{2 B^2 d^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{g^2 i^2 (c+d x) (b c-a d)^3}+\frac{2 B^2 d^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 7.17614, antiderivative size = 1521, normalized size of antiderivative = 4.17, number of steps used = 113, 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)^2),x]

[Out]

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

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}{(97 c+97 d x)^2 (a g+b g x)^2} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)^2}-\frac{2 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)^2}+\frac{2 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{\left (2 b^2 d\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{9409 (b c-a d)^3 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 )^2}{c+d x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{b^2 \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{9409 (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)^2} \, dx}{9409 (b c-a d)^2 g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{(4 b 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}{9409 (b c-a d)^3 g^2}-\frac{(4 b 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}{9409 (b c-a d)^3 g^2}+\frac{(2 b B) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{9409 (b c-a d)^2 g^2}+\frac{(2 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) (c+d x)^2} \, dx}{9409 (b c-a d)^2 g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 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 (c+d x)} \, dx}{9409 (b c-a d) g^2}+\frac{(2 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{9409 (b c-a d) g^2}+\frac{(4 b 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}{9409 (b c-a d)^3 e g^2}-\frac{(4 b 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}{9409 (b c-a d)^3 e g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{(2 b 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}{9409 (b c-a d) g^2}+\frac{(2 B d) \int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9409 (b c-a d) g^2}+\frac{(4 b 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}{9409 (b c-a d)^3 e g^2}-\frac{(4 b 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}{9409 (b c-a d)^3 e g^2}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b^2 B\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{9409 (b c-a d)^2 g^2}+\frac{(4 b 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}{9409 (b c-a d)^2 g^2}-\frac{(4 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) (c+d x)} \, dx}{9409 (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)^2} \, dx}{9409 (b c-a d)^2 g^2}\\ &=-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b B^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{9409 (b c-a d)^2 g^2}+\frac{(4 b 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}{9409 (b c-a d)^2 g^2}-\frac{(4 b 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}{9409 (b c-a d)^2 g^2}-\frac{\left (2 B^2 d\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{9409 (b c-a d)^2 g^2}\\ &=-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b^2 B d\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^3 g^2}+\frac{(4 A b B d) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{9409 (b c-a d)^2 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^2 g^2}+\frac{\left (2 b B^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{9409 (b c-a d) g^2}-\frac{\left (2 B^2 d\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{9409 (b c-a d) g^2}\\ &=\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b^2 B d\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}{9409 (b c-a d)^3 g^2}-\frac{\left (2 b^2 B^2 d\right ) \int \frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^3 g^2}+\frac{(4 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 )}{9409 (b c-a d)^2 g^2}+\frac{\left (2 b 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}{9409 (b c-a d) g^2}-\frac{\left (2 B^2 d\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9409 (b c-a d) g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{(4 A b B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 A b^2 B d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b^2 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}{9409 (b c-a d)^3 g^2}-\frac{\left (4 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 A b B d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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}{9409 (b c-a d)^2 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{(4 A b B d) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{9409 (b c-a d)^3 g^2}+\frac{(4 A b 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (2 b^2 B^2 d\right ) \int \frac{\log ^2(c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b^2 B^2 d\right ) \int \frac{\log (a+b x) \log (c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b^2 B^2 d\right ) \int \frac{\log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (4 A b B d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b B^2 d^2\right ) \int \frac{\log ^2(c+d x)}{c+d x} \, dx}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^2 g^2}-\frac{\left (4 b^2 B^2 d \left (-\log (a+b x)-\log \left (\frac{1}{c+d x}\right )+\log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}+\frac{(4 A b 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,c+d x\right )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{x} \, dx,x,a+b x\right )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b B^2 d\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (2 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 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 ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^3(c+d x)}{28227 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 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 ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^3(c+d x)}{28227 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 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 ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b^2 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 )}{9409 (b c-a d)^3 g^2}+\frac{\left (2 b^2 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^3(c+d x)}{28227 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 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 ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (c+d x) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}+\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^3(c+d x)}{28227 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{4 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 ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}-2 \frac{\left (4 b 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 )}{9409 (b c-a d)^3 g^2}\\ &=-\frac{2 b B^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{2 B^2 d}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{4 b B^2 d \log (a+b x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(a+b x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac{1}{c+d x}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log (a+b x) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{9409 (b c-a d)^3 g^2}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (a+b x)}+\frac{2 B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^2 g^2 (c+d x)}-\frac{2 b d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log ^2(a+b x) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \log (a+b x) \log \left (\frac{1}{c+d x}\right ) \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{4 b 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)}{9409 (b c-a d)^3 g^2}+\frac{2 b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 A b B d \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}-\frac{2 b B^2 d \log (a+b x) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log \left (\frac{e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^3(c+d x)}{28227 (b c-a d)^3 g^2}-\frac{4 A b B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{2 b B^2 d \log ^2(a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}+\frac{4 b B^2 d \log (a+b x) \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 A b B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}+\frac{4 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 ) \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b 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 )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (-\frac{d (a+b x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (\frac{b (c+d x)}{b c-a d}\right )}{9409 (b c-a d)^3 g^2}-\frac{4 b B^2 d \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{9409 (b c-a d)^3 g^2}\\ \end{align*}

Mathematica [A]  time = 1.04172, size = 307, normalized size = 0.84 \[ -\frac{3 B \left (-a^2 B d^2+2 a b d (A (c+d x)-B d x)+b^2 (2 A d x (c+d x)+B c (c+2 d x))\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )-3 d \left (A^2-2 A B+2 B^2\right ) (a+b x) (a d-b c)+3 b \left (A^2+2 A B+2 B^2\right ) (c+d x) (b c-a d)+6 b d \left (A^2+2 B^2\right ) (a+b x) (c+d x) \log (a+b x)-6 b d \left (A^2+2 B^2\right ) (a+b x) (c+d x) \log (c+d x)+6 B (b c-a d) (a A d-a B d+A b c+2 A b d x+b B c) \log \left (\frac{e (a+b x)}{c+d x}\right )+2 b B^2 d (a+b x) (c+d x) \log ^3\left (\frac{e (a+b x)}{c+d x}\right )}{3 g^2 i^2 (a+b x) (c+d x) (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.053, size = 2572, normalized size = 7.1 \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.92064, size = 2693, normalized size = 7.38 \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.558737, size = 1062, normalized size = 2.91 \begin{align*} \frac{12 \, A B a b c d - 3 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{2} c^{2} + 3 \,{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{2} d^{2} - 2 \,{\left (B^{2} b^{2} d^{2} x^{2} + B^{2} a b c d +{\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{3} - 3 \,{\left (2 \, A B b^{2} d^{2} x^{2} + B^{2} b^{2} c^{2} + 2 \, A B a b c d - B^{2} a^{2} d^{2} + 2 \,{\left ({\left (A B + B^{2}\right )} b^{2} c d +{\left (A B - B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 6 \,{\left ({\left (A^{2} + 2 \, B^{2}\right )} b^{2} c d -{\left (A^{2} + 2 \, B^{2}\right )} a b d^{2}\right )} x - 6 \,{\left ({\left (A^{2} + 2 \, B^{2}\right )} b^{2} d^{2} x^{2} + A^{2} a b c d +{\left (A B + B^{2}\right )} b^{2} c^{2} -{\left (A B - B^{2}\right )} a^{2} d^{2} +{\left ({\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{2} c d +{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{3 \,{\left ({\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x +{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 11.0714, size = 1404, normalized size = 3.85 \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)**2/(d*i*x+c*i)**2,x)

[Out]

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

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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 )}^{2}}\,{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)^2,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)^2), x)