3.185 \(\int \frac{(c i+d i x)^3 (A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=561 \[ \frac{2 B d^3 i^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4}+\frac{2 B^2 d^3 i^3 n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^4}-\frac{d^2 i^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g^4 (a+b x)}-\frac{2 B d^2 i^3 n (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)}-\frac{d^3 i^3 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^4 g^4}-\frac{d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b^2 g^4 (a+b x)^2}-\frac{B d i^3 n (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac{i^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac{2 B i^3 n (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 b g^4 (a+b x)^3}-\frac{2 B^2 d^2 i^3 n^2 (c+d x)}{b^3 g^4 (a+b x)}-\frac{B^2 d i^3 n^2 (c+d x)^2}{4 b^2 g^4 (a+b x)^2}-\frac{2 B^2 i^3 n^2 (c+d x)^3}{27 b g^4 (a+b x)^3} \]

[Out]

(-2*B^2*d^2*i^3*n^2*(c + d*x))/(b^3*g^4*(a + b*x)) - (B^2*d*i^3*n^2*(c + d*x)^2)/(4*b^2*g^4*(a + b*x)^2) - (2*
B^2*i^3*n^2*(c + d*x)^3)/(27*b*g^4*(a + b*x)^3) - (2*B*d^2*i^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/(b^3*g^4*(a + b*x)) - (B*d*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^4*(a + b*x)
^2) - (2*B*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*b*g^4*(a + b*x)^3) - (d^2*i^3*(c + d*x
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^4*(a + b*x)) - (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])^2)/(2*b^2*g^4*(a + b*x)^2) - (i^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b*g^4
*(a + b*x)^3) - (d^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g
^4) + (2*B*d^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4)
+ (2*B^2*d^3*i^3*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4)

________________________________________________________________________________________

Rubi [B]  time = 5.09333, antiderivative size = 1170, normalized size of antiderivative = 2.09, number of steps used = 100, number of rules used = 20, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317, 2507, 2488, 2506, 6610} \[ \frac{11 B^2 d^3 n^2 \log ^2(a+b x) i^3}{6 b^4 g^4}-\frac{A B d^3 n \log ^2(a+b x) i^3}{b^4 g^4}-\frac{B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) i^3}{b^4 g^4}-\frac{B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) i^3}{b^4 g^4}+\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^3}{b^4 g^4}-\frac{3 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^3}{b^4 g^4 (a+b x)}-\frac{3 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^3}{2 b^4 g^4 (a+b x)^2}-\frac{(b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 i^3}{3 b^4 g^4 (a+b x)^3}+\frac{11 B^2 d^3 n^2 \log ^2(c+d x) i^3}{6 b^4 g^4}-\frac{49 B^2 d^3 n^2 \log (a+b x) i^3}{18 b^4 g^4}-\frac{11 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^3}{3 b^4 g^4}-\frac{11 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^3}{3 b^4 g^4 (a+b x)}-\frac{7 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^3}{6 b^4 g^4 (a+b x)^2}-\frac{2 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) i^3}{9 b^4 g^4 (a+b x)^3}+\frac{49 B^2 d^3 n^2 \log (c+d x) i^3}{18 b^4 g^4}-\frac{11 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) i^3}{3 b^4 g^4}+\frac{11 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) i^3}{3 b^4 g^4}-\frac{11 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^3}{3 b^4 g^4}+\frac{2 A B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) i^3}{b^4 g^4}-\frac{11 B^2 d^3 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^3}{3 b^4 g^4}+\frac{2 A B d^3 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) i^3}{b^4 g^4}-\frac{11 B^2 d^3 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) i^3}{3 b^4 g^4}+\frac{2 B^2 d^3 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) i^3}{b^4 g^4}+\frac{2 B^2 d^3 n^2 \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right ) i^3}{b^4 g^4}-\frac{49 B^2 d^2 (b c-a d) n^2 i^3}{18 b^4 g^4 (a+b x)}-\frac{17 B^2 d (b c-a d)^2 n^2 i^3}{36 b^4 g^4 (a+b x)^2}-\frac{2 B^2 (b c-a d)^3 n^2 i^3}{27 b^4 g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^4,x]

[Out]

(-2*B^2*(b*c - a*d)^3*i^3*n^2)/(27*b^4*g^4*(a + b*x)^3) - (17*B^2*d*(b*c - a*d)^2*i^3*n^2)/(36*b^4*g^4*(a + b*
x)^2) - (49*B^2*d^2*(b*c - a*d)*i^3*n^2)/(18*b^4*g^4*(a + b*x)) - (49*B^2*d^3*i^3*n^2*Log[a + b*x])/(18*b^4*g^
4) - (A*B*d^3*i^3*n*Log[a + b*x]^2)/(b^4*g^4) + (11*B^2*d^3*i^3*n^2*Log[a + b*x]^2)/(6*b^4*g^4) - (B^2*d^3*i^3
*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*((a + b*x)/(c + d*x))^n]^2)/(b^4*g^4) - (B^2*d^3*i^3*Log[a + b*x]*Log
[e*((a + b*x)/(c + d*x))^n]^2)/(b^4*g^4) - (2*B*(b*c - a*d)^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9
*b^4*g^4*(a + b*x)^3) - (7*B*d*(b*c - a*d)^2*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^4*g^4*(a + b*x
)^2) - (11*B*d^2*(b*c - a*d)*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^4*g^4*(a + b*x)) - (11*B*d^3*i
^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^4*g^4) - ((b*c - a*d)^3*i^3*(A + B*Log[e*((a +
b*x)/(c + d*x))^n])^2)/(3*b^4*g^4*(a + b*x)^3) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
^2)/(2*b^4*g^4*(a + b*x)^2) - (3*d^2*(b*c - a*d)*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g^4*(a + b
*x)) + (d^3*i^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^4*g^4) + (49*B^2*d^3*i^3*n^2*Log[c +
d*x])/(18*b^4*g^4) - (11*B^2*d^3*i^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b^4*g^4) + (11*B*
d^3*i^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(3*b^4*g^4) + (11*B^2*d^3*i^3*n^2*Log[c + d*x]^
2)/(6*b^4*g^4) + (2*A*B*d^3*i^3*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^4*g^4) - (11*B^2*d^3*i^3*n^2
*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b^4*g^4) + (2*A*B*d^3*i^3*n*PolyLog[2, -((d*(a + b*x))/(b*c -
a*d))])/(b^4*g^4) - (11*B^2*d^3*i^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b^4*g^4) - (11*B^2*d^3*i
^3*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b^4*g^4) + (2*B^2*d^3*i^3*n*Log[e*((a + b*x)/(c + d*x))^n]*Po
lyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(b^4*g^4) + (2*B^2*d^3*i^3*n^2*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x
))])/(b^4*g^4)

Rule 2528

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

Rule 2525

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

Rule 12

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

Rule 44

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

Rule 2524

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

Rule 2418

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

Rule 2390

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

Rule 2301

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

Rule 2394

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

Rule 2393

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

Rule 2391

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

Rule 6688

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

Rule 6742

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

Rule 2411

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

Rule 2344

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

Rule 2317

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

Rule 2507

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

Rule 2488

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

Rule 2506

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

Rule 6610

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

Rubi steps

\begin{align*} \int \frac{(185 c+185 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)^4}+\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)^3}+\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)^2}+\frac{6331625 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}\right ) \, dx\\ &=\frac{\left (6331625 d^3\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{b^3 g^4}+\frac{\left (18994875 d^2 (b c-a d)\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{b^3 g^4}+\frac{\left (18994875 d (b c-a d)^2\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{b^3 g^4}+\frac{\left (6331625 (b c-a d)^3\right ) \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b^3 g^4}\\ &=-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}-\frac{\left (12663250 B d^3 n\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b^4 g^4}+\frac{\left (37989750 B d^2 (b c-a d) n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (18994875 B d (b c-a d)^2 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (12663250 B (b c-a d)^3 n\right ) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}-\frac{\left (12663250 B d^3 n\right ) \int \frac{(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^4}+\frac{\left (37989750 B d^2 (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (18994875 B d (b c-a d)^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (12663250 B (b c-a d)^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}-\frac{\left (12663250 B d^3 (b c-a d) n\right ) \int \frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^4}+\frac{\left (37989750 B d^2 (b c-a d)^2 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^4 g^4}+\frac{\left (18994875 B d (b c-a d)^3 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^3}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^4 g^4}+\frac{\left (12663250 B (b c-a d)^4 n\right ) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^4 g^4}\\ &=-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}-\frac{\left (12663250 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 b^3 g^4}+\frac{\left (18994875 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^4}-\frac{\left (37989750 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^4}+\frac{\left (12663250 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b^4 g^4}-\frac{\left (18994875 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^4 g^4}+\frac{\left (37989750 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^4 g^4}+\frac{\left (12663250 B d^2 (b c-a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3 b^3 g^4}-\frac{\left (18994875 B d^2 (b c-a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^4}+\frac{\left (37989750 B d^2 (b c-a d) n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^3 g^4}-\frac{\left (12663250 B d^3 (b c-a d) n\right ) \int \left (\frac{A \log (a+b x)}{(a+b x) (c+d x)}+\frac{B \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{b^4 g^4}-\frac{\left (12663250 B d (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3 b^3 g^4}+\frac{\left (18994875 B d (b c-a d)^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^3 g^4}+\frac{\left (12663250 B (b c-a d)^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3 b^3 g^4}\\ &=-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}-\frac{\left (12663250 A B d^3 (b c-a d) n\right ) \int \frac{\log (a+b x)}{(a+b x) (c+d x)} \, dx}{b^4 g^4}-\frac{\left (12663250 B^2 d^3 (b c-a d) n\right ) \int \frac{\log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3 b^4 g^4}-\frac{\left (12663250 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 g^4}+\frac{\left (18994875 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 g^4}-\frac{\left (37989750 B^2 d^3 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^4 g^4}+\frac{\left (12663250 B^2 d^2 (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^2 (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^2 (b c-a d) n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}-\frac{\left (6331625 B^2 d (b c-a d)^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^4 g^4}+\frac{\left (18994875 B^2 d (b c-a d)^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^4}+\frac{\left (12663250 B^2 (b c-a d)^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^4 g^4}\\ &=-\frac{6331625 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}+\frac{\left (6331625 B^2 d^3\right ) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^3 g^4}-\frac{\left (12663250 A B d^3 (b c-a d) n\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 )}{b^5 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{3 b^4 g^4}-\frac{\left (12663250 B^2 d^3 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 g^4}+\frac{\left (18994875 B^2 d^3 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 g^4}-\frac{\left (37989750 B^2 d^3 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{b^4 g^4}+\frac{\left (12663250 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^2 (b c-a d)^2 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}-\frac{\left (6331625 B^2 d (b c-a d)^3 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b^4 g^4}+\frac{\left (18994875 B^2 d (b c-a d)^3 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^4}+\frac{\left (12663250 B^2 (b c-a d)^4 n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b^4 g^4}\\ &=-\frac{6331625 B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{6331625 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}-\frac{\left (12663250 A B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^4}+\frac{\left (12663250 A B d^4 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b^5 g^4}+\frac{\left (12663250 B^2 d^3 (b c-a d) n\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b^3 g^4}-\frac{\left (12663250 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b^3 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^4}+\frac{\left (18994875 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^3 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^4}-\frac{\left (37989750 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b^3 g^4}-\frac{\left (12663250 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^4 g^4}+\frac{\left (12663250 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^4 g^4}+\frac{\left (18994875 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^4}-\frac{\left (18994875 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^4 g^4}-\frac{\left (37989750 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^4 g^4}+\frac{\left (12663250 B^2 d^2 (b c-a d)^2 n^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}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^2 (b c-a d)^2 n^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}{b^4 g^4}+\frac{\left (37989750 B^2 d^2 (b c-a d)^2 n^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}{b^4 g^4}-\frac{\left (6331625 B^2 d (b c-a d)^3 n^2\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}{3 b^4 g^4}+\frac{\left (18994875 B^2 d (b c-a d)^3 n^2\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}{2 b^4 g^4}+\frac{\left (12663250 B^2 (b c-a d)^4 n^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}{9 b^4 g^4}\\ &=-\frac{12663250 B^2 (b c-a d)^3 n^2}{27 b^4 g^4 (a+b x)^3}-\frac{107637625 B^2 d (b c-a d)^2 n^2}{36 b^4 g^4 (a+b x)^2}-\frac{310249625 B^2 d^2 (b c-a d) n^2}{18 b^4 g^4 (a+b x)}-\frac{310249625 B^2 d^3 n^2 \log (a+b x)}{18 b^4 g^4}-\frac{6331625 A B d^3 n \log ^2(a+b x)}{b^4 g^4}-\frac{6331625 B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{6331625 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{310249625 B^2 d^3 n^2 \log (c+d x)}{18 b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}+\frac{12663250 A B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^4 g^4}+\frac{12663250 B^2 d^3 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^4}-\frac{\left (12663250 A B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^3 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^4}+\frac{\left (12663250 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 g^4}+\frac{\left (37989750 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^4 g^4}-\frac{\left (12663250 B^2 d^3 (b c-a d) n^2\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}{b^4 g^4}\\ &=-\frac{12663250 B^2 (b c-a d)^3 n^2}{27 b^4 g^4 (a+b x)^3}-\frac{107637625 B^2 d (b c-a d)^2 n^2}{36 b^4 g^4 (a+b x)^2}-\frac{310249625 B^2 d^2 (b c-a d) n^2}{18 b^4 g^4 (a+b x)}-\frac{310249625 B^2 d^3 n^2 \log (a+b x)}{18 b^4 g^4}-\frac{6331625 A B d^3 n \log ^2(a+b x)}{b^4 g^4}+\frac{69647875 B^2 d^3 n^2 \log ^2(a+b x)}{6 b^4 g^4}-\frac{6331625 B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{6331625 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{310249625 B^2 d^3 n^2 \log (c+d x)}{18 b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}+\frac{69647875 B^2 d^3 n^2 \log ^2(c+d x)}{6 b^4 g^4}+\frac{12663250 A B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^4 g^4}+\frac{12663250 A B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^4}+\frac{12663250 B^2 d^3 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^4}+\frac{12663250 B^2 d^3 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^4 g^4}+\frac{\left (12663250 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^4}-\frac{\left (18994875 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 g^4}+\frac{\left (37989750 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^4 g^4}\\ &=-\frac{12663250 B^2 (b c-a d)^3 n^2}{27 b^4 g^4 (a+b x)^3}-\frac{107637625 B^2 d (b c-a d)^2 n^2}{36 b^4 g^4 (a+b x)^2}-\frac{310249625 B^2 d^2 (b c-a d) n^2}{18 b^4 g^4 (a+b x)}-\frac{310249625 B^2 d^3 n^2 \log (a+b x)}{18 b^4 g^4}-\frac{6331625 A B d^3 n \log ^2(a+b x)}{b^4 g^4}+\frac{69647875 B^2 d^3 n^2 \log ^2(a+b x)}{6 b^4 g^4}-\frac{6331625 B^2 d^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{6331625 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 g^4}-\frac{12663250 B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b^4 g^4 (a+b x)^3}-\frac{44321375 B d (b c-a d)^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b^4 g^4 (a+b x)^2}-\frac{69647875 B d^2 (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)}-\frac{69647875 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4}-\frac{6331625 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^4 g^4 (a+b x)^3}-\frac{18994875 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^4 g^4 (a+b x)^2}-\frac{18994875 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4 (a+b x)}+\frac{6331625 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g^4}+\frac{310249625 B^2 d^3 n^2 \log (c+d x)}{18 b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^4 g^4}+\frac{69647875 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^4 g^4}+\frac{69647875 B^2 d^3 n^2 \log ^2(c+d x)}{6 b^4 g^4}+\frac{12663250 A B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^4 g^4}+\frac{12663250 A B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^4 g^4}-\frac{69647875 B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^4 g^4}+\frac{12663250 B^2 d^3 n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^4}+\frac{12663250 B^2 d^3 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b^4 g^4}\\ \end{align*}

Mathematica [B]  time = 8.15392, size = 8775, normalized size = 15.64 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^4,x]

[Out]

Result too large to show

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Maple [F]  time = 0.695, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{3}}{ \left ( bgx+ag \right ) ^{4}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/3*A*B*c*d^2*i^3*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^
2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7
*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*b^5*c
^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*
b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((b^6*c
^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4)) - 1/9*A*B*c^3*i^3*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*
a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^
2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2
- 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3
)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/6*A*B*c^2*d*i^3
*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*
a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4
*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
- 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*
c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) + 1/6*A^2*d^3*i^3
*((18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3)/(b^7*g^4*x^3 + 3*a*b^6*g^4*x^2 + 3*a^2*b^5*g^4*x + a^3*b^4*g^4) + 6*log
(b*x + a)/(b^4*g^4)) - (3*b*x + a)*A*B*c^2*d*i^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^4*x^3 + 3*a*b^4
*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 2*(3*b^2*x^2 + 3*a*b*x + a^2)*A*B*c*d^2*i^3*log(e*(b*x/(d*x + c) +
a/(d*x + c))^n)/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/2*(3*b*x + a)*A^2*c^2*d*i
^3/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - (3*b^2*x^2 + 3*a*b*x + a^2)*A^2*c*d^2*i^3
/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 2/3*A*B*c^3*i^3*log(e*(b*x/(d*x + c) + a/(d
*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A^2*c^3*i^3/(b^4*g^4*x^3 + 3*a
*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/6*(18*(b^3*c*d^2*i^3 - a*b^2*d^3*i^3)*B^2*x^2 + 9*(b^3*c^2*d*i
^3 + 2*a*b^2*c*d^2*i^3 - 3*a^2*b*d^3*i^3)*B^2*x + (2*b^3*c^3*i^3 + 3*a*b^2*c^2*d*i^3 + 6*a^2*b*c*d^2*i^3 - 11*
a^3*d^3*i^3)*B^2 - 6*(B^2*b^3*d^3*i^3*x^3 + 3*B^2*a*b^2*d^3*i^3*x^2 + 3*B^2*a^2*b*d^3*i^3*x + B^2*a^3*d^3*i^3)
*log(b*x + a))*log((d*x + c)^n)^2/(b^7*g^4*x^3 + 3*a*b^6*g^4*x^2 + 3*a^2*b^5*g^4*x + a^3*b^4*g^4) - integrate(
-1/3*(18*B^2*b^4*c^2*d^2*i^3*x^2*log(e)^2 + 12*B^2*b^4*c^3*d*i^3*x*log(e)^2 + 3*B^2*b^4*c^4*i^3*log(e)^2 + 3*(
B^2*b^4*d^4*i^3*log(e)^2 + 2*A*B*b^4*d^4*i^3*log(e))*x^4 + 6*(2*B^2*b^4*c*d^3*i^3*log(e)^2 + A*B*b^4*c*d^3*i^3
*log(e))*x^3 + 3*(B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*
i^3*x + B^2*b^4*c^4*i^3)*log((b*x + a)^n)^2 + 6*(6*B^2*b^4*c^2*d^2*i^3*x^2*log(e) + 4*B^2*b^4*c^3*d*i^3*x*log(
e) + B^2*b^4*c^4*i^3*log(e) + (B^2*b^4*d^4*i^3*log(e) + A*B*b^4*d^4*i^3)*x^4 + (4*B^2*b^4*c*d^3*i^3*log(e) + A
*B*b^4*c*d^3*i^3)*x^3)*log((b*x + a)^n) + (9*(4*a*b^3*c*d^3*i^3*n - 5*a^2*b^2*d^4*i^3*n + (i^3*n - 4*i^3*log(e
))*b^4*c^2*d^2)*B^2*x^2 - 6*(B^2*b^4*d^4*i^3*log(e) + A*B*b^4*d^4*i^3)*x^4 + 2*(6*a*b^3*c^2*d^2*i^3*n + 12*a^2
*b^2*c*d^3*i^3*n - 19*a^3*b*d^4*i^3*n + (i^3*n - 12*i^3*log(e))*b^4*c^3*d)*B^2*x - 6*(A*B*b^4*c*d^3*i^3 + (3*a
*b^3*d^4*i^3*n - (3*i^3*n - 4*i^3*log(e))*b^4*c*d^3)*B^2)*x^3 + (2*a*b^3*c^3*d*i^3*n + 3*a^2*b^2*c^2*d^2*i^3*n
+ 6*a^3*b*c*d^3*i^3*n - 11*a^4*d^4*i^3*n - 6*b^4*c^4*i^3*log(e))*B^2 - 6*(B^2*b^4*d^4*i^3*n*x^4 + 4*B^2*a*b^3
*d^4*i^3*n*x^3 + 6*B^2*a^2*b^2*d^4*i^3*n*x^2 + 4*B^2*a^3*b*d^4*i^3*n*x + B^2*a^4*d^4*i^3*n)*log(b*x + a) - 6*(
B^2*b^4*d^4*i^3*x^4 + 4*B^2*b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B^2*b^4*c^
4*i^3)*log((b*x + a)^n))*log((d*x + c)^n))/(b^8*d*g^4*x^5 + a^4*b^4*c*g^4 + (b^8*c*g^4 + 4*a*b^7*d*g^4)*x^4 +
2*(2*a*b^7*c*g^4 + 3*a^2*b^6*d*g^4)*x^3 + 2*(3*a^2*b^6*c*g^4 + 2*a^3*b^5*d*g^4)*x^2 + (4*a^3*b^5*c*g^4 + a^4*b
^4*d*g^4)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A^{2} d^{3} i^{3} x^{3} + 3 \, A^{2} c d^{2} i^{3} x^{2} + 3 \, A^{2} c^{2} d i^{3} x + A^{2} c^{3} i^{3} +{\left (B^{2} d^{3} i^{3} x^{3} + 3 \, B^{2} c d^{2} i^{3} x^{2} + 3 \, B^{2} c^{2} d i^{3} x + B^{2} c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d^{3} i^{3} x^{3} + 3 \, A B c d^{2} i^{3} x^{2} + 3 \, A B c^{2} d i^{3} x + A B c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b^{4} g^{4} x^{4} + 4 \, a b^{3} g^{4} x^{3} + 6 \, a^{2} b^{2} g^{4} x^{2} + 4 \, a^{3} b g^{4} x + a^{4} g^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{3}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

integrate((d*i*x + c*i)^3*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(b*g*x + a*g)^4, x)