3.171 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^5} \, dx\)

Optimal. Leaf size=830 \[ -\frac{b^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^4}{4 (b c-a d)^4 (a+b x)^4}-\frac{3 b^3 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^4}{16 (b c-a d)^4 (a+b x)^4}-\frac{3 b^3 B^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^4}{32 (b c-a d)^4 (a+b x)^4}-\frac{3 b^3 B^3 n^3 (c+d x)^4}{128 (b c-a d)^4 (a+b x)^4}+\frac{b^2 d \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^3}{(b c-a d)^4 (a+b x)^3}+\frac{b^2 B d n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^3}{(b c-a d)^4 (a+b x)^3}+\frac{2 b^2 B^2 d n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^3}{3 (b c-a d)^4 (a+b x)^3}+\frac{2 b^2 B^3 d n^3 (c+d x)^3}{9 (b c-a d)^4 (a+b x)^3}-\frac{3 b d^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)^2}{2 (b c-a d)^4 (a+b x)^2}-\frac{9 b B d^2 n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)^2}{4 (b c-a d)^4 (a+b x)^2}-\frac{9 b B^2 d^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)^2}{4 (b c-a d)^4 (a+b x)^2}-\frac{9 b B^3 d^2 n^3 (c+d x)^2}{8 (b c-a d)^4 (a+b x)^2}+\frac{d^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 (c+d x)}{(b c-a d)^4 (a+b x)}+\frac{3 B d^3 n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 (c+d x)}{(b c-a d)^4 (a+b x)}+\frac{6 B^2 d^3 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (c+d x)}{(b c-a d)^4 (a+b x)}+\frac{6 B^3 d^3 n^3 (c+d x)}{(b c-a d)^4 (a+b x)} \]

[Out]

(6*B^3*d^3*n^3*(c + d*x))/((b*c - a*d)^4*(a + b*x)) - (9*b*B^3*d^2*n^3*(c + d*x)^2)/(8*(b*c - a*d)^4*(a + b*x)
^2) + (2*b^2*B^3*d*n^3*(c + d*x)^3)/(9*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^3*n^3*(c + d*x)^4)/(128*(b*c - a*
d)^4*(a + b*x)^4) + (6*B^2*d^3*n^2*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^4*(a + b*x
)) - (9*b*B^2*d^2*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(4*(b*c - a*d)^4*(a + b*x)^2) + (2
*b^2*B^2*d*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(3*(b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B^
2*n^2*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(32*(b*c - a*d)^4*(a + b*x)^4) + (3*B*d^3*n*(c + d
*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)) - (9*b*B*d^2*n*(c + d*x)^2*(A + B*Lo
g[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(4*(b*c - a*d)^4*(a + b*x)^2) + (b^2*B*d*n*(c + d*x)^3*(A + B*Log[(e*(a + b
*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)^4*(a + b*x)^3) - (3*b^3*B*n*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d
*x)^n])^2)/(16*(b*c - a*d)^4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c -
a*d)^4*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(2*(b*c - a*d)^4*(a + b*
x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^4*(a + b*x)^3) - (b^3*(c +
d*x)^4*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/(4*(b*c - a*d)^4*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 4.67497, antiderivative size = 2173, normalized size of antiderivative = 2.62, number of steps used = 93, number of rules used = 16, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.485, Rules used = {6742, 2492, 44, 2514, 2490, 32, 2488, 2411, 2343, 2333, 2315, 2491, 2509, 37, 2506, 6610} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]

[Out]

-A^3/(4*b*(a + b*x)^4) - (3*A^2*B*n)/(16*b*(a + b*x)^4) - (3*A*B^2*n^2)/(32*b*(a + b*x)^4) - (3*B^3*n^3)/(128*
b*(a + b*x)^4) + (A^2*B*d*n)/(4*b*(b*c - a*d)*(a + b*x)^3) + (7*A*B^2*d*n^2)/(24*b*(b*c - a*d)*(a + b*x)^3) +
(37*B^3*d*n^3)/(288*b*(b*c - a*d)*(a + b*x)^3) - (3*A^2*B*d^2*n)/(8*b*(b*c - a*d)^2*(a + b*x)^2) - (13*A*B^2*d
^2*n^2)/(16*b*(b*c - a*d)^2*(a + b*x)^2) - (79*B^3*d^2*n^3)/(192*b*(b*c - a*d)^2*(a + b*x)^2) + (3*A^2*B*d^3*n
)/(4*b*(b*c - a*d)^3*(a + b*x)) + (25*A*B^2*d^3*n^2)/(8*b*(b*c - a*d)^3*(a + b*x)) + (451*B^3*d^3*n^3)/(96*b*(
b*c - a*d)^3*(a + b*x)) - (3*b*B^3*d^2*n^3*(c + d*x)^2)/(16*(b*c - a*d)^4*(a + b*x)^2) + (3*A^2*B*d^4*n*Log[a
+ b*x])/(4*b*(b*c - a*d)^4) + (13*A*B^2*d^4*n^2*Log[a + b*x])/(8*b*(b*c - a*d)^4) + (79*B^3*d^4*n^3*Log[a + b*
x])/(96*b*(b*c - a*d)^4) - (3*A^2*B*d^4*n*Log[c + d*x])/(4*b*(b*c - a*d)^4) - (13*A*B^2*d^4*n^2*Log[c + d*x])/
(8*b*(b*c - a*d)^4) - (79*B^3*d^4*n^3*Log[c + d*x])/(96*b*(b*c - a*d)^4) - (3*A^2*B*Log[(e*(a + b*x)^n)/(c + d
*x)^n])/(4*b*(a + b*x)^4) - (3*A*B^2*n*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(8*b*(a + b*x)^4) - (3*B^3*n^2*Log[(e
*(a + b*x)^n)/(c + d*x)^n])/(32*b*(a + b*x)^4) + (A*B^2*d*n*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*(b*c - a*d)
*(a + b*x)^3) + (7*B^3*d*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(24*b*(b*c - a*d)*(a + b*x)^3) - (3*A*B^2*d^2*n
*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(4*b*(b*c - a*d)^2*(a + b*x)^2) - (7*B^3*d^2*n^2*Log[(e*(a + b*x)^n)/(c + d
*x)^n])/(16*b*(b*c - a*d)^2*(a + b*x)^2) + (3*A*B^2*d^3*n*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*(b*c
- a*d)^4*(a + b*x)) + (31*B^3*d^3*n^2*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(8*(b*c - a*d)^4*(a + b*x))
- (3*b*B^3*d^2*n^2*(c + d*x)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(8*(b*c - a*d)^4*(a + b*x)^2) - (3*A*B^2*d^4*
n*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*(b*c - a*d)^4) - (7*B^3*d^4*n^2*Log
[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(8*b*(b*c - a*d)^4) + (3*A*B^2*d^4*n*Log[(b*c
- a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(2*b*(b*c - a*d)^4) + (7*B^3*d^4*n^2*Log[(b*c - a*d)/
(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(8*b*(b*c - a*d)^4) - (3*A*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^
n]^2)/(4*b*(a + b*x)^4) - (3*B^3*n*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(16*b*(a + b*x)^4) + (B^3*d*n*Log[(e*(a
+ b*x)^n)/(c + d*x)^n]^2)/(4*b*(b*c - a*d)*(a + b*x)^3) + (3*B^3*d^3*n*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x
)^n]^2)/(2*(b*c - a*d)^4*(a + b*x)) - (3*b*B^3*d^2*n*(c + d*x)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(8*(b*c -
a*d)^4*(a + b*x)^2) - (3*B^3*d^4*n*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(4*b
*(b*c - a*d)^4) + (3*B^3*d^4*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(4*b*(b*c -
a*d)^4) - (B^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3)/(4*b*(a + b*x)^4) + (3*A*B^2*d^4*n^2*PolyLog[2, (d*(a + b*x
))/(b*(c + d*x))])/(2*b*(b*c - a*d)^4) + (7*B^3*d^4*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(8*b*(b*c - a
*d)^4) + (3*A*B^2*d^4*n^2*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(2*b*(b*c - a*d)^4) + (7*B^3*d^4*n^3*Poly
Log[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(8*b*(b*c - a*d)^4) + (3*B^3*d^4*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*P
olyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/(2*b*(b*c - a*d)^4) + (3*B^3*d^4*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n
]*PolyLog[2, 1 - (b*c - a*d)/(b*(c + d*x))])/(2*b*(b*c - a*d)^4) + (3*B^3*d^4*n^3*PolyLog[3, 1 + (b*c - a*d)/(
d*(a + b*x))])/(2*b*(b*c - a*d)^4) - (3*B^3*d^4*n^3*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/(2*b*(b*c - a*d
)^4)

Rule 6742

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

Rule 2492

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*((g_.) + (h_.)*(x_))^
(m_.), x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(h*(m + 1)), x] - Dist[(p*
r*s*(b*c - a*d))/(h*(m + 1)), Int[((g + h*x)^(m + 1)*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, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0]
&& IGtQ[s, 0] && 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 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 2490

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -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 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 2343

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

Rule 2333

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2491

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)/((g_.) + (h_.)*(x_))^3
, x_Symbol] :> Dist[d/(d*g - c*h), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(g + h*x)^2, x], x] - Dist[h/(d*
g - c*h), Int[((c + d*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(g + h*x)^3, 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] && NeQ[d*g - c*h, 0] && IG
tQ[s, 0]

Rule 2509

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^5} \, dx &=\int \left (\frac{A^3}{(a+b x)^5}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5}\right ) \, dx\\ &=-\frac{A^3}{4 b (a+b x)^4}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\left (3 A B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{\left (3 A^2 B (b c-a d) n\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b}+\frac{\left (3 A B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5 (c+d x)} \, dx}{4 b}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{\left (3 A^2 B (b c-a d) n\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b}+\frac{\left (3 A B^2 (b c-a d) n\right ) \int \left (\frac{b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^5}-\frac{b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \left (\frac{b \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^5}-\frac{b d \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{1}{2} \left (3 A B^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\frac{1}{4} \left (3 B^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\frac{\left (3 A B^2 d^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 (b c-a d)^4}+\frac{\left (3 B^3 d^4 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{4 (b c-a d)^4}-\frac{\left (3 A B^2 d^5 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}-\frac{\left (3 B^3 d^5 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{4 b (b c-a d)^4}-\frac{\left (3 A B^2 d^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac{\left (3 B^3 d^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{4 (b c-a d)^3}+\frac{\left (3 A B^2 d^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2}+\frac{\left (3 B^3 d^2 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{4 (b c-a d)^2}-\frac{\left (3 A B^2 d n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{2 (b c-a d)}-\frac{\left (3 B^3 d n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{4 (b c-a d)}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 (b c-a d)^4 (a+b x)}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{\left (3 b B^3 d^2 n\right ) \int \frac{(c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{4 (b c-a d)^3}-\frac{\left (3 B^3 d^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{4 (b c-a d)^3}-\frac{\left (A B^2 d n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{2 b}-\frac{\left (B^3 d n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4 (c+d x)} \, dx}{2 b}-\frac{\left (3 A B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac{\left (3 B^3 d^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac{\left (3 A B^2 d^4 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (3 A B^2 d^4 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}+\frac{\left (3 B^3 d^4 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}+\frac{\left (3 A B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)}+\frac{\left (3 A B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{8 b}+\frac{\left (3 B^3 (b c-a d) n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5 (c+d x)} \, dx}{8 b}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{3 A B^2 d^3 n^2}{2 b (b c-a d)^3 (a+b x)}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{3 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}-\frac{\left (A B^2 d 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}{2 b}-\frac{\left (B^3 d n^2\right ) \int \left (\frac{b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^4}-\frac{b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{2 b}+\frac{\left (3 b B^3 d^2 n^2\right ) \int \frac{(c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{4 (b c-a d)^3}-\frac{\left (3 A B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac{\left (3 A B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3}+\frac{\left (3 A B^2 d^2 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}{4 b (b c-a d)}+\frac{\left (3 A B^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b}+\frac{\left (3 B^3 (b c-a d) n^2\right ) \int \left (\frac{b \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d) (a+b x)^5}-\frac{b d \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b}-\frac{\left (3 B^3 d^3 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^3\right ) \int \frac{\text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{3 B^3 d^3 n^3}{2 b (b c-a d)^3 (a+b x)}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{3 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{1}{8} \left (3 B^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx+\frac{\left (3 B^3 d^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{8 (b c-a d)^4}+\frac{\left (B^3 d^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{2 (b c-a d)^4}-\frac{\left (3 B^3 d^5 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{8 b (b c-a d)^4}-\frac{\left (B^3 d^5 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{2 b (b c-a d)^4}+\frac{\left (3 A B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{8 (b c-a d)^3}-\frac{\left (B^3 d^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3}-\frac{\left (3 A B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}+\frac{\left (3 B^3 d^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{8 (b c-a d)^2}+\frac{\left (B^3 d^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2}-\frac{\left (3 B^3 d n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{8 (b c-a d)}-\frac{\left (B^3 d n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx}{2 (b c-a d)}+\frac{\left (3 b B^3 d^2 n^3\right ) \int \frac{c+d x}{(a+b x)^3} \, dx}{8 (b c-a d)^3}-\frac{\left (3 B^3 d^3 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{3 B^3 d^3 n^3}{b (b c-a d)^3 (a+b x)}-\frac{3 b B^3 d^2 n^3 (c+d x)^2}{16 (b c-a d)^4 (a+b x)^2}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{32 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}+\frac{7 B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{7 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{31 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{7 B^3 d^4 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{\left (3 A B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (3 A B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}-\frac{\left (B^3 d n^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{8 b}-\frac{\left (B^3 d n^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{6 b}-\frac{\left (3 B^3 d^3 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{8 (b c-a d)^3}-\frac{\left (B^3 d^3 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{2 (b c-a d)^3}+\frac{\left (3 B^3 d^4 n^3\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{8 b (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{8 b (b c-a d)^3}+\frac{\left (B^3 d^4 n^3\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}-\frac{\left (B^3 d^4 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{2 b (b c-a d)^3}+\frac{\left (3 B^3 d^2 n^3\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{16 b (b c-a d)}+\frac{\left (B^3 d^2 n^3\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)}+\frac{\left (3 B^3 (b c-a d) n^3\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{32 b}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{31 B^3 d^3 n^3}{8 b (b c-a d)^3 (a+b x)}-\frac{3 b B^3 d^2 n^3 (c+d x)^2}{16 (b c-a d)^4 (a+b x)^2}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{32 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}+\frac{7 B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{7 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{31 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{7 B^3 d^4 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}-\frac{\left (B^3 d n^3\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}{8 b}-\frac{\left (B^3 d n^3\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}{6 b}-\frac{\left (3 B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{8 b (b c-a d)^3}-\frac{\left (B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{-b c+a d}{b x}\right )}{x \left (\frac{-b c+a d}{d}+\frac{b x}{d}\right )} \, dx,x,c+d x\right )}{2 b (b c-a d)^3}+\frac{\left (3 B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{8 b^2 (b c-a d)^3}+\frac{\left (B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b c-a d}{d x}\right )}{x \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )} \, dx,x,a+b x\right )}{2 b^2 (b c-a d)^3}+\frac{\left (3 B^3 d^2 n^3\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}{16 b (b c-a d)}+\frac{\left (B^3 d^2 n^3\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}{4 b (b c-a d)}+\frac{\left (3 B^3 (b c-a d) n^3\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{32 b}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}-\frac{3 B^3 n^3}{128 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}+\frac{37 B^3 d n^3}{288 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{79 B^3 d^2 n^3}{192 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{451 B^3 d^3 n^3}{96 b (b c-a d)^3 (a+b x)}-\frac{3 b B^3 d^2 n^3 (c+d x)^2}{16 (b c-a d)^4 (a+b x)^2}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{79 B^3 d^4 n^3 \log (a+b x)}{96 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{79 B^3 d^4 n^3 \log (c+d x)}{96 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{32 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}+\frac{7 B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{7 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{31 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{7 B^3 d^4 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{\left (3 B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{8 b (b c-a d)^3}+\frac{\left (B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\left (\frac{-b c+a d}{d}+\frac{b}{d x}\right ) x} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{8 b^2 (b c-a d)^3}-\frac{\left (B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\left (\frac{b c-a d}{b}+\frac{d}{b x}\right ) x} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}-\frac{3 B^3 n^3}{128 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}+\frac{37 B^3 d n^3}{288 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{79 B^3 d^2 n^3}{192 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{451 B^3 d^3 n^3}{96 b (b c-a d)^3 (a+b x)}-\frac{3 b B^3 d^2 n^3 (c+d x)^2}{16 (b c-a d)^4 (a+b x)^2}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{79 B^3 d^4 n^3 \log (a+b x)}{96 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{79 B^3 d^4 n^3 \log (c+d x)}{96 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{32 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}+\frac{7 B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{7 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{31 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{7 B^3 d^4 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{\left (3 B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{8 b (b c-a d)^3}+\frac{\left (B^3 d^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b}\right )}{\frac{b}{d}+\frac{(-b c+a d) x}{d}} \, dx,x,\frac{1}{c+d x}\right )}{2 b (b c-a d)^3}-\frac{\left (3 B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{8 b^2 (b c-a d)^3}-\frac{\left (B^3 d^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(b c-a d) x}{d}\right )}{\frac{d}{b}+\frac{(b c-a d) x}{b}} \, dx,x,\frac{1}{a+b x}\right )}{2 b^2 (b c-a d)^3}\\ &=-\frac{A^3}{4 b (a+b x)^4}-\frac{3 A^2 B n}{16 b (a+b x)^4}-\frac{3 A B^2 n^2}{32 b (a+b x)^4}-\frac{3 B^3 n^3}{128 b (a+b x)^4}+\frac{A^2 B d n}{4 b (b c-a d) (a+b x)^3}+\frac{7 A B^2 d n^2}{24 b (b c-a d) (a+b x)^3}+\frac{37 B^3 d n^3}{288 b (b c-a d) (a+b x)^3}-\frac{3 A^2 B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}-\frac{13 A B^2 d^2 n^2}{16 b (b c-a d)^2 (a+b x)^2}-\frac{79 B^3 d^2 n^3}{192 b (b c-a d)^2 (a+b x)^2}+\frac{3 A^2 B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac{25 A B^2 d^3 n^2}{8 b (b c-a d)^3 (a+b x)}+\frac{451 B^3 d^3 n^3}{96 b (b c-a d)^3 (a+b x)}-\frac{3 b B^3 d^2 n^3 (c+d x)^2}{16 (b c-a d)^4 (a+b x)^2}+\frac{3 A^2 B d^4 n \log (a+b x)}{4 b (b c-a d)^4}+\frac{13 A B^2 d^4 n^2 \log (a+b x)}{8 b (b c-a d)^4}+\frac{79 B^3 d^4 n^3 \log (a+b x)}{96 b (b c-a d)^4}-\frac{3 A^2 B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac{13 A B^2 d^4 n^2 \log (c+d x)}{8 b (b c-a d)^4}-\frac{79 B^3 d^4 n^3 \log (c+d x)}{96 b (b c-a d)^4}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (a+b x)^4}-\frac{3 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{32 b (a+b x)^4}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d) (a+b x)^3}+\frac{7 B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{24 b (b c-a d) (a+b x)^3}-\frac{3 A B^2 d^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^2 (a+b x)^2}-\frac{7 B^3 d^2 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (b c-a d)^2 (a+b x)^2}+\frac{3 A B^2 d^3 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}+\frac{31 B^3 d^3 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 A B^2 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}-\frac{7 B^3 d^4 n^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{8 b (b c-a d)^4}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}-\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{16 b (a+b x)^4}+\frac{B^3 d n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d) (a+b x)^3}+\frac{3 B^3 d^3 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^4 (a+b x)}-\frac{3 b B^3 d^2 n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{8 (b c-a d)^4 (a+b x)^2}-\frac{3 B^3 d^4 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}+\frac{3 B^3 d^4 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (b c-a d)^4}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{8 b (b c-a d)^4}+\frac{3 A B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{7 B^3 d^4 n^3 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{8 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}+\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{2 b (b c-a d)^4}-\frac{3 B^3 d^4 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{2 b (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 2.251, size = 1370, normalized size = 1.65 \[ -\frac{-288 B^3 d^4 n^3 \log ^3(a+b x) (a+b x)^4+288 B^3 d^4 n^3 \log ^3(c+d x) (a+b x)^4+72 B^2 d^4 n^2 \log ^2(c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^4+72 B^2 d^4 n^2 \log ^2(a+b x) \left (12 A+25 B n+12 B n \log (c+d x)+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^4+12 B d^4 n \log (c+d x) \left (72 A^2+300 B n A+415 B^2 n^2+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^4-12 B d^4 n \log (a+b x) \left (72 A^2+300 B n A+415 B^2 n^2+72 B^2 n^2 \log ^2(c+d x)+72 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+12 B (12 A+25 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+12 B n \log (c+d x) \left (12 A+25 B n+12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (a+b x)^4+(b c-a d) \left (288 b^3 c^3 A^3-288 a^3 d^3 A^3+864 a^2 b c d^2 A^3-864 a b^2 c^2 d A^3-864 b^3 B d^3 n x^3 A^2-3024 a b^2 B d^3 n x^2 A^2+432 b^3 B c d^2 n x^2 A^2+216 b^3 B c^3 n A^2-1800 a^3 B d^3 n A^2+1656 a^2 b B c d^2 n A^2-936 a b^2 B c^2 d n A^2-3744 a^2 b B d^3 n x A^2+1440 a b^2 B c d^2 n x A^2-288 b^3 B c^2 d n x A^2-3600 b^3 B^2 d^3 n^2 x^3 A+108 b^3 B^2 c^3 n^2 A-4980 a^3 B^2 d^3 n^2 A+1932 a^2 b B^2 c d^2 n^2 A-660 a b^2 B^2 c^2 d n^2 A-11736 a b^2 B^2 d^3 n^2 x^2 A+936 b^3 B^2 c d^2 n^2 x^2 A-13008 a^2 b B^2 d^3 n^2 x A+2544 a b^2 B^2 c d^2 n^2 x A-336 b^3 B^2 c^2 d n^2 x A+27 b^3 B^3 c^3 n^3-5845 a^3 B^3 d^3 n^3+1067 a^2 b B^3 c d^2 n^3-229 a b^2 B^3 c^2 d n^3-4980 b^3 B^3 d^3 n^3 x^3+288 B^3 (b c-a d)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )-15630 a b^2 B^3 d^3 n^3 x^2+690 b^3 B^3 c d^2 n^3 x^2+72 B^2 \left (12 A (b c-a d)^3+B n \left (\left (3 c^3-4 d x c^2+6 d^2 x^2 c-12 d^3 x^3\right ) b^3+a d \left (-13 c^2+20 d x c-42 d^2 x^2\right ) b^2+a^2 d^2 (23 c-52 d x) b-25 a^3 d^3\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )-16468 a^2 b B^3 d^3 n^3 x+1676 a b^2 B^3 c d^2 n^3 x-148 b^3 B^3 c^2 d n^3 x+12 B \left (72 A^2 (b c-a d)^3+B^2 n^2 \left (\left (9 c^3-28 d x c^2+78 d^2 x^2 c-300 d^3 x^3\right ) b^3+a d \left (-55 c^2+212 d x c-978 d^2 x^2\right ) b^2+a^2 d^2 (161 c-1084 d x) b-415 a^3 d^3\right )+12 A B n \left (\left (3 c^3-4 d x c^2+6 d^2 x^2 c-12 d^3 x^3\right ) b^3+a d \left (-13 c^2+20 d x c-42 d^2 x^2\right ) b^2+a^2 d^2 (23 c-52 d x) b-25 a^3 d^3\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{1152 b (b c-a d)^4 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(a + b*x)^5,x]

[Out]

-(-288*B^3*d^4*n^3*(a + b*x)^4*Log[a + b*x]^3 + 288*B^3*d^4*n^3*(a + b*x)^4*Log[c + d*x]^3 + 72*B^2*d^4*n^2*(a
+ b*x)^4*Log[c + d*x]^2*(12*A + 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 72*B^2*d^4*n^2*(a + b*x)^4*
Log[a + b*x]^2*(12*A + 25*B*n + 12*B*n*Log[c + d*x] + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 12*B*d^4*n*(a +
b*x)^4*Log[c + d*x]*(72*A^2 + 300*A*B*n + 415*B^2*n^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n]
+ 72*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2) + (b*c - a*d)*(288*A^3*b^3*c^3 - 864*a*A^3*b^2*c^2*d + 864*a^2*A
^3*b*c*d^2 - 288*a^3*A^3*d^3 + 216*A^2*b^3*B*c^3*n - 936*a*A^2*b^2*B*c^2*d*n + 1656*a^2*A^2*b*B*c*d^2*n - 1800
*a^3*A^2*B*d^3*n + 108*A*b^3*B^2*c^3*n^2 - 660*a*A*b^2*B^2*c^2*d*n^2 + 1932*a^2*A*b*B^2*c*d^2*n^2 - 4980*a^3*A
*B^2*d^3*n^2 + 27*b^3*B^3*c^3*n^3 - 229*a*b^2*B^3*c^2*d*n^3 + 1067*a^2*b*B^3*c*d^2*n^3 - 5845*a^3*B^3*d^3*n^3
- 288*A^2*b^3*B*c^2*d*n*x + 1440*a*A^2*b^2*B*c*d^2*n*x - 3744*a^2*A^2*b*B*d^3*n*x - 336*A*b^3*B^2*c^2*d*n^2*x
+ 2544*a*A*b^2*B^2*c*d^2*n^2*x - 13008*a^2*A*b*B^2*d^3*n^2*x - 148*b^3*B^3*c^2*d*n^3*x + 1676*a*b^2*B^3*c*d^2*
n^3*x - 16468*a^2*b*B^3*d^3*n^3*x + 432*A^2*b^3*B*c*d^2*n*x^2 - 3024*a*A^2*b^2*B*d^3*n*x^2 + 936*A*b^3*B^2*c*d
^2*n^2*x^2 - 11736*a*A*b^2*B^2*d^3*n^2*x^2 + 690*b^3*B^3*c*d^2*n^3*x^2 - 15630*a*b^2*B^3*d^3*n^3*x^2 - 864*A^2
*b^3*B*d^3*n*x^3 - 3600*A*b^3*B^2*d^3*n^2*x^3 - 4980*b^3*B^3*d^3*n^3*x^3 + 12*B*(72*A^2*(b*c - a*d)^3 + B^2*n^
2*(-415*a^3*d^3 + a^2*b*d^2*(161*c - 1084*d*x) + a*b^2*d*(-55*c^2 + 212*c*d*x - 978*d^2*x^2) + b^3*(9*c^3 - 28
*c^2*d*x + 78*c*d^2*x^2 - 300*d^3*x^3)) + 12*A*B*n*(-25*a^3*d^3 + a^2*b*d^2*(23*c - 52*d*x) + a*b^2*d*(-13*c^2
+ 20*c*d*x - 42*d^2*x^2) + b^3*(3*c^3 - 4*c^2*d*x + 6*c*d^2*x^2 - 12*d^3*x^3)))*Log[(e*(a + b*x)^n)/(c + d*x)
^n] + 72*B^2*(12*A*(b*c - a*d)^3 + B*n*(-25*a^3*d^3 + a^2*b*d^2*(23*c - 52*d*x) + a*b^2*d*(-13*c^2 + 20*c*d*x
- 42*d^2*x^2) + b^3*(3*c^3 - 4*c^2*d*x + 6*c*d^2*x^2 - 12*d^3*x^3)))*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 288*
B^3*(b*c - a*d)^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 12*B*d^4*n*(a + b*x)^4*Log[a + b*x]*(72*A^2 + 300*A*B*
n + 415*B^2*n^2 + 72*B^2*n^2*Log[c + d*x]^2 + 12*B*(12*A + 25*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 72*B^2*L
og[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 12*B*n*Log[c + d*x]*(12*A + 25*B*n + 12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]
)))/(1152*b*(b*c - a*d)^4*(a + b*x)^4)

________________________________________________________________________________________

Maple [C]  time = 33.085, size = 236754, normalized size = 285.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 3.30885, size = 7128, normalized size = 8.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/4*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/11
52*(72*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) -
12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*
d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*
a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)*x)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*
d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3
- 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 -
a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*log((b*x + a)^n*e/(d
*x + c)^n)^2/e - (12*(9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c^2*d^2*e^2*n^2 - 576*a^3*b*c*d
^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c*d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x^3 + 6*(13*b^4*c^2*d^2*e^2*n^2
- 176*a*b^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3
+ 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a)^2 + 72*(b^4*d^4*e^2*n^2*x^
4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(d*x + c
)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2*e^2*n^2 + 324*a^2*b^2*c*d^3*e^2*n^2 - 271*a^3*b*d^4*e^2*n^2)*x
- 300*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^
4*d^4*e^2*n^2)*log(b*x + a) + 12*(25*b^4*d^4*e^2*n^2*x^4 + 100*a*b^3*d^4*e^2*n^2*x^3 + 150*a^2*b^2*d^4*e^2*n^2
*x^2 + 100*a^3*b*d^4*e^2*n^2*x + 25*a^4*d^4*e^2*n^2 - 12*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^
2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e
/(d*x + c)^n)/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4
*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3
*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a
^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7
*b^2*d^4)*x)*e) + (27*b^4*c^4*e^3*n^3 - 256*a*b^3*c^3*d*e^3*n^3 + 1296*a^2*b^2*c^2*d^2*e^3*n^3 - 6912*a^3*b*c*
d^3*e^3*n^3 + 5845*a^4*d^4*e^3*n^3 - 4980*(b^4*c*d^3*e^3*n^3 - a*b^3*d^4*e^3*n^3)*x^3 - 288*(b^4*d^4*e^3*n^3*x
^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x +
a)^3 + 288*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x
+ a^4*d^4*e^3*n^3)*log(d*x + c)^3 + 30*(23*b^4*c^2*d^2*e^3*n^3 - 544*a*b^3*c*d^3*e^3*n^3 + 521*a^2*b^2*d^4*e^3
*n^3)*x^2 + 1800*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*
n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a)^2 + 72*(25*b^4*d^4*e^3*n^3*x^4 + 100*a*b^3*d^4*e^3*n^3*x^3 + 150*a^2*b^2
*d^4*e^3*n^3*x^2 + 100*a^3*b*d^4*e^3*n^3*x + 25*a^4*d^4*e^3*n^3 - 12*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^
3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 - 4*
(37*b^4*c^3*d*e^3*n^3 - 456*a*b^3*c^2*d^2*e^3*n^3 + 4536*a^2*b^2*c*d^3*e^3*n^3 - 4117*a^3*b*d^4*e^3*n^3)*x - 4
980*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d
^4*e^3*n^3)*log(b*x + a) + 12*(415*b^4*d^4*e^3*n^3*x^4 + 1660*a*b^3*d^4*e^3*n^3*x^3 + 2490*a^2*b^2*d^4*e^3*n^3
*x^2 + 1660*a^3*b*d^4*e^3*n^3*x + 415*a^4*d^4*e^3*n^3 + 72*(b^4*d^4*e^3*n^3*x^4 + 4*a*b^3*d^4*e^3*n^3*x^3 + 6*
a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a)^2 - 300*(b^4*d^4*e^3*n^3*x^4 +
4*a*b^3*d^4*e^3*n^3*x^3 + 6*a^2*b^2*d^4*e^3*n^3*x^2 + 4*a^3*b*d^4*e^3*n^3*x + a^4*d^4*e^3*n^3)*log(b*x + a))*
log(d*x + c))/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4
*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3
*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a
^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7
*b^2*d^4)*x)*e^2))/e)*B^3 + 1/96*A*B^2*(12*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d
^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a
^3*b^2*c*d^3 + a^4*b*d^4) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25
*a^3*d^3*e*n - 6*(b^3*c*d^2*e*n - 7*a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e
*n)*x)/(a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d
^2 - a^3*b^5*d^3)*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a
^6*b^2*d^3)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e - (9*b^4*c^4*e^2*n^2 - 64*a*b^3*c^3*d*e^2*n^2 + 216*a^2*b^2*c
^2*d^2*e^2*n^2 - 576*a^3*b*c*d^3*e^2*n^2 + 415*a^4*d^4*e^2*n^2 - 300*(b^4*c*d^3*e^2*n^2 - a*b^3*d^4*e^2*n^2)*x
^3 + 6*(13*b^4*c^2*d^2*e^2*n^2 - 176*a*b^3*c*d^3*e^2*n^2 + 163*a^2*b^2*d^4*e^2*n^2)*x^2 + 72*(b^4*d^4*e^2*n^2*
x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x +
a)^2 + 72*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x
+ a^4*d^4*e^2*n^2)*log(d*x + c)^2 - 4*(7*b^4*c^3*d*e^2*n^2 - 60*a*b^3*c^2*d^2*e^2*n^2 + 324*a^2*b^2*c*d^3*e^2*
n^2 - 271*a^3*b*d^4*e^2*n^2)*x - 300*(b^4*d^4*e^2*n^2*x^4 + 4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^
2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a) + 12*(25*b^4*d^4*e^2*n^2*x^4 + 100*a*b^3*d^4*e^2*n^2
*x^3 + 150*a^2*b^2*d^4*e^2*n^2*x^2 + 100*a^3*b*d^4*e^2*n^2*x + 25*a^4*d^4*e^2*n^2 - 12*(b^4*d^4*e^2*n^2*x^4 +
4*a*b^3*d^4*e^2*n^2*x^3 + 6*a^2*b^2*d^4*e^2*n^2*x^2 + 4*a^3*b*d^4*e^2*n^2*x + a^4*d^4*e^2*n^2)*log(b*x + a))*l
og(d*x + c))/((a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4*
a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*
b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^
5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*
b^2*d^4)*x)*e^2)) - 3/4*A*B^2*log((b*x + a)^n*e/(d*x + c)^n)^2/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*
b^2*x + a^4*b) + 1/16*(12*d^4*e*n*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3
+ a^4*b*d^4) - 12*d^4*e*n*log(d*x + c)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*
d^4) + (12*b^3*d^3*e*n*x^3 - 3*b^3*c^3*e*n + 13*a*b^2*c^2*d*e*n - 23*a^2*b*c*d^2*e*n + 25*a^3*d^3*e*n - 6*(b^3
*c*d^2*e*n - 7*a*b^2*d^3*e*n)*x^2 + 4*(b^3*c^2*d*e*n - 5*a*b^2*c*d^2*e*n + 13*a^2*b*d^3*e*n)*x)/(a^4*b^4*c^3 -
3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3 + (b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*x^4
+ 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*
a^4*b^4*c*d^2 - a^5*b^3*d^3)*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*x))*A^2*B
/e - 3/4*A^2*B*log((b*x + a)^n*e/(d*x + c)^n)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) -
1/4*A^3/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b)

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Fricas [B]  time = 2.45033, size = 12382, normalized size = 14.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/1152*(288*A^3*b^4*c^4 - 1152*A^3*a*b^3*c^3*d + 1728*A^3*a^2*b^2*c^2*d^2 - 1152*A^3*a^3*b*c*d^3 + 288*A^3*a^
4*d^4 + (27*B^3*b^4*c^4 - 256*B^3*a*b^3*c^3*d + 1296*B^3*a^2*b^2*c^2*d^2 - 6912*B^3*a^3*b*c*d^3 + 5845*B^3*a^4
*d^4)*n^3 - 12*(415*(B^3*b^4*c*d^3 - B^3*a*b^3*d^4)*n^3 + 300*(A*B^2*b^4*c*d^3 - A*B^2*a*b^3*d^4)*n^2 + 72*(A^
2*B*b^4*c*d^3 - A^2*B*a*b^3*d^4)*n)*x^3 - 288*(B^3*b^4*d^4*n^3*x^4 + 4*B^3*a*b^3*d^4*n^3*x^3 + 6*B^3*a^2*b^2*d
^4*n^3*x^2 + 4*B^3*a^3*b*d^4*n^3*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*
d^3)*n^3)*log(b*x + a)^3 + 288*(B^3*b^4*d^4*n^3*x^4 + 4*B^3*a*b^3*d^4*n^3*x^3 + 6*B^3*a^2*b^2*d^4*n^3*x^2 + 4*
B^3*a^3*b*d^4*n^3*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n^3)*log(d
*x + c)^3 + 288*(B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3 + B^3*a^4*d^4)*lo
g(e)^3 + 12*(9*A*B^2*b^4*c^4 - 64*A*B^2*a*b^3*c^3*d + 216*A*B^2*a^2*b^2*c^2*d^2 - 576*A*B^2*a^3*b*c*d^3 + 415*
A*B^2*a^4*d^4)*n^2 + 6*(5*(23*B^3*b^4*c^2*d^2 - 544*B^3*a*b^3*c*d^3 + 521*B^3*a^2*b^2*d^4)*n^3 + 12*(13*A*B^2*
b^4*c^2*d^2 - 176*A*B^2*a*b^3*c*d^3 + 163*A*B^2*a^2*b^2*d^4)*n^2 + 72*(A^2*B*b^4*c^2*d^2 - 8*A^2*B*a*b^3*c*d^3
+ 7*A^2*B*a^2*b^2*d^4)*n)*x^2 - 72*((25*B^3*b^4*d^4*n^3 + 12*A*B^2*b^4*d^4*n^2)*x^4 - (3*B^3*b^4*c^4 - 16*B^3
*a*b^3*c^3*d + 36*B^3*a^2*b^2*c^2*d^2 - 48*B^3*a^3*b*c*d^3)*n^3 + 4*(12*A*B^2*a*b^3*d^4*n^2 + (3*B^3*b^4*c*d^3
+ 22*B^3*a*b^3*d^4)*n^3)*x^3 - 12*(A*B^2*b^4*c^4 - 4*A*B^2*a*b^3*c^3*d + 6*A*B^2*a^2*b^2*c^2*d^2 - 4*A*B^2*a^
3*b*c*d^3)*n^2 + 6*(12*A*B^2*a^2*b^2*d^4*n^2 - (B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*d^3 - 18*B^3*a^2*b^2*d^4)*n^3)
*x^2 + 4*(12*A*B^2*a^3*b*d^4*n^2 + (B^3*b^4*c^3*d - 6*B^3*a*b^3*c^2*d^2 + 18*B^3*a^2*b^2*c*d^3 + 12*B^3*a^3*b*
d^4)*n^3)*x + 12*(B^3*b^4*d^4*n^2*x^4 + 4*B^3*a*b^3*d^4*n^2*x^3 + 6*B^3*a^2*b^2*d^4*n^2*x^2 + 4*B^3*a^3*b*d^4*
n^2*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n^2)*log(e))*log(b*x + a
)^2 - 72*((25*B^3*b^4*d^4*n^3 + 12*A*B^2*b^4*d^4*n^2)*x^4 - (3*B^3*b^4*c^4 - 16*B^3*a*b^3*c^3*d + 36*B^3*a^2*b
^2*c^2*d^2 - 48*B^3*a^3*b*c*d^3)*n^3 + 4*(12*A*B^2*a*b^3*d^4*n^2 + (3*B^3*b^4*c*d^3 + 22*B^3*a*b^3*d^4)*n^3)*x
^3 - 12*(A*B^2*b^4*c^4 - 4*A*B^2*a*b^3*c^3*d + 6*A*B^2*a^2*b^2*c^2*d^2 - 4*A*B^2*a^3*b*c*d^3)*n^2 + 6*(12*A*B^
2*a^2*b^2*d^4*n^2 - (B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*d^3 - 18*B^3*a^2*b^2*d^4)*n^3)*x^2 + 4*(12*A*B^2*a^3*b*d^
4*n^2 + (B^3*b^4*c^3*d - 6*B^3*a*b^3*c^2*d^2 + 18*B^3*a^2*b^2*c*d^3 + 12*B^3*a^3*b*d^4)*n^3)*x + 12*(B^3*b^4*d
^4*n^3*x^4 + 4*B^3*a*b^3*d^4*n^3*x^3 + 6*B^3*a^2*b^2*d^4*n^3*x^2 + 4*B^3*a^3*b*d^4*n^3*x - (B^3*b^4*c^4 - 4*B^
3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n^3)*log(b*x + a) + 12*(B^3*b^4*d^4*n^2*x^4 + 4*B^3
*a*b^3*d^4*n^2*x^3 + 6*B^3*a^2*b^2*d^4*n^2*x^2 + 4*B^3*a^3*b*d^4*n^2*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*
B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n^2)*log(e))*log(d*x + c)^2 + 72*(12*A*B^2*b^4*c^4 - 48*A*B^2*a*b^3*c
^3*d + 72*A*B^2*a^2*b^2*c^2*d^2 - 48*A*B^2*a^3*b*c*d^3 + 12*A*B^2*a^4*d^4 - 12*(B^3*b^4*c*d^3 - B^3*a*b^3*d^4)
*n*x^3 + 6*(B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*d^3 + 7*B^3*a^2*b^2*d^4)*n*x^2 - 4*(B^3*b^4*c^3*d - 6*B^3*a*b^3*c^
2*d^2 + 18*B^3*a^2*b^2*c*d^3 - 13*B^3*a^3*b*d^4)*n*x + (3*B^3*b^4*c^4 - 16*B^3*a*b^3*c^3*d + 36*B^3*a^2*b^2*c^
2*d^2 - 48*B^3*a^3*b*c*d^3 + 25*B^3*a^4*d^4)*n)*log(e)^2 + 72*(3*A^2*B*b^4*c^4 - 16*A^2*B*a*b^3*c^3*d + 36*A^2
*B*a^2*b^2*c^2*d^2 - 48*A^2*B*a^3*b*c*d^3 + 25*A^2*B*a^4*d^4)*n - 4*((37*B^3*b^4*c^3*d - 456*B^3*a*b^3*c^2*d^2
+ 4536*B^3*a^2*b^2*c*d^3 - 4117*B^3*a^3*b*d^4)*n^3 + 12*(7*A*B^2*b^4*c^3*d - 60*A*B^2*a*b^3*c^2*d^2 + 324*A*B
^2*a^2*b^2*c*d^3 - 271*A*B^2*a^3*b*d^4)*n^2 + 72*(A^2*B*b^4*c^3*d - 6*A^2*B*a*b^3*c^2*d^2 + 18*A^2*B*a^2*b^2*c
*d^3 - 13*A^2*B*a^3*b*d^4)*n)*x - 12*((415*B^3*b^4*d^4*n^3 + 300*A*B^2*b^4*d^4*n^2 + 72*A^2*B*b^4*d^4*n)*x^4 -
(9*B^3*b^4*c^4 - 64*B^3*a*b^3*c^3*d + 216*B^3*a^2*b^2*c^2*d^2 - 576*B^3*a^3*b*c*d^3)*n^3 + 4*(72*A^2*B*a*b^3*
d^4*n + 5*(15*B^3*b^4*c*d^3 + 68*B^3*a*b^3*d^4)*n^3 + 12*(3*A*B^2*b^4*c*d^3 + 22*A*B^2*a*b^3*d^4)*n^2)*x^3 - 1
2*(3*A*B^2*b^4*c^4 - 16*A*B^2*a*b^3*c^3*d + 36*A*B^2*a^2*b^2*c^2*d^2 - 48*A*B^2*a^3*b*c*d^3)*n^2 + 6*(72*A^2*B
*a^2*b^2*d^4*n - (13*B^3*b^4*c^2*d^2 - 176*B^3*a*b^3*c*d^3 - 252*B^3*a^2*b^2*d^4)*n^3 - 12*(A*B^2*b^4*c^2*d^2
- 8*A*B^2*a*b^3*c*d^3 - 18*A*B^2*a^2*b^2*d^4)*n^2)*x^2 + 72*(B^3*b^4*d^4*n*x^4 + 4*B^3*a*b^3*d^4*n*x^3 + 6*B^3
*a^2*b^2*d^4*n*x^2 + 4*B^3*a^3*b*d^4*n*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^
3*b*c*d^3)*n)*log(e)^2 - 72*(A^2*B*b^4*c^4 - 4*A^2*B*a*b^3*c^3*d + 6*A^2*B*a^2*b^2*c^2*d^2 - 4*A^2*B*a^3*b*c*d
^3)*n + 4*(72*A^2*B*a^3*b*d^4*n + (7*B^3*b^4*c^3*d - 60*B^3*a*b^3*c^2*d^2 + 324*B^3*a^2*b^2*c*d^3 + 144*B^3*a^
3*b*d^4)*n^3 + 12*(A*B^2*b^4*c^3*d - 6*A*B^2*a*b^3*c^2*d^2 + 18*A*B^2*a^2*b^2*c*d^3 + 12*A*B^2*a^3*b*d^4)*n^2)
*x + 12*((25*B^3*b^4*d^4*n^2 + 12*A*B^2*b^4*d^4*n)*x^4 + 4*(12*A*B^2*a*b^3*d^4*n + (3*B^3*b^4*c*d^3 + 22*B^3*a
*b^3*d^4)*n^2)*x^3 - (3*B^3*b^4*c^4 - 16*B^3*a*b^3*c^3*d + 36*B^3*a^2*b^2*c^2*d^2 - 48*B^3*a^3*b*c*d^3)*n^2 +
6*(12*A*B^2*a^2*b^2*d^4*n - (B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*d^3 - 18*B^3*a^2*b^2*d^4)*n^2)*x^2 - 12*(A*B^2*b^
4*c^4 - 4*A*B^2*a*b^3*c^3*d + 6*A*B^2*a^2*b^2*c^2*d^2 - 4*A*B^2*a^3*b*c*d^3)*n + 4*(12*A*B^2*a^3*b*d^4*n + (B^
3*b^4*c^3*d - 6*B^3*a*b^3*c^2*d^2 + 18*B^3*a^2*b^2*c*d^3 + 12*B^3*a^3*b*d^4)*n^2)*x)*log(e))*log(b*x + a) + 12
*((415*B^3*b^4*d^4*n^3 + 300*A*B^2*b^4*d^4*n^2 + 72*A^2*B*b^4*d^4*n)*x^4 - (9*B^3*b^4*c^4 - 64*B^3*a*b^3*c^3*d
+ 216*B^3*a^2*b^2*c^2*d^2 - 576*B^3*a^3*b*c*d^3)*n^3 + 4*(72*A^2*B*a*b^3*d^4*n + 5*(15*B^3*b^4*c*d^3 + 68*B^3
*a*b^3*d^4)*n^3 + 12*(3*A*B^2*b^4*c*d^3 + 22*A*B^2*a*b^3*d^4)*n^2)*x^3 - 12*(3*A*B^2*b^4*c^4 - 16*A*B^2*a*b^3*
c^3*d + 36*A*B^2*a^2*b^2*c^2*d^2 - 48*A*B^2*a^3*b*c*d^3)*n^2 + 6*(72*A^2*B*a^2*b^2*d^4*n - (13*B^3*b^4*c^2*d^2
- 176*B^3*a*b^3*c*d^3 - 252*B^3*a^2*b^2*d^4)*n^3 - 12*(A*B^2*b^4*c^2*d^2 - 8*A*B^2*a*b^3*c*d^3 - 18*A*B^2*a^2
*b^2*d^4)*n^2)*x^2 + 72*(B^3*b^4*d^4*n^3*x^4 + 4*B^3*a*b^3*d^4*n^3*x^3 + 6*B^3*a^2*b^2*d^4*n^3*x^2 + 4*B^3*a^3
*b*d^4*n^3*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n^3)*log(b*x + a)
^2 + 72*(B^3*b^4*d^4*n*x^4 + 4*B^3*a*b^3*d^4*n*x^3 + 6*B^3*a^2*b^2*d^4*n*x^2 + 4*B^3*a^3*b*d^4*n*x - (B^3*b^4*
c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c*d^3)*n)*log(e)^2 - 72*(A^2*B*b^4*c^4 - 4*A^2*B
*a*b^3*c^3*d + 6*A^2*B*a^2*b^2*c^2*d^2 - 4*A^2*B*a^3*b*c*d^3)*n + 4*(72*A^2*B*a^3*b*d^4*n + (7*B^3*b^4*c^3*d -
60*B^3*a*b^3*c^2*d^2 + 324*B^3*a^2*b^2*c*d^3 + 144*B^3*a^3*b*d^4)*n^3 + 12*(A*B^2*b^4*c^3*d - 6*A*B^2*a*b^3*c
^2*d^2 + 18*A*B^2*a^2*b^2*c*d^3 + 12*A*B^2*a^3*b*d^4)*n^2)*x + 12*((25*B^3*b^4*d^4*n^3 + 12*A*B^2*b^4*d^4*n^2)
*x^4 - (3*B^3*b^4*c^4 - 16*B^3*a*b^3*c^3*d + 36*B^3*a^2*b^2*c^2*d^2 - 48*B^3*a^3*b*c*d^3)*n^3 + 4*(12*A*B^2*a*
b^3*d^4*n^2 + (3*B^3*b^4*c*d^3 + 22*B^3*a*b^3*d^4)*n^3)*x^3 - 12*(A*B^2*b^4*c^4 - 4*A*B^2*a*b^3*c^3*d + 6*A*B^
2*a^2*b^2*c^2*d^2 - 4*A*B^2*a^3*b*c*d^3)*n^2 + 6*(12*A*B^2*a^2*b^2*d^4*n^2 - (B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*
d^3 - 18*B^3*a^2*b^2*d^4)*n^3)*x^2 + 4*(12*A*B^2*a^3*b*d^4*n^2 + (B^3*b^4*c^3*d - 6*B^3*a*b^3*c^2*d^2 + 18*B^3
*a^2*b^2*c*d^3 + 12*B^3*a^3*b*d^4)*n^3)*x + 12*(B^3*b^4*d^4*n^2*x^4 + 4*B^3*a*b^3*d^4*n^2*x^3 + 6*B^3*a^2*b^2*
d^4*n^2*x^2 + 4*B^3*a^3*b*d^4*n^2*x - (B^3*b^4*c^4 - 4*B^3*a*b^3*c^3*d + 6*B^3*a^2*b^2*c^2*d^2 - 4*B^3*a^3*b*c
*d^3)*n^2)*log(e))*log(b*x + a) + 12*((25*B^3*b^4*d^4*n^2 + 12*A*B^2*b^4*d^4*n)*x^4 + 4*(12*A*B^2*a*b^3*d^4*n
+ (3*B^3*b^4*c*d^3 + 22*B^3*a*b^3*d^4)*n^2)*x^3 - (3*B^3*b^4*c^4 - 16*B^3*a*b^3*c^3*d + 36*B^3*a^2*b^2*c^2*d^2
- 48*B^3*a^3*b*c*d^3)*n^2 + 6*(12*A*B^2*a^2*b^2*d^4*n - (B^3*b^4*c^2*d^2 - 8*B^3*a*b^3*c*d^3 - 18*B^3*a^2*b^2
*d^4)*n^2)*x^2 - 12*(A*B^2*b^4*c^4 - 4*A*B^2*a*b^3*c^3*d + 6*A*B^2*a^2*b^2*c^2*d^2 - 4*A*B^2*a^3*b*c*d^3)*n +
4*(12*A*B^2*a^3*b*d^4*n + (B^3*b^4*c^3*d - 6*B^3*a*b^3*c^2*d^2 + 18*B^3*a^2*b^2*c*d^3 + 12*B^3*a^3*b*d^4)*n^2)
*x)*log(e))*log(d*x + c) + 12*(72*A^2*B*b^4*c^4 - 288*A^2*B*a*b^3*c^3*d + 432*A^2*B*a^2*b^2*c^2*d^2 - 288*A^2*
B*a^3*b*c*d^3 + 72*A^2*B*a^4*d^4 - 12*(25*(B^3*b^4*c*d^3 - B^3*a*b^3*d^4)*n^2 + 12*(A*B^2*b^4*c*d^3 - A*B^2*a*
b^3*d^4)*n)*x^3 + (9*B^3*b^4*c^4 - 64*B^3*a*b^3*c^3*d + 216*B^3*a^2*b^2*c^2*d^2 - 576*B^3*a^3*b*c*d^3 + 415*B^
3*a^4*d^4)*n^2 + 6*((13*B^3*b^4*c^2*d^2 - 176*B^3*a*b^3*c*d^3 + 163*B^3*a^2*b^2*d^4)*n^2 + 12*(A*B^2*b^4*c^2*d
^2 - 8*A*B^2*a*b^3*c*d^3 + 7*A*B^2*a^2*b^2*d^4)*n)*x^2 + 12*(3*A*B^2*b^4*c^4 - 16*A*B^2*a*b^3*c^3*d + 36*A*B^2
*a^2*b^2*c^2*d^2 - 48*A*B^2*a^3*b*c*d^3 + 25*A*B^2*a^4*d^4)*n - 4*((7*B^3*b^4*c^3*d - 60*B^3*a*b^3*c^2*d^2 + 3
24*B^3*a^2*b^2*c*d^3 - 271*B^3*a^3*b*d^4)*n^2 + 12*(A*B^2*b^4*c^3*d - 6*A*B^2*a*b^3*c^2*d^2 + 18*A*B^2*a^2*b^2
*c*d^3 - 13*A*B^2*a^3*b*d^4)*n)*x)*log(e))/(a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^
3 + a^8*b*d^4 + (b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b^8*c
^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3
*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2
*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^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((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(b*x+a)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3/(b*x+a)^5,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^3/(b*x + a)^5, x)