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

Optimal. Leaf size=682 \[ -\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 i^2 (a+b x)^3 (b c-a d)^5}-\frac {2 b^4 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}+\frac {2 b^3 B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac {12 b^2 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^5}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac {2 A B d^4 (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac {4 b d^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{3 B g^4 i^2 (b c-a d)^5}-\frac {2 b^4 B^2 (c+d x)^3}{27 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {b^3 B^2 d (c+d x)^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {12 b^2 B^2 d^2 (c+d x)}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac {2 B^2 d^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g^4 i^2 (c+d x) (b c-a d)^5}+\frac {2 B^2 d^4 (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5} \]

[Out]

-2*A*B*d^4*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)+2*B^2*d^4*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-12*b^2*B^2*d^2*
(d*x+c)/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+b^3*B^2*d*(d*x+c)^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/27*b^4*B^2*(d*x+c)^3
/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-2*B^2*d^4*(b*x+a)*ln(e*(b*x+a)/(d*x+c))/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-12*b^2*B*
d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*B*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*
x+c)))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/9*b^4*B*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^5/g^4/i^2/(b*
x+a)^3+d^4*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-6*b^2*d^2*(d*x+c)*(A+B*ln(e*(b*x
+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^4/
i^2/(b*x+a)^2-1/3*b^4*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-4/3*b*d^3*(A+B*ln
(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c)^5/g^4/i^2

________________________________________________________________________________________

Rubi [C]  time = 9.48, antiderivative size = 2222, normalized size of antiderivative = 3.26, number of steps used = 177, number of rules used = 31, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.738, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*B^2)/(27*(b*c - a*d)^2*g^4*i^2*(a + b*x)^3) + (7*b*B^2*d)/(9*(b*c - a*d)^3*g^4*i^2*(a + b*x)^2) - (92*b*
B^2*d^2)/(9*(b*c - a*d)^4*g^4*i^2*(a + b*x)) - (2*B^2*d^3)/((b*c - a*d)^4*g^4*i^2*(c + d*x)) - (110*b*B^2*d^3*
Log[a + b*x])/(9*(b*c - a*d)^5*g^4*i^2) + (4*A*b*B*d^3*Log[a + b*x]^2)/((b*c - a*d)^5*g^4*i^2) + (10*b*B^2*d^3
*Log[a + b*x]^2)/(3*(b*c - a*d)^5*g^4*i^2) - (4*b*B^2*d^3*Log[a + b*x]*Log[(c + d*x)^(-1)]^2)/((b*c - a*d)^5*g
^4*i^2) + (4*b*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/((b*c - a*d)^5*g^4*i^2) + (4*b
*B^2*d^3*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^5*g^4*i^2) + (4*b*B^2*
d^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^5*g^4*i^2) - (2*b*B*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(9*(b*c - a*d)^2*g^4*i^2*(a + b*x)^3) + (4*b*B*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d
)^3*g^4*i^2*(a + b*x)^2) - (26*b*B*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^4*g^4*i^2*(a + b*x
)) + (2*B*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i^2*(c + d*x)) - (20*b*B*d^3*Log[a + b*
x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^5*g^4*i^2) - (b*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2
)/(3*(b*c - a*d)^2*g^4*i^2*(a + b*x)^3) + (b*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^3*g^4*i^2*
(a + b*x)^2) - (3*b*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^4*i^2*(a + b*x)) - (d^3*(A +
B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^4*g^4*i^2*(c + d*x)) - (4*b*d^3*Log[a + b*x]*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^4*i^2) + (110*b*B^2*d^3*Log[c + d*x])/(9*(b*c - a*d)^5*g^4*i^2) - (4*b
*B^2*d^3*Log[a + b*x]^2*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) - (8*A*b*B*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]
*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) - (20*b*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*(b*c
- a*d)^5*g^4*i^2) - (8*b*B^2*d^3*Log[a + b*x]*Log[(c + d*x)^(-1)]*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) + (8*
b*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)]
)*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) + (20*b*B*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(3*(b
*c - a*d)^5*g^4*i^2) + (4*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) +
(4*A*b*B*d^3*Log[c + d*x]^2)/((b*c - a*d)^5*g^4*i^2) + (10*b*B^2*d^3*Log[c + d*x]^2)/(3*(b*c - a*d)^5*g^4*i^2
) - (4*b*B^2*d^3*Log[a + b*x]*Log[c + d*x]^2)/((b*c - a*d)^5*g^4*i^2) + (4*b*B^2*d^3*Log[(e*(a + b*x))/(c + d*
x)]*Log[c + d*x]^2)/((b*c - a*d)^5*g^4*i^2) + (4*b*B^2*d^3*Log[c + d*x]^3)/(3*(b*c - a*d)^5*g^4*i^2) - (8*A*b*
B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^4*i^2) - (20*b*B^2*d^3*Log[a + b*x]*Log[(b
*(c + d*x))/(b*c - a*d)])/(3*(b*c - a*d)^5*g^4*i^2) + (4*b*B^2*d^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d
)])/((b*c - a*d)^5*g^4*i^2) - (8*A*b*B*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^5*g^4*i^2) -
(20*b*B^2*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*(b*c - a*d)^5*g^4*i^2) + (8*b*B^2*d^3*Log[a + b*x]
*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^5*g^4*i^2) - (8*A*b*B*d^3*PolyLog[2, (b*(c + d*x))/(b*
c - a*d)])/((b*c - a*d)^5*g^4*i^2) - (20*b*B^2*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*(b*c - a*d)^5*g^4
*i^2) - (8*b*B^2*d^3*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^4*i^2) + (8*b
*B^2*d^3*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c + d*x))/(b*c - a
*d)])/((b*c - a*d)^5*g^4*i^2) - (8*b*B^2*d^3*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b
*x))])/((b*c - a*d)^5*g^4*i^2) - (8*b*B^2*d^3*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^5*g^4*i^2
) - (8*b*B^2*d^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^4*i^2) - (8*b*B^2*d^3*PolyLog[3, 1 +
(b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^5*g^4*i^2)

Rule 12

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

Rule 30

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2301

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

Rule 2302

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

Rule 2317

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

Rule 2344

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

Rule 2374

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

Rule 2375

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

Rule 2390

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

Rule 2391

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

Rule 2393

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

Rule 2394

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

Rule 2396

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2418

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

Rule 2433

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

Rule 2434

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

Rule 2440

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

Rule 2488

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]

Rule 2499

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

Rule 2500

Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]

Rule 2506

Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

Rule 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]

Rule 2524

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

Rule 2525

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6589

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(99 c+99 d x)^2 (a g+b g x)^4} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^2 g^4 (a+b x)^4}-\frac {2 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^3}+\frac {b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)^2}-\frac {4 b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)^2}+\frac {4 b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (4 b^2 d^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (4 b d^4\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (b^2 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{3267 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{9801 (b c-a d)^4 g^4}-\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)^3} \, dx}{9801 (b c-a d)^3 g^4}+\frac {b^2 \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{9801 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{e (a+b x)} \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{e (a+b x)} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B d^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{3267 (b c-a d)^4 g^4}+\frac {\left (2 B d^3\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)^2} \, dx}{9801 (b c-a d)^4 g^4}-\frac {(2 b B d) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{9801 (b c-a d)^3 g^4}+\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)^4 (c+d x)} \, dx}{29403 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2 (c+d x)} \, dx}{3267 (b c-a d)^3 g^4}+\frac {\left (2 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{9801 (b c-a d)^3 g^4}-\frac {(2 b B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3 (c+d x)} \, dx}{9801 (b c-a d)^2 g^4}+\frac {(2 b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{29403 (b c-a d) g^4}+\frac {\left (8 b B d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x} \, dx}{9801 (b c-a d)^5 e g^4}-\frac {\left (8 b B d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 e g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B d^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3267 (b c-a d)^3 g^4}+\frac {\left (2 B d^3\right ) \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}{9801 (b c-a d)^3 g^4}-\frac {(2 b B d) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{9801 (b c-a d)^2 g^4}+\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)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{29403 (b c-a d) g^4}+\frac {\left (8 b B d^3\right ) \int \frac {(b c-a d) e \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^5 e g^4}-\frac {\left (8 b B d^3\right ) \int \frac {(b c-a d) e \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^5 e g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {\left (2 b^2 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{29403 (b c-a d)^5 g^4}-\frac {\left (2 b^2 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b^2 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{29403 (b c-a d)^4 g^4}+\frac {\left (2 b^2 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{9801 (b c-a d)^4 g^4}+\frac {\left (2 b^2 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3267 (b c-a d)^4 g^4}+\frac {\left (8 b B d^3\right ) \int \frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (8 b B d^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (2 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (2 b^2 B d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{29403 (b c-a d)^3 g^4}-\frac {\left (2 b^2 B d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{9801 (b c-a d)^3 g^4}+\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)^4} \, dx}{29403 (b c-a d)^2 g^4}\\ &=-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{29403 (b c-a d)^5 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{3267 (b c-a d)^5 g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{29403 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3267 (b c-a d)^4 g^4}+\frac {\left (8 b B d^3\right ) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)}\right ) \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (8 b B d^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (2 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (b B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{29403 (b c-a d)^3 g^4}-\frac {\left (b B^2 d\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{9801 (b c-a d)^3 g^4}+\frac {\left (2 b B^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{88209 (b c-a d)^2 g^4}\\ &=-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b^2 B d^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B d^4\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^3\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}+\frac {\left (8 b B^2 d^3\right ) \int \frac {\log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{29403 (b c-a d)^3 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{9801 (b c-a d)^3 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3267 (b c-a d)^3 g^4}-\frac {\left (2 B^2 d^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{9801 (b c-a d)^3 g^4}-\frac {\left (b B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{29403 (b c-a d)^2 g^4}-\frac {\left (b B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{9801 (b c-a d)^2 g^4}+\frac {\left (2 b B^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{88209 (b c-a d) g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{29403 (b c-a d)^5 e g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{29403 (b c-a d)^5 e g^4}+\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3267 (b c-a d)^5 e g^4}-\frac {\left (2 b B^2 d^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3267 (b c-a d)^5 e g^4}\\ &=\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b^2 B d^3\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x}\right ) \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (4 b^2 B^2 d^3\right ) \int \frac {\log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B d^4\right ) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x}\right ) \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{9801 (b c-a d)^4 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{29403 (b c-a d)^3 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9801 (b c-a d)^3 g^4}+\frac {\left (2 b B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3267 (b c-a d)^3 g^4}-\frac {\left (2 B^2 d^3\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}{9801 (b c-a d)^3 g^4}-\frac {\left (b B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{29403 (b c-a d)^2 g^4}-\frac {\left (b B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{9801 (b c-a d)^2 g^4}+\frac {\left (2 b B^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{88209 (b c-a d) g^4}+\frac {\left (2 b B^2 d^3\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{29403 (b c-a d)^5 e g^4}-\frac {\left (2 b B^2 d^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{29403 (b c-a d)^5 e g^4}+\frac {\left (2 b B^2 d^3\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{3267 (b c-a d)^5 e g^4}-\frac {\left (2 b B^2 d^3\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3267 (b c-a d)^5 e g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 A b^2 B d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{29403 (b c-a d)^5 g^4}-\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3267 (b c-a d)^5 g^4}-\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3267 (b c-a d)^5 g^4}-\frac {\left (8 b^2 B^2 d^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (8 A B d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (2 b B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{29403 (b c-a d)^5 g^4}-\frac {\left (2 b B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^4\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{29403 (b c-a d)^5 g^4}-\frac {\left (4 b^2 B^2 d^3\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b^2 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3267 (b c-a d)^5 g^4}-\frac {\left (8 b^2 B^2 d^3\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b^2 B^2 d^3\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{29403 (b c-a d)^5 g^4}+\frac {\left (4 b B^2 d^4\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3267 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{9801 (b c-a d)^4 g^4}-\frac {\left (8 b^2 B^2 d^3 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 A b B d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{29403 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{29403 (b c-a d)^5 g^4}+\frac {\left (4 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3267 (b c-a d)^5 g^4}+\frac {\left (2 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3267 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right ) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^4 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (4 b B^2 d^3\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (4 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {1}{-\frac {-b c+a d}{b}+\frac {d x}{b}}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 B^2 d^4\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3 \left (-\log (a+b x)-\log \left (\frac {1}{c+d x}\right )+\log \left (\frac {e (a+b x)}{c+d x}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^3(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^3(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (4 b^2 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (\frac {1}{x}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (4 b^2 B^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^3(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}+\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{x}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}-\frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^3(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-2 \frac {\left (8 b B^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9801 (b c-a d)^5 g^4}\\ &=-\frac {2 b B^2}{264627 (b c-a d)^2 g^4 (a+b x)^3}+\frac {7 b B^2 d}{88209 (b c-a d)^3 g^4 (a+b x)^2}-\frac {92 b B^2 d^2}{88209 (b c-a d)^4 g^4 (a+b x)}-\frac {2 B^2 d^3}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {10 b B^2 d^3 \log (a+b x)}{8019 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(a+b x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(a+b x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{9801 (b c-a d)^5 g^4}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{88209 (b c-a d)^2 g^4 (a+b x)^3}+\frac {4 b B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^3 g^4 (a+b x)^2}-\frac {26 b B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^4 g^4 (a+b x)}+\frac {2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {20 b B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{29403 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{29403 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3267 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^4 g^4 (c+d x)}-\frac {4 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log (c+d x)}{8019 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log ^2(a+b x) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {20 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{29403 (b c-a d)^5 g^4}+\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 A b B d^3 \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {10 b B^2 d^3 \log ^2(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {4 b B^2 d^3 \log (a+b x) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{9801 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^3(c+d x)}{29403 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {4 b B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 A b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {20 b B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{29403 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}+\frac {8 b B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{9801 (b c-a d)^5 g^4}-\frac {8 b B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9801 (b c-a d)^5 g^4}\\ \end {align*}

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Mathematica [A]  time = 2.25, size = 613, normalized size = 0.90 \[ -\frac {9 B \left (-3 a^4 B d^4+12 a^3 b d^3 (A (c+d x)-B d x)+18 a^2 b^2 d^2 (2 A d x (c+d x)+B c (c+2 d x))+6 a b^3 d \left (6 A d^2 x^2 (c+d x)+B \left (-c^3+3 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )\right )+b^4 \left (12 A d^3 x^3 (c+d x)+B \left (c^4-2 c^3 d x+6 c^2 d^2 x^2+22 c d^3 x^3+10 d^4 x^4\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )-27 d^3 \left (A^2-2 A B+2 B^2\right ) (a+b x)^3 (a d-b c)+6 b d^3 \left (18 A^2+30 A B+55 B^2\right ) (a+b x)^3 (c+d x) \log (a+b x)-6 b d^3 \left (18 A^2+30 A B+55 B^2\right ) (a+b x)^3 (c+d x) \log (c+d x)+3 b d^2 \left (27 A^2+78 A B+92 B^2\right ) (a+b x)^2 (c+d x) (b c-a d)+b \left (9 A^2+6 A B+2 B^2\right ) (c+d x) (b c-a d)^3-3 b d \left (9 A^2+12 A B+7 B^2\right ) (a+b x) (c+d x) (b c-a d)^2+6 B (b c-a d) \left (3 b d^2 (9 A+13 B) (a+b x)^2 (c+d x)-3 b d (3 A+2 B) (a+b x) (c+d x) (b c-a d)+b (3 A+B) (c+d x) (b c-a d)^2+9 d^3 (A-B) (a+b x)^3\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 b B^2 d^3 (a+b x)^3 (c+d x) \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{27 g^4 i^2 (a+b x)^3 (c+d x) (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/27*(-27*(A^2 - 2*A*B + 2*B^2)*d^3*(-(b*c) + a*d)*(a + b*x)^3 + b*(9*A^2 + 6*A*B + 2*B^2)*(b*c - a*d)^3*(c +
d*x) - 3*b*(9*A^2 + 12*A*B + 7*B^2)*d*(b*c - a*d)^2*(a + b*x)*(c + d*x) + 3*b*(27*A^2 + 78*A*B + 92*B^2)*d^2*
(b*c - a*d)*(a + b*x)^2*(c + d*x) + 6*b*(18*A^2 + 30*A*B + 55*B^2)*d^3*(a + b*x)^3*(c + d*x)*Log[a + b*x] + 6*
B*(b*c - a*d)*(9*(A - B)*d^3*(a + b*x)^3 + b*(3*A + B)*(b*c - a*d)^2*(c + d*x) - 3*b*(3*A + 2*B)*d*(b*c - a*d)
*(a + b*x)*(c + d*x) + 3*b*(9*A + 13*B)*d^2*(a + b*x)^2*(c + d*x))*Log[(e*(a + b*x))/(c + d*x)] + 9*B*(-3*a^4*
B*d^4 + 12*a^3*b*d^3*(-(B*d*x) + A*(c + d*x)) + 18*a^2*b^2*d^2*(2*A*d*x*(c + d*x) + B*c*(c + 2*d*x)) + 6*a*b^3
*d*(6*A*d^2*x^2*(c + d*x) + B*(-c^3 + 3*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3)) + b^4*(12*A*d^3*x^3*(c + d*x) + B*
(c^4 - 2*c^3*d*x + 6*c^2*d^2*x^2 + 22*c*d^3*x^3 + 10*d^4*x^4)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 36*b*B^2*d^3*
(a + b*x)^3*(c + d*x)*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*b*(18*A^2 + 30*A*B + 55*B^2)*d^3*(a + b*x)^3*(c + d*x
)*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2*(a + b*x)^3*(c + d*x))

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fricas [B]  time = 1.00, size = 1534, normalized size = 2.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*((9*A^2 + 6*A*B + 2*B^2)*b^4*c^4 - 27*(2*A^2 + 2*A*B + B^2)*a*b^3*c^3*d + 162*(A^2 + 2*A*B + 2*B^2)*a^2*
b^2*c^2*d^2 - 5*(18*A^2 + 66*A*B + 49*B^2)*a^3*b*c*d^3 - 27*(A^2 - 2*A*B + 2*B^2)*a^4*d^4 + 6*((18*A^2 + 30*A*
B + 55*B^2)*b^4*c*d^3 - (18*A^2 + 30*A*B + 55*B^2)*a*b^3*d^4)*x^3 + 36*(B^2*b^4*d^4*x^4 + B^2*a^3*b*c*d^3 + (B
^2*b^4*c*d^3 + 3*B^2*a*b^3*d^4)*x^3 + 3*(B^2*a*b^3*c*d^3 + B^2*a^2*b^2*d^4)*x^2 + (3*B^2*a^2*b^2*c*d^3 + B^2*a
^3*b*d^4)*x)*log((b*e*x + a*e)/(d*x + c))^3 + 3*((18*A^2 + 66*A*B + 85*B^2)*b^4*c^2*d^2 + 8*(9*A^2 + 6*A*B + 2
0*B^2)*a*b^3*c*d^3 - (90*A^2 + 114*A*B + 245*B^2)*a^2*b^2*d^4)*x^2 + 9*(2*(6*A*B + 5*B^2)*b^4*d^4*x^4 + B^2*b^
4*c^4 - 6*B^2*a*b^3*c^3*d + 18*B^2*a^2*b^2*c^2*d^2 + 12*A*B*a^3*b*c*d^3 - 3*B^2*a^4*d^4 + 2*((6*A*B + 11*B^2)*
b^4*c*d^3 + 9*(2*A*B + B^2)*a*b^3*d^4)*x^3 + 6*(B^2*b^4*c^2*d^2 + 6*A*B*a^2*b^2*d^4 + 3*(2*A*B + 3*B^2)*a*b^3*
c*d^3)*x^2 - 2*(B^2*b^4*c^3*d - 9*B^2*a*b^3*c^2*d^2 - 18*(A*B + B^2)*a^2*b^2*c*d^3 - 6*(A*B - B^2)*a^3*b*d^4)*
x)*log((b*e*x + a*e)/(d*x + c))^2 - ((18*A^2 + 30*A*B + 19*B^2)*b^4*c^3*d - 81*(2*A^2 + 6*A*B + 7*B^2)*a*b^3*c
^2*d^2 - 3*(18*A^2 - 114*A*B - 29*B^2)*a^2*b^2*c*d^3 + (198*A^2 + 114*A*B + 461*B^2)*a^3*b*d^4)*x + 6*((18*A^2
+ 30*A*B + 55*B^2)*b^4*d^4*x^4 + 18*A^2*a^3*b*c*d^3 + (3*A*B + B^2)*b^4*c^4 - 9*(2*A*B + B^2)*a*b^3*c^3*d + 5
4*(A*B + B^2)*a^2*b^2*c^2*d^2 - 9*(A*B - B^2)*a^4*d^4 + ((18*A^2 + 66*A*B + 85*B^2)*b^4*c*d^3 + 27*(2*A^2 + 2*
A*B + 5*B^2)*a*b^3*d^4)*x^3 + 3*((6*A*B + 11*B^2)*b^4*c^2*d^2 + 9*(2*A^2 + 6*A*B + 7*B^2)*a*b^3*c*d^3 + 18*(A^
2 + 2*B^2)*a^2*b^2*d^4)*x^2 - ((6*A*B + 5*B^2)*b^4*c^3*d - 27*(2*A*B + 3*B^2)*a*b^3*c^2*d^2 - 54*(A^2 + 2*A*B
+ 2*B^2)*a^2*b^2*c*d^3 - 18*(A^2 - 2*A*B + 2*B^2)*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^8*c^5*d - 5*
a*b^7*c^4*d^2 + 10*a^2*b^6*c^3*d^3 - 10*a^3*b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d^6)*g^4*i^2*x^4 + (b^8*c^
6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 + 20*a^3*b^5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2
*d^6)*g^4*i^2*x^3 + 3*(a*b^7*c^6 - 4*a^2*b^6*c^5*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c*d^5 -
a^7*b*d^6)*g^4*i^2*x^2 + (3*a^2*b^6*c^6 - 14*a^3*b^5*c^5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*a^6*
b^2*c^2*d^4 + 2*a^7*b*c*d^5 - a^8*d^6)*g^4*i^2*x + (a^3*b^5*c^6 - 5*a^4*b^4*c^5*d + 10*a^5*b^3*c^4*d^2 - 10*a^
6*b^2*c^3*d^3 + 5*a^7*b*c^2*d^4 - a^8*c*d^5)*g^4*i^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.06, size = 4487, normalized size = 6.58 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d^5/i^2/(a*d-b*c)^6/g^4*B^2/(d*x+c)*a^2-d^5/i^2/(a*d-b*c)^6/g^4*A^2/(d*x+c)*a^2+2*d^3/i^2/(a*d-b*c)^6/g^4*B
^2*b^2*c-2*d^4/i^2/(a*d-b*c)^6/g^4*B^2*b*a+d^3/i^2/(a*d-b*c)^6/g^4*A^2*b^2*c-d^4/i^2/(a*d-b*c)^6/g^4*A^2*b*a-1
2*d^2*e/i^2/(a*d-b*c)^6/g^4*A*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c+
2/3*d*e^3/i^2/(a*d-b*c)^6/g^4*A*B*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e
)*a+4*d^4/i^2/(a*d-b*c)^6/g^4*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*b*c*a-4*d^2*e^2/i^2/(a*d-b*c)^6/g^4*
A*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a+12*d^3*e/i^2/(a*d-b*c)^6/g
^4*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-4*d^4/i^2/(a*d-b*c)^6/g^4
*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*b*c*a-2*d^3/i^2/(a*d-b*c)^6/g^4*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)/(d*x+c)*b^2*c^2+2*d^4/i^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+c)*b*c*a+2*d*e^2/i^2/(
a*d-b*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*c+6*d^3*e/i
^2/(a*d-b*c)^6/g^4*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a-6*d^2*e
/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*c+12*d^
3*e/i^2/(a*d-b*c)^6/g^4*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-12*d
^2*e/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c-2/3
*e^3/i^2/(a*d-b*c)^6/g^4*A*B*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c+1
/3*d*e^3/i^2/(a*d-b*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
^2*a+2/9*d*e^3/i^2/(a*d-b*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c
)/d*e)*a-2*d^2*e^2/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b*c)/(d
*x+c)/d*e)^2*a-2*d^2*e^2/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d-b
*c)/(d*x+c)/d*e)*a+2*d*e^2/i^2/(a*d-b*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a*d
-b*c)/(d*x+c)/d*e)*c+4*d*e^2/i^2/(a*d-b*c)^6/g^4*A*B*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*ln(b/d*e+(a
*d-b*c)/(d*x+c)/d*e)*c+4/3*d^4/i^2/(a*d-b*c)^6/g^4*B^2*b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^3*a+2*d^5/i^2/(a*d-b*
c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*a^2-4/3*d^3/i^2/(a*d-b*c)^6/g^4*B^2*b^2*ln(b/d*e+(a*d-b*c
)/(d*x+c)/d*e)^3*c+d^3/i^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b^2*c-1/3*e^3/i^2/(a*d-b*c)^6
/g^4*A^2*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*c-2*d^3/i^2/(a*d-b*c)^6/g^4*B^2/(d*x+c)*b^2*c^2+2*d^5/i
^2/(a*d-b*c)^6/g^4*A*B/(d*x+c)*a^2-d^3/i^2/(a*d-b*c)^6/g^4*A^2/(d*x+c)*b^2*c^2-2*d^3/i^2/(a*d-b*c)^6/g^4*B^2*l
n(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b^2*c-4*d^3/i^2/(a*d-b*c)^6/g^4*A^2*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c-d^4/i
^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*b*a+4*d^4/i^2/(a*d-b*c)^6/g^4*A^2*b*ln(b/d*e+(a*d-b*c
)/(d*x+c)/d*e)*a+2*d^4/i^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*a-d^5/i^2/(a*d-b*c)^6/g^4*B^2
*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2/(d*x+c)*a^2-2/27*e^3/i^2/(a*d-b*c)^6/g^4*B^2*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b
*c/d*e+b/d*e)^3*c+2*d^4/i^2/(a*d-b*c)^6/g^4*A*B*b*a-2*d^3/i^2/(a*d-b*c)^6/g^4*A*B*b^2*c-2*d^2*e^2/i^2/(a*d-b*c
)^6/g^4*A^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-2*d^4/i^2/(a*d-b*c)^6/g^4*A*B*ln(b/d*e+(a*d-b*c)/(
d*x+c)/d*e)*b*a-1/3*e^3/i^2/(a*d-b*c)^6/g^4*B^2*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*
c)/(d*x+c)/d*e)^2*c-d^2*e^2/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*a-4*d^3/i^2/
(a*d-b*c)^6/g^4*A*B*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*c-d^3/i^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*
x+c)/d*e)^2/(d*x+c)*b^2*c^2+2*d^3/i^2/(a*d-b*c)^6/g^4*B^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*b^2*c^2+2*d^
4/i^2/(a*d-b*c)^6/g^4*A^2/(d*x+c)*a*b*c+2/9*d*e^3/i^2/(a*d-b*c)^6/g^4*A*B*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e
+b/d*e)^3*a-4*d^4/i^2/(a*d-b*c)^6/g^4*A*B/(d*x+c)*b*c*a-2*d^2*e^2/i^2/(a*d-b*c)^6/g^4*A*B*b^3/(1/(d*x+c)*a*e-1
/(d*x+c)*b*c/d*e+b/d*e)^2*a+2*d*e^2/i^2/(a*d-b*c)^6/g^4*A*B*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*c+12
*d^3*e/i^2/(a*d-b*c)^6/g^4*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*a-12*d^2*e/i^2/(a*d-b*c)^6/g^4*A*B*
b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*c+2*d^3/i^2/(a*d-b*c)^6/g^4*A*B/(d*x+c)*b^2*c^2-2/9*e^3/i^2/(a*d-b
*c)^6/g^4*B^2*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c+d*e^2/i^2/(a*d-b
*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*c+12*d^3*e/i^2/(a*d-b*c)^6/g^4*B^2*b^2/(1/(d*x+c)*
a*e-1/(d*x+c)*b*c/d*e+b/d*e)*a+2/27*d*e^3/i^2/(a*d-b*c)^6/g^4*B^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^
3*a+6*d^3*e/i^2/(a*d-b*c)^6/g^4*A^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*a-6*d^2*e/i^2/(a*d-b*c)^6/g^4*
A^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)*c-12*d^2*e/i^2/(a*d-b*c)^6/g^4*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c
)*b*c/d*e+b/d*e)*c+4*d^4/i^2/(a*d-b*c)^6/g^4*A*B*b*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a-2*d^5/i^2/(a*d-b*c)^6/g
^4*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(d*x+c)*a^2+2*d^3/i^2/(a*d-b*c)^6/g^4*A*B*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*b^2*c+1/3*d*e^3/i^2/(a*d-b*c)^6/g^4*A^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a+2*d*e^2/i^2/(a*d-b*
c)^6/g^4*A^2*b^4/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^2*c+4*d^4/i^2/(a*d-b*c)^6/g^4*B^2/(d*x+c)*a*b*c-2/9*e
^3/i^2/(a*d-b*c)^6/g^4*A*B*b^5/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*c

________________________________________________________________________________________

maxima [B]  time = 7.92, size = 6160, normalized size = 9.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*B^2*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)
*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a
^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11
*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c
^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6
*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a
^6*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10
*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^
2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2
- 2/3*A*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^
3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4
*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 -
11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3
*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 -
6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4
*a^6*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 -
10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*
a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- 1/27*B^2*(6*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 - a*
b^3*d^4)*x^3 + 3*(11*b^4*c^2*d^2 + 8*a*b^3*c*d^3 - 19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*
c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2
- 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*
b^2*c*d^3 + a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c^3*d - 81*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 + 19*a^3*b*d^4)*
x + 30*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2
*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a) - 6*(5*b^4*d^4*x^4 + 5*a^3*b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 +
15*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*c*d^3 + a^3*b*d^4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*
c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*
log(d*x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a^3*b^5*c^6*g^4*i^2 - 5*a^4*b^4*c^5*d*g^4*i^2 + 10*a^5*b^3
*c^4*d^2*g^4*i^2 - 10*a^6*b^2*c^3*d^3*g^4*i^2 + 5*a^7*b*c^2*d^4*g^4*i^2 - a^8*c*d^5*g^4*i^2 + (b^8*c^5*d*g^4*i
^2 - 5*a*b^7*c^4*d^2*g^4*i^2 + 10*a^2*b^6*c^3*d^3*g^4*i^2 - 10*a^3*b^5*c^2*d^4*g^4*i^2 + 5*a^4*b^4*c*d^5*g^4*i
^2 - a^5*b^3*d^6*g^4*i^2)*x^4 + (b^8*c^6*g^4*i^2 - 2*a*b^7*c^5*d*g^4*i^2 - 5*a^2*b^6*c^4*d^2*g^4*i^2 + 20*a^3*
b^5*c^3*d^3*g^4*i^2 - 25*a^4*b^4*c^2*d^4*g^4*i^2 + 14*a^5*b^3*c*d^5*g^4*i^2 - 3*a^6*b^2*d^6*g^4*i^2)*x^3 + 3*(
a*b^7*c^6*g^4*i^2 - 4*a^2*b^6*c^5*d*g^4*i^2 + 5*a^3*b^5*c^4*d^2*g^4*i^2 - 5*a^5*b^3*c^2*d^4*g^4*i^2 + 4*a^6*b^
2*c*d^5*g^4*i^2 - a^7*b*d^6*g^4*i^2)*x^2 + (3*a^2*b^6*c^6*g^4*i^2 - 14*a^3*b^5*c^5*d*g^4*i^2 + 25*a^4*b^4*c^4*
d^2*g^4*i^2 - 20*a^5*b^3*c^3*d^3*g^4*i^2 + 5*a^6*b^2*c^2*d^4*g^4*i^2 + 2*a^7*b*c*d^5*g^4*i^2 - a^8*d^6*g^4*i^2
)*x) + (2*b^4*c^4 - 27*a*b^3*c^3*d + 324*a^2*b^2*c^2*d^2 - 245*a^3*b*c*d^3 - 54*a^4*d^4 + 330*(b^4*c*d^3 - a*b
^3*d^4)*x^3 + 36*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^
2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^3 - 36*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4
)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(d*x + c)^3 + 15*(17*b^4*c^2*d
^2 + 32*a*b^3*c*d^3 - 49*a^2*b^2*d^4)*x^2 - 90*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*
(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 - 18*(5*b^4*d^4*x^4 + 5*a^3*
b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 15*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*c*d^3 + a^3*b*d^
4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a
^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*log(d*x + c)^2 - (19*b^4*c^3*d - 567*a*b^3*c^2*d^2 + 87*a^2*b^2*c*d
^3 + 461*a^3*b*d^4)*x + 330*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*
b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a) - 6*(55*b^4*d^4*x^4 + 55*a^3*b*c*d^3 + 55*(b^4*c*
d^3 + 3*a*b^3*d^4)*x^3 + 165*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 + 55*(3*a
^2*b^2*c*d^3 + a^3*b*d^4)*x - 30*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 +
a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^3*b^5*c^6*g^4*i^2 - 5*a^4*
b^4*c^5*d*g^4*i^2 + 10*a^5*b^3*c^4*d^2*g^4*i^2 - 10*a^6*b^2*c^3*d^3*g^4*i^2 + 5*a^7*b*c^2*d^4*g^4*i^2 - a^8*c*
d^5*g^4*i^2 + (b^8*c^5*d*g^4*i^2 - 5*a*b^7*c^4*d^2*g^4*i^2 + 10*a^2*b^6*c^3*d^3*g^4*i^2 - 10*a^3*b^5*c^2*d^4*g
^4*i^2 + 5*a^4*b^4*c*d^5*g^4*i^2 - a^5*b^3*d^6*g^4*i^2)*x^4 + (b^8*c^6*g^4*i^2 - 2*a*b^7*c^5*d*g^4*i^2 - 5*a^2
*b^6*c^4*d^2*g^4*i^2 + 20*a^3*b^5*c^3*d^3*g^4*i^2 - 25*a^4*b^4*c^2*d^4*g^4*i^2 + 14*a^5*b^3*c*d^5*g^4*i^2 - 3*
a^6*b^2*d^6*g^4*i^2)*x^3 + 3*(a*b^7*c^6*g^4*i^2 - 4*a^2*b^6*c^5*d*g^4*i^2 + 5*a^3*b^5*c^4*d^2*g^4*i^2 - 5*a^5*
b^3*c^2*d^4*g^4*i^2 + 4*a^6*b^2*c*d^5*g^4*i^2 - a^7*b*d^6*g^4*i^2)*x^2 + (3*a^2*b^6*c^6*g^4*i^2 - 14*a^3*b^5*c
^5*d*g^4*i^2 + 25*a^4*b^4*c^4*d^2*g^4*i^2 - 20*a^5*b^3*c^3*d^3*g^4*i^2 + 5*a^6*b^2*c^2*d^4*g^4*i^2 + 2*a^7*b*c
*d^5*g^4*i^2 - a^8*d^6*g^4*i^2)*x)) - 1/3*A^2*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*
a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*
b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*
b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^
4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a
^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^
4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*
a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x +
c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2)) -
2/9*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 - a*b^3*d^4)*x
^3 + 3*(11*b^4*c^2*d^2 + 8*a*b^3*c*d^3 - 19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 - 18*(b^4
*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3
+ a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c^3*d - 81*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 + 19*a^3*b*d^4)*x + 30*(b^
4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3
+ a^3*b*d^4)*x)*log(b*x + a) - 6*(5*b^4*d^4*x^4 + 5*a^3*b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 15*(a*b^3
*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*c*d^3 + a^3*b*d^4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*log(d*x +
c))*A*B/(a^3*b^5*c^6*g^4*i^2 - 5*a^4*b^4*c^5*d*g^4*i^2 + 10*a^5*b^3*c^4*d^2*g^4*i^2 - 10*a^6*b^2*c^3*d^3*g^4*i
^2 + 5*a^7*b*c^2*d^4*g^4*i^2 - a^8*c*d^5*g^4*i^2 + (b^8*c^5*d*g^4*i^2 - 5*a*b^7*c^4*d^2*g^4*i^2 + 10*a^2*b^6*c
^3*d^3*g^4*i^2 - 10*a^3*b^5*c^2*d^4*g^4*i^2 + 5*a^4*b^4*c*d^5*g^4*i^2 - a^5*b^3*d^6*g^4*i^2)*x^4 + (b^8*c^6*g^
4*i^2 - 2*a*b^7*c^5*d*g^4*i^2 - 5*a^2*b^6*c^4*d^2*g^4*i^2 + 20*a^3*b^5*c^3*d^3*g^4*i^2 - 25*a^4*b^4*c^2*d^4*g^
4*i^2 + 14*a^5*b^3*c*d^5*g^4*i^2 - 3*a^6*b^2*d^6*g^4*i^2)*x^3 + 3*(a*b^7*c^6*g^4*i^2 - 4*a^2*b^6*c^5*d*g^4*i^2
+ 5*a^3*b^5*c^4*d^2*g^4*i^2 - 5*a^5*b^3*c^2*d^4*g^4*i^2 + 4*a^6*b^2*c*d^5*g^4*i^2 - a^7*b*d^6*g^4*i^2)*x^2 +
(3*a^2*b^6*c^6*g^4*i^2 - 14*a^3*b^5*c^5*d*g^4*i^2 + 25*a^4*b^4*c^4*d^2*g^4*i^2 - 20*a^5*b^3*c^3*d^3*g^4*i^2 +
5*a^6*b^2*c^2*d^4*g^4*i^2 + 2*a^7*b*c*d^5*g^4*i^2 - a^8*d^6*g^4*i^2)*x)

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mupad [B]  time = 13.57, size = 2701, normalized size = 3.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((e*(a + b*x))/(c + d*x))*(x^2*((4*B^2*b*d)/(g^4*i^2*(a*d - b*c)^3) - (4*b*d^3*(b*d*((2*a^2*d^2 + b^2*c^2
- 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + ((a*d + b*c)*(a*d - b*c))/d^2)*(5*B^2 + 6*A*B))/(3*g^4*i
^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + x*((8*(2*B^2 - 3*A*B))/(9*g^4*i^2*(a^
2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B^2*(a*d + b*c))/(g^4*i^2*(a*d - b*c)^3) - (4*b*d^3*(((2*a^2*d^2 + b^2*c^2
- 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2))*(a*d + b*c) + (a*c*(a*d - b*c))/d^2)*(5*B^2 + 6*A*B))/(3*g
^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (2*(B^2*b*c - 9*B^2*a*d + 9*A*B*a
*d + 3*A*B*b*c))/(9*g^4*i^2*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (4*B^2*a*c)/(g^4*i^2*(a*d - b*c)^3) - (
4*b^2*d^2*x^3*(5*B^2 + 6*A*B))/(3*g^4*i^2*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (
4*a*b*c*d^3*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2))*(5*B^2 + 6*A*B))/(3*g^4*
i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/(b^2*x^4 + (a^3*c)/(b*d) + (x*(a^3*d
+ 3*a^2*b*c))/(b*d) + (x^3*(b^3*c + 3*a*b^2*d))/(b*d) + (x^2*(3*a*b^2*c + 3*a^2*b*d))/(b*d)) - ((27*A^2*a^3*d^
3 + 9*A^2*b^3*c^3 + 54*B^2*a^3*d^3 + 2*B^2*b^3*c^3 - 54*A*B*a^3*d^3 + 6*A*B*b^3*c^3 - 45*A^2*a*b^2*c^2*d + 117
*A^2*a^2*b*c*d^2 - 25*B^2*a*b^2*c^2*d + 299*B^2*a^2*b*c*d^2 - 48*A*B*a*b^2*c^2*d + 276*A*B*a^2*b*c*d^2)/(3*(a*
d - b*c)) + (2*x^3*(18*A^2*b^3*d^3 + 55*B^2*b^3*d^3 + 30*A*B*b^3*d^3))/(a*d - b*c) + (x*(198*A^2*a^2*b*d^3 + 4
61*B^2*a^2*b*d^3 - 18*A^2*b^3*c^2*d - 19*B^2*b^3*c^2*d + 144*A^2*a*b^2*c*d^2 + 548*B^2*a*b^2*c*d^2 + 114*A*B*a
^2*b*d^3 - 30*A*B*b^3*c^2*d + 456*A*B*a*b^2*c*d^2))/(3*(a*d - b*c)) + (x^2*(90*A^2*a*b^2*d^3 + 245*B^2*a*b^2*d
^3 + 18*A^2*b^3*c*d^2 + 85*B^2*b^3*c*d^2 + 114*A*B*a*b^2*d^3 + 66*A*B*b^3*c*d^2))/(a*d - b*c))/(x*(9*a^6*d^4*g
^4*i^2 - 27*a^2*b^4*c^4*g^4*i^2 + 72*a^3*b^3*c^3*d*g^4*i^2 - 54*a^4*b^2*c^2*d^2*g^4*i^2) - x^2*(27*a*b^5*c^4*g
^4*i^2 - 27*a^5*b*d^4*g^4*i^2 - 54*a^2*b^4*c^3*d*g^4*i^2 + 54*a^4*b^2*c*d^3*g^4*i^2) - x^3*(9*b^6*c^4*g^4*i^2
- 27*a^4*b^2*d^4*g^4*i^2 + 72*a^3*b^3*c*d^3*g^4*i^2 - 54*a^2*b^4*c^2*d^2*g^4*i^2) + x^4*(9*a^3*b^3*d^4*g^4*i^2
- 9*b^6*c^3*d*g^4*i^2 + 27*a*b^5*c^2*d^2*g^4*i^2 - 27*a^2*b^4*c*d^3*g^4*i^2) - 9*a^3*b^3*c^4*g^4*i^2 + 9*a^6*
c*d^3*g^4*i^2 + 27*a^4*b^2*c^3*d*g^4*i^2 - 27*a^5*b*c^2*d^2*g^4*i^2) - log((e*(a + b*x))/(c + d*x))^2*((x*((4*
B^2)/(3*g^4*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B^2*b*d^3*(((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3)
+ (a*(a*d - b*c))/(2*b*d^2))*(a*d + b*c) + (a*c*(a*d - b*c))/d^2))/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (B^2*(3*a*d + b*c))/(3*g^4*i^2*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) +
(4*B^2*b^2*d^2*x^3)/(g^4*i^2*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (4*B^2*b*d^3*x
^2*(b*d*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + ((a*d + b*c)*(a*d - b*c))/
d^2))/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (4*B^2*a*b*c*d^3*((2*a^2*d
^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2)))/(b^2*x^4 + (a^3*c)/(b*d) + (x*(a^3*d + 3*a^2*b*c))/(b*d) + (x^3*(b^3*c + 3*a*
b^2*d))/(b*d) + (x^2*(3*a*b^2*c + 3*a^2*b*d))/(b*d)) - (2*b*d^3*(5*B^2 + 6*A*B))/(3*g^4*i^2*(a*d - b*c)^2*(a^3
*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (b*d^3*atan((b*d^3*(18*A^2 + 55*B^2 + 30*A*B)*(9*a^5*d^5*g
^4*i^2 + 9*b^5*c^5*g^4*i^2 - 27*a*b^4*c^4*d*g^4*i^2 - 27*a^4*b*c*d^4*g^4*i^2 + 18*a^2*b^3*c^3*d^2*g^4*i^2 + 18
*a^3*b^2*c^2*d^3*g^4*i^2)*2i)/(9*g^4*i^2*(a*d - b*c)^5*(36*A^2*b*d^3 + 110*B^2*b*d^3 + 60*A*B*b*d^3)) + (b^2*d
^4*x*(18*A^2 + 55*B^2 + 30*A*B)*(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4*a^3*b*c*d^3*g^4
*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2)*4i)/(g^4*i^2*(a*d - b*c)^5*(36*A^2*b*d^3 + 110*B^2*b*d^3 + 60*A*B*b*d^3)))*(
18*A^2 + 55*B^2 + 30*A*B)*4i)/(9*g^4*i^2*(a*d - b*c)^5) + (4*B^2*b*d^3*log((e*(a + b*x))/(c + d*x))^3)/(3*g^4*
i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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