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

Optimal. Leaf size=635 \[ \frac {6 b^2 B g^3 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^3}+\frac {3 b^2 g^3 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^4 i^3}+\frac {2 b^2 B g^3 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^3}+\frac {b^2 g^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i^3}+\frac {2 b g^3 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i^3 (c+d x)}-\frac {4 A b B g^3 (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac {B g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i^3 (c+d x)^2}+\frac {2 b^2 B^2 g^3 (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}-\frac {6 b^2 B^2 g^3 (b c-a d) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}-\frac {4 b B^2 g^3 (a+b x) (b c-a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^3 (c+d x)}+\frac {4 b B^2 g^3 (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {B^2 g^3 (a+b x)^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 6.24, antiderivative size = 1890, normalized size of antiderivative = 2.98, number of steps used = 124, number of rules used = 26, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {2528, 2523, 12, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 2525, 44, 6688, 6742, 2500, 2433, 2375, 2317, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*(b*c - a*d)^3*g^3)/(4*d^4*i^3*(c + d*x)^2) - (9*b*B^2*(b*c - a*d)^2*g^3)/(2*d^4*i^3*(c + d*x)) - (9*b^2*B
^2*(b*c - a*d)*g^3*Log[a + b*x])/(2*d^4*i^3) - (a*b^2*B^2*g^3*Log[a + b*x]^2)/(d^3*i^3) - (5*b^2*B^2*(b*c - a*
d)*g^3*Log[a + b*x]^2)/(2*d^4*i^3) + (3*b^2*B^2*(b*c - a*d)*g^3*Log[a + b*x]*Log[(c + d*x)^(-1)]^2)/(d^4*i^3)
- (3*b^2*B^2*(b*c - a*d)*g^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/(d^4*i^3) - (B*(b*c - a*
d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^4*i^3*(c + d*x)^2) + (5*b*B*(b*c - a*d)^2*g^3*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(d^4*i^3*(c + d*x)) + (2*a*b^2*B*g^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)
]))/(d^3*i^3) + (5*b^2*B*(b*c - a*d)*g^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^3) + (b^3*g
^3*x*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^3) + ((b*c - a*d)^3*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)
])^2)/(2*d^4*i^3*(c + d*x)^2) - (3*b*(b*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i^3*(c + d
*x)) + (9*b^2*B^2*(b*c - a*d)*g^3*Log[c + d*x])/(2*d^4*i^3) + (3*b^2*B^2*(b*c - a*d)*g^3*Log[a + b*x]^2*Log[c
+ d*x])/(d^4*i^3) + (2*b^3*B^2*c*g^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^4*i^3) + (6*A*b^2*B*(b
*c - a*d)*g^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^4*i^3) + (5*b^2*B^2*(b*c - a*d)*g^3*Log[-((d*
(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^4*i^3) + (6*b^2*B^2*(b*c - a*d)*g^3*Log[a + b*x]*Log[(c + d*x)^(-1)]
*Log[c + d*x])/(d^4*i^3) - (6*b^2*B^2*(b*c - a*d)*g^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])/(d^4*i^3) - (2*b^3*B*c*g^3*(A + B*Log[(e*(a + b*x)
)/(c + d*x)])*Log[c + d*x])/(d^4*i^3) - (5*b^2*B*(b*c - a*d)*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c +
d*x])/(d^4*i^3) - (3*b^2*(b*c - a*d)*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[c + d*x])/(d^4*i^3) - (b^3
*B^2*c*g^3*Log[c + d*x]^2)/(d^4*i^3) - (3*A*b^2*B*(b*c - a*d)*g^3*Log[c + d*x]^2)/(d^4*i^3) - (5*b^2*B^2*(b*c
- a*d)*g^3*Log[c + d*x]^2)/(2*d^4*i^3) + (3*b^2*B^2*(b*c - a*d)*g^3*Log[a + b*x]*Log[c + d*x]^2)/(d^4*i^3) - (
3*b^2*B^2*(b*c - a*d)*g^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x]^2)/(d^4*i^3) - (b^2*B^2*(b*c - a*d)*g^3*Lo
g[c + d*x]^3)/(d^4*i^3) + (2*a*b^2*B^2*g^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(d^3*i^3) + (5*b^2*B^2
*(b*c - a*d)*g^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(d^4*i^3) - (3*b^2*B^2*(b*c - a*d)*g^3*Log[a + b
*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(d^4*i^3) + (2*a*b^2*B^2*g^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(
d^3*i^3) + (5*b^2*B^2*(b*c - a*d)*g^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^4*i^3) - (6*b^2*B^2*(b*c -
a*d)*g^3*Log[a + b*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d^4*i^3) + (2*b^3*B^2*c*g^3*PolyLog[2, (b*(c
+ d*x))/(b*c - a*d)])/(d^4*i^3) + (6*A*b^2*B*(b*c - a*d)*g^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^3)
+ (5*b^2*B^2*(b*c - a*d)*g^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^3) + (6*b^2*B^2*(b*c - a*d)*g^3*Log
[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^3) - (6*b^2*B^2*(b*c - a*d)*g^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)])/(d^4*i^3) + (6*b^2
*B^2*(b*c - a*d)*g^3*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/(d^4*i^3) + (6*b^2*B^2*(b*c - a*d)*g^3*PolyLog[
3, (b*(c + d*x))/(b*c - a*d)])/(d^4*i^3)

Rule 12

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

Rule 30

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

Rule 44

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

Rule 2301

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

Rule 2302

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

Rule 2317

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

Rule 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 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 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 2523

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

Rule 2524

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

Rule 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 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 {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(100 c+100 d x)^3} \, dx &=\int \left (\frac {b^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3 (c+d x)^3}+\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3 (c+d x)^2}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx}{1000000 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c+d x} \, dx}{1000000 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^2} \, dx}{1000000 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c+d x)^3} \, dx}{1000000 d^3}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {\left (b^3 B g^3\right ) \int \frac {(b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{500000 d^3}+\frac {\left (3 b^2 B (b c-a d) g^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}{500000 d^4}+\frac {\left (3 b B (b c-a d)^2 g^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}{500000 d^4}-\frac {\left (B (b c-a d)^3 g^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)^3} \, dx}{1000000 d^4}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {\left (b^3 B (b c-a d) g^3\right ) \int \frac {x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x) (c+d x)} \, dx}{500000 d^3}+\frac {\left (3 b B (b c-a d)^3 g^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}{500000 d^4}-\frac {\left (B (b c-a d)^4 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^3} \, dx}{1000000 d^4}+\frac {\left (3 b^2 B (b c-a d) g^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}{500000 d^4 e}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {\left (b^3 B (b c-a d) g^3\right ) \int \left (-\frac {a \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)}+\frac {c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx}{500000 d^3}+\frac {\left (3 b B (b c-a d)^3 g^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}{500000 d^4}-\frac {\left (B (b c-a d)^4 g^3\right ) \int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{1000000 d^4}+\frac {\left (3 b^2 B (b c-a d) g^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}{500000 d^4 e}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}+\frac {\left (a b^3 B g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{500000 d^3}-\frac {\left (b^3 B c g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{500000 d^3}-\frac {\left (b^3 B (b c-a d) g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1000000 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{500000 d^4}+\frac {\left (b^2 B (b c-a d) g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1000000 d^3}-\frac {\left (3 b^2 B (b c-a d) g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{500000 d^3}+\frac {\left (3 b^2 B (b c-a d)^2 g^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}{500000 d^4}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1000000 d^3}-\frac {\left (3 b B (b c-a d)^2 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{500000 d^3}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1000000 d^3}\\ &=-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}+\frac {\left (b^3 B^2 c g^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}{500000 d^4}-\frac {\left (a b^2 B^2 g^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}{500000 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4}-\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4}+\frac {\left (3 b^2 B (b c-a d)^2 g^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}{500000 d^4}+\frac {\left (b B^2 (b c-a d)^2 g^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{1000000 d^4}-\frac {\left (3 b B^2 (b c-a d)^2 g^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{500000 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{2000000 d^4}\\ &=-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}+\frac {\left (3 b^3 B (b c-a d) g^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}{500000 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3\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}{500000 d^3}+\frac {\left (b B^2 (b c-a d)^3 g^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{1000000 d^4}-\frac {\left (3 b B^2 (b c-a d)^3 g^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{500000 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{2000000 d^4}+\frac {\left (b^3 B^2 c g^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}{500000 d^4 e}-\frac {\left (a b^2 B^2 g^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}{500000 d^3 e}+\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4 e}-\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4 e}-\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4 e}+\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4 e}\\ &=-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}+\frac {\left (3 b^3 B (b c-a d) g^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}{500000 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3\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}{500000 d^3}+\frac {\left (b B^2 (b c-a d)^3 g^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}{1000000 d^4}-\frac {\left (3 b B^2 (b c-a d)^3 g^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}{500000 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2000000 d^4}+\frac {\left (b^3 B^2 c g^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}{500000 d^4 e}-\frac {\left (a b^2 B^2 g^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}{500000 d^3 e}+\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4 e}-\frac {\left (b^2 B^2 (b c-a d) g^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}{1000000 d^4 e}-\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4 e}+\frac {\left (3 b^2 B^2 (b c-a d) g^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}{500000 d^4 e}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}+\frac {\left (b^4 B^2 c g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{500000 d^4}-\frac {\left (a b^3 B^2 g^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{500000 d^3}-\frac {\left (b^3 B^2 c g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{500000 d^3}+\frac {\left (a b^2 B^2 g^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{500000 d^2}+\frac {\left (3 A b^3 B (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{500000 d^4}+\frac {\left (b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1000000 d^4}-\frac {\left (b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1000000 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{500000 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{500000 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{500000 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{500000 d^3}-\frac {\left (b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1000000 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1000000 d^3}+\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{500000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{500000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)}{c+d x} \, dx}{500000 d^3}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {\left (b^3 B^2 c g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{500000 d^4}-\frac {\left (a b^2 B^2 g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{500000 d^3}-\frac {\left (a b^3 B^2 g^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{500000 d^3}-\frac {\left (b^3 B^2 c g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{500000 d^3}-\frac {\left (3 A b^2 B (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{500000 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1000000 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1000000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{500000 d^4}+\frac {\left (b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1000000 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{1000000 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log (a+b x) \log (c+d x)}{a+b x} \, dx}{500000 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{500000 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{500000 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{500000 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1000000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{1000000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{500000 d^3}+\frac {\left (3 b^3 B^2 (b c-a d) g^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}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {\left (b^3 B^2 c g^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 )}{500000 d^4}-\frac {\left (a b^2 B^2 g^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 )}{500000 d^3}-\frac {\left (3 A b^2 B (b c-a d) g^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 )}{500000 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^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 )}{1000000 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^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 )}{1000000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{1000000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{500000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{500000 d^3}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{1000000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}-\frac {\left (3 b B^2 (b c-a d) g^3\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 )}{1000000 d^3}-\frac {\left (3 b B^2 (b c-a d) g^3\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 )}{500000 d^3}+\frac {\left (3 b B^2 (b c-a d) g^3\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 )}{500000 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^3(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1000000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2(c+d x)}{1000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^3(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1000000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3\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 )}{1000000 d^5}-\frac {\left (3 b^3 B^2 (b c-a d) g^3\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 )}{1000000 d^5}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^3(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1000000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^3(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1000000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+2 \frac {\left (3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3}{4000000 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3}{2000000 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 \log (a+b x)}{2000000 d^4}-\frac {a b^2 B^2 g^3 \log ^2(a+b x)}{1000000 d^3}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(a+b x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{1000000 d^4}-\frac {B (b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2000000 d^4 (c+d x)^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4 (c+d x)}+\frac {a b^2 B g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{500000 d^3}+\frac {b^2 B (b c-a d) g^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{200000 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2000000 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1000000 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 \log (c+d x)}{2000000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log (c+d x)}{1000000 d^4}+\frac {b^3 B^2 c g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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)}{500000 d^4}-\frac {b^3 B c g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{500000 d^4}-\frac {b^2 B (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{200000 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1000000 d^4}-\frac {b^3 B^2 c g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {3 A b^2 B (b c-a d) g^3 \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^2(c+d x)}{400000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2(c+d x)}{1000000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1000000 d^4}-\frac {b^2 B^2 (b c-a d) g^3 \log ^3(c+d x)}{1000000 d^4}+\frac {a b^2 B^2 g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1000000 d^4}+\frac {a b^2 B^2 g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^3}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{200000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}+\frac {b^3 B^2 c g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {3 A b^2 B (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}+\frac {b^2 B^2 (b c-a d) g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{200000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}-\frac {3 b^2 B^2 (b c-a d) g^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 )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{500000 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{500000 d^4}\\ \end {align*}

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Mathematica [B]  time = 7.86, size = 6052, normalized size = 9.53 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A^{2} b^{3} g^{3} x^{3} + 3 \, A^{2} a b^{2} g^{3} x^{2} + 3 \, A^{2} a^{2} b g^{3} x + A^{2} a^{3} g^{3} + {\left (B^{2} b^{3} g^{3} x^{3} + 3 \, B^{2} a b^{2} g^{3} x^{2} + 3 \, B^{2} a^{2} b g^{3} x + B^{2} a^{3} g^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left (A B b^{3} g^{3} x^{3} + 3 \, A B a b^{2} g^{3} x^{2} + 3 \, A B a^{2} b g^{3} x + A B a^{3} g^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [F]  time = 2.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{3} \left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}{\left (d i x +c i \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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