3.170 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{(a+b x)^4} \, dx\)

Optimal. Leaf size=611 \[ -\frac{2 b^2 B^2 n^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{9 (a+b x)^3 (b c-a d)^3}-\frac{b^2 B n (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 (a+b x)^3 (b c-a d)^3}-\frac{b^2 (c+d x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 (a+b x)^3 (b c-a d)^3}-\frac{6 B^2 d^2 n^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(a+b x) (b c-a d)^3}+\frac{3 b B^2 d n^2 (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{2 (a+b x)^2 (b c-a d)^3}-\frac{3 B d^2 n (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(a+b x) (b c-a d)^3}-\frac{d^2 (c+d x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x) (b c-a d)^3}+\frac{3 b B d n (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{b d (c+d x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^2 (b c-a d)^3}-\frac{2 b^2 B^3 n^3 (c+d x)^3}{27 (a+b x)^3 (b c-a d)^3}-\frac{6 B^3 d^2 n^3 (c+d x)}{(a+b x) (b c-a d)^3}+\frac{3 b B^3 d n^3 (c+d x)^2}{4 (a+b x)^2 (b c-a d)^3} \]

[Out]

(-6*B^3*d^2*n^3*(c + d*x))/((b*c - a*d)^3*(a + b*x)) + (3*b*B^3*d*n^3*(c + d*x)^2)/(4*(b*c - a*d)^3*(a + b*x)^
2) - (2*b^2*B^3*n^3*(c + d*x)^3)/(27*(b*c - a*d)^3*(a + b*x)^3) - (6*B^2*d^2*n^2*(c + d*x)*(A + B*Log[(e*(a +
b*x)^n)/(c + d*x)^n]))/((b*c - a*d)^3*(a + b*x)) + (3*b*B^2*d*n^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c +
d*x)^n]))/(2*(b*c - a*d)^3*(a + b*x)^2) - (2*b^2*B^2*n^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))
/(9*(b*c - a*d)^3*(a + b*x)^3) - (3*B*d^2*n*(c + d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*c - a*d)
^3*(a + b*x)) + (3*b*B*d*n*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(2*(b*c - a*d)^3*(a + b*x)^
2) - (b^2*B*n*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/(3*(b*c - a*d)^3*(a + b*x)^3) - (d^2*(c
+ d*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)) + (b*d*(c + d*x)^2*(A + B*Log[(e*
(a + b*x)^n)/(c + d*x)^n])^3)/((b*c - a*d)^3*(a + b*x)^2) - (b^2*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^n)/(c + d
*x)^n])^3)/(3*(b*c - a*d)^3*(a + b*x)^3)

________________________________________________________________________________________

Rubi [C]  time = 3.43242, antiderivative size = 1876, normalized size of antiderivative = 3.07, number of steps used = 66, 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)^4,x]

[Out]

-A^3/(3*b*(a + b*x)^3) - (A^2*B*n)/(3*b*(a + b*x)^3) - (2*A*B^2*n^2)/(9*b*(a + b*x)^3) - (2*B^3*n^3)/(27*b*(a
+ b*x)^3) + (A^2*B*d*n)/(2*b*(b*c - a*d)*(a + b*x)^2) + (5*A*B^2*d*n^2)/(6*b*(b*c - a*d)*(a + b*x)^2) + (5*B^3
*d*n^3)/(18*b*(b*c - a*d)*(a + b*x)^2) - (A^2*B*d^2*n)/(b*(b*c - a*d)^2*(a + b*x)) - (11*A*B^2*d^2*n^2)/(3*b*(
b*c - a*d)^2*(a + b*x)) - (47*B^3*d^2*n^3)/(9*b*(b*c - a*d)^2*(a + b*x)) + (b*B^3*d*n^3*(c + d*x)^2)/(4*(b*c -
a*d)^3*(a + b*x)^2) - (A^2*B*d^3*n*Log[a + b*x])/(b*(b*c - a*d)^3) - (5*A*B^2*d^3*n^2*Log[a + b*x])/(3*b*(b*c
- a*d)^3) - (5*B^3*d^3*n^3*Log[a + b*x])/(9*b*(b*c - a*d)^3) + (A^2*B*d^3*n*Log[c + d*x])/(b*(b*c - a*d)^3) +
(5*A*B^2*d^3*n^2*Log[c + d*x])/(3*b*(b*c - a*d)^3) + (5*B^3*d^3*n^3*Log[c + d*x])/(9*b*(b*c - a*d)^3) - (A^2*
B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*(a + b*x)^3) - (2*A*B^2*n*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(a + b
*x)^3) - (2*B^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(9*b*(a + b*x)^3) + (A*B^2*d*n*Log[(e*(a + b*x)^n)/(c +
d*x)^n])/(b*(b*c - a*d)*(a + b*x)^2) + (B^3*d*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*(b*c - a*d)*(a + b*x)
^2) - (2*A*B^2*d^2*n*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/((b*c - a*d)^3*(a + b*x)) - (14*B^3*d^2*n^2*(
c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*(b*c - a*d)^3*(a + b*x)) + (b*B^3*d*n^2*(c + d*x)^2*Log[(e*(a +
b*x)^n)/(c + d*x)^n])/(2*(b*c - a*d)^3*(a + b*x)^2) + (2*A*B^2*d^3*n*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*
(a + b*x)^n)/(c + d*x)^n])/(b*(b*c - a*d)^3) + (2*B^3*d^3*n^2*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*
x)^n)/(c + d*x)^n])/(3*b*(b*c - a*d)^3) - (2*A*B^2*d^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c
+ d*x)^n])/(b*(b*c - a*d)^3) - (2*B^3*d^3*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]
)/(3*b*(b*c - a*d)^3) - (A*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(b*(a + b*x)^3) - (B^3*n*Log[(e*(a + b*x)^n
)/(c + d*x)^n]^2)/(3*b*(a + b*x)^3) - (2*B^3*d^2*n*(c + d*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/((b*c - a*d)^
3*(a + b*x)) + (b*B^3*d*n*(c + d*x)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(2*(b*c - a*d)^3*(a + b*x)^2) + (B^3
*d^3*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)^3) - (B^3*d^3*n*Lo
g[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(b*(b*c - a*d)^3) - (B^3*Log[(e*(a + b*x)^n)/
(c + d*x)^n]^3)/(3*b*(a + b*x)^3) - (2*A*B^2*d^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*(b*c - a*d)^3
) - (2*B^3*d^3*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*b*(b*c - a*d)^3) - (2*A*B^2*d^3*n^2*PolyLog[2,
1 + (b*c - a*d)/(d*(a + b*x))])/(b*(b*c - a*d)^3) - (2*B^3*d^3*n^3*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/
(3*b*(b*c - a*d)^3) - (2*B^3*d^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))
])/(b*(b*c - a*d)^3) - (2*B^3*d^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[2, 1 - (b*c - a*d)/(b*(c + d*x)
)])/(b*(b*c - a*d)^3) - (2*B^3*d^3*n^3*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/(b*(b*c - a*d)^3) + (2*B^3*d
^3*n^3*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/(b*(b*c - a*d)^3)

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)^4} \, dx &=\int \left (\frac{A^3}{(a+b x)^4}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}\right ) \, dx\\ &=-\frac{A^3}{3 b (a+b x)^3}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, 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)^4} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{\left (A^2 B (b c-a d) n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b}+\frac{\left (2 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)^4 (c+d x)} \, dx}{b}+\frac{\left (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)^4 (c+d x)} \, dx}{b}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{\left (A^2 B (b c-a d) n\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}{b}+\frac{\left (2 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)^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}{b}+\frac{\left (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)^4}-\frac{b d \log ^2\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 ^2\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 ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\left (2 A B^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx+\left (B^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac{\left (2 A B^2 d^3 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{(b c-a d)^3}-\frac{\left (B^3 d^3 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{(b c-a d)^3}+\frac{\left (2 A B^2 d^4 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b (b c-a d)^3}+\frac{\left (B^3 d^4 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{b (b c-a d)^3}+\frac{\left (2 A B^2 d^2 n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac{\left (B^3 d^2 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac{\left (2 A B^2 d n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}-\frac{\left (B^3 d n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{b c-a d}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{\left (b B^3 d 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}{(b c-a d)^2}+\frac{\left (B^3 d^2 n\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac{\left (A B^2 d n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b}+\frac{\left (2 A B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac{\left (2 B^3 d^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac{\left (2 A B^2 d^3 n^2\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}+\frac{\left (2 A B^2 d^3 n^2\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{b (b c-a d)^2}-\frac{\left (2 B^3 d^3 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}{b (b c-a d)^2}+\frac{\left (2 B^3 d^3 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}{b (b c-a d)^2}+\frac{\left (2 A B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b}+\frac{\left (2 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)^4 (c+d x)} \, dx}{3 b}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{2 A B^2 d^2 n^2}{b (b c-a d)^2 (a+b x)}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{2 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{\left (A B^2 d 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}{b}-\frac{\left (b B^3 d 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}{(b c-a d)^2}+\frac{\left (2 A B^2 d^2 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 )}{b (b c-a d)^2}+\frac{\left (2 B^3 d^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{(b c-a d)^2}-\frac{\left (2 A B^2 d^3 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 )}{b^2 (b c-a d)^2}+\frac{\left (2 A B^2 (b c-a 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}{3 b}+\frac{\left (2 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)^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}{3 b}+\frac{\left (2 B^3 d^2 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{(b c-a d)^2}+\frac{\left (2 B^3 d^3 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}{b (b c-a d)^2}+\frac{\left (2 B^3 d^3 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}{b (b c-a d)^2}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{2 B^3 d^2 n^3}{b (b c-a d)^2 (a+b x)}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{4 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}+\frac{1}{3} \left (2 B^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4} \, dx-\frac{\left (2 B^3 d^3 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx}{3 (b c-a d)^3}+\frac{\left (2 B^3 d^4 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{3 b (b c-a d)^3}-\frac{\left (2 A B^2 d^2 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 )}{b (b c-a d)^2}+\frac{\left (2 B^3 d^2 n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2}+\frac{\left (2 A B^2 d^3 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 )}{b^2 (b c-a d)^2}-\frac{\left (2 B^3 d n^2\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^3} \, dx}{3 (b c-a d)}-\frac{\left (b B^3 d n^3\right ) \int \frac{c+d x}{(a+b x)^3} \, dx}{2 (b c-a d)^2}+\frac{\left (2 B^3 d^2 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{(b c-a d)^2}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{4 B^3 d^2 n^3}{b (b c-a d)^2 (a+b x)}+\frac{b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac{B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}+\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{\left (2 A B^2 d^2 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 )}{b (b c-a d)^2}+\frac{\left (2 A B^2 d^3 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 )}{b^2 (b c-a d)^2}-\frac{\left (B^3 d n^3\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b}+\frac{\left (2 B^3 d^2 n^3\right ) \int \frac{1}{(a+b x)^2} \, dx}{3 (b c-a d)^2}-\frac{\left (2 B^3 d^3 n^3\right ) \int \frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac{\left (2 B^3 d^3 n^3\right ) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{3 b (b c-a d)^2}+\frac{\left (2 B^3 (b c-a d) n^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{14 B^3 d^2 n^3}{3 b (b c-a d)^2 (a+b x)}+\frac{b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac{B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}+\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{\left (B^3 d 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}{3 b}+\frac{\left (2 B^3 d^2 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 )}{3 b (b c-a d)^2}-\frac{\left (2 B^3 d^3 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 )}{3 b^2 (b c-a d)^2}+\frac{\left (2 B^3 (b c-a 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}{9 b}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}-\frac{2 B^3 n^3}{27 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac{5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac{b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac{B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}+\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{\left (2 B^3 d^2 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 )}{3 b (b c-a d)^2}+\frac{\left (2 B^3 d^3 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 )}{3 b^2 (b c-a d)^2}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}-\frac{2 B^3 n^3}{27 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac{5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac{b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac{B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}+\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{\left (2 B^3 d^2 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 )}{3 b (b c-a d)^2}+\frac{\left (2 B^3 d^3 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 )}{3 b^2 (b c-a d)^2}\\ &=-\frac{A^3}{3 b (a+b x)^3}-\frac{A^2 B n}{3 b (a+b x)^3}-\frac{2 A B^2 n^2}{9 b (a+b x)^3}-\frac{2 B^3 n^3}{27 b (a+b x)^3}+\frac{A^2 B d n}{2 b (b c-a d) (a+b x)^2}+\frac{5 A B^2 d n^2}{6 b (b c-a d) (a+b x)^2}+\frac{5 B^3 d n^3}{18 b (b c-a d) (a+b x)^2}-\frac{A^2 B d^2 n}{b (b c-a d)^2 (a+b x)}-\frac{11 A B^2 d^2 n^2}{3 b (b c-a d)^2 (a+b x)}-\frac{47 B^3 d^2 n^3}{9 b (b c-a d)^2 (a+b x)}+\frac{b B^3 d n^3 (c+d x)^2}{4 (b c-a d)^3 (a+b x)^2}-\frac{A^2 B d^3 n \log (a+b x)}{b (b c-a d)^3}-\frac{5 A B^2 d^3 n^2 \log (a+b x)}{3 b (b c-a d)^3}-\frac{5 B^3 d^3 n^3 \log (a+b x)}{9 b (b c-a d)^3}+\frac{A^2 B d^3 n \log (c+d x)}{b (b c-a d)^3}+\frac{5 A B^2 d^3 n^2 \log (c+d x)}{3 b (b c-a d)^3}+\frac{5 B^3 d^3 n^3 \log (c+d x)}{9 b (b c-a d)^3}-\frac{A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{2 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{9 b (a+b x)^3}+\frac{A B^2 d n \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b (b c-a d) (a+b x)^2}+\frac{B^3 d n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (b c-a d) (a+b x)^2}-\frac{2 A B^2 d^2 n (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}-\frac{14 B^3 d^2 n^2 (c+d x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 (b c-a d)^3 (a+b x)}+\frac{b B^3 d n^2 (c+d x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}+\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{3 b (b c-a d)^3}-\frac{A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b (a+b x)^3}-\frac{B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 B^3 d^2 n (c+d x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b c-a d)^3 (a+b x)}+\frac{b B^3 d n (c+d x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 (b c-a d)^3 (a+b x)^2}+\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 d^3 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 )}{b (b c-a d)^3}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b (a+b x)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_2\left (\frac{d (a+b x)}{b (c+d x)}\right )}{3 b (b c-a d)^3}-\frac{2 A B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{3 b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 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 )}{b (b c-a d)^3}-\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b (b c-a d)^3}+\frac{2 B^3 d^3 n^3 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 1.47614, size = 1003, normalized size = 1.64 \[ \frac{-36 B^3 d^3 n^3 \log ^3(a+b x) (a+b x)^3+36 B^3 d^3 n^3 \log ^3(c+d x) (a+b x)^3+18 B^2 d^3 n^2 \log ^2(c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^3+18 B^2 d^3 n^2 \log ^2(a+b x) \left (6 A+11 B n+6 B n \log (c+d x)+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^3+6 B d^3 n \log (c+d x) \left (18 A^2+66 B n A+85 B^2 n^2+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) (a+b x)^3-6 B d^3 n \log (a+b x) \left (18 A^2+66 B n A+85 B^2 n^2+18 B^2 n^2 \log ^2(c+d x)+18 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+6 B (6 A+11 B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+6 B n \log (c+d x) \left (6 A+11 B n+6 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) (a+b x)^3-(b c-a d) \left (36 b^2 c^2 A^3+36 a^2 d^2 A^3-72 a b c d A^3+108 b^2 B d^2 n x^2 A^2+36 b^2 B c^2 n A^2+198 a^2 B d^2 n A^2-126 a b B c d n A^2+270 a b B d^2 n x A^2-54 b^2 B c d n x A^2+24 b^2 B^2 c^2 n^2 A+510 a^2 B^2 d^2 n^2 A-138 a b B^2 c d n^2 A+396 b^2 B^2 d^2 n^2 x^2 A+882 a b B^2 d^2 n^2 x A-90 b^2 B^2 c d n^2 x A+8 b^2 B^3 c^2 n^3+575 a^2 B^3 d^2 n^3-73 a b B^3 c d n^3+36 B^3 (b c-a d)^2 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+510 b^2 B^3 d^2 n^3 x^2+18 B^2 \left (6 A (b c-a d)^2+B n \left (\left (2 c^2-3 d x c+6 d^2 x^2\right ) b^2+a d (15 d x-7 c) b+11 a^2 d^2\right )\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+1077 a b B^3 d^2 n^3 x-57 b^2 B^3 c d n^3 x+6 B \left (18 A^2 (b c-a d)^2+6 A B n \left (\left (2 c^2-3 d x c+6 d^2 x^2\right ) b^2+a d (15 d x-7 c) b+11 a^2 d^2\right )+B^2 n^2 \left (\left (4 c^2-15 d x c+66 d^2 x^2\right ) b^2+a d (147 d x-23 c) b+85 a^2 d^2\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{108 b (b c-a d)^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-36*B^3*d^3*n^3*(a + b*x)^3*Log[a + b*x]^3 + 36*B^3*d^3*n^3*(a + b*x)^3*Log[c + d*x]^3 + 18*B^2*d^3*n^2*(a +
b*x)^3*Log[c + d*x]^2*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 18*B^2*d^3*n^2*(a + b*x)^3*Log[a
+ b*x]^2*(6*A + 11*B*n + 6*B*n*Log[c + d*x] + 6*B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 6*B*d^3*n*(a + b*x)^3*L
og[c + d*x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 6*B*(6*A + 11*B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*Log
[(e*(a + b*x)^n)/(c + d*x)^n]^2) - (b*c - a*d)*(36*A^3*b^2*c^2 - 72*a*A^3*b*c*d + 36*a^2*A^3*d^2 + 36*A^2*b^2*
B*c^2*n - 126*a*A^2*b*B*c*d*n + 198*a^2*A^2*B*d^2*n + 24*A*b^2*B^2*c^2*n^2 - 138*a*A*b*B^2*c*d*n^2 + 510*a^2*A
*B^2*d^2*n^2 + 8*b^2*B^3*c^2*n^3 - 73*a*b*B^3*c*d*n^3 + 575*a^2*B^3*d^2*n^3 - 54*A^2*b^2*B*c*d*n*x + 270*a*A^2
*b*B*d^2*n*x - 90*A*b^2*B^2*c*d*n^2*x + 882*a*A*b*B^2*d^2*n^2*x - 57*b^2*B^3*c*d*n^3*x + 1077*a*b*B^3*d^2*n^3*
x + 108*A^2*b^2*B*d^2*n*x^2 + 396*A*b^2*B^2*d^2*n^2*x^2 + 510*b^2*B^3*d^2*n^3*x^2 + 6*B*(18*A^2*(b*c - a*d)^2
+ 6*A*B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + B^2*n^2*(85*a^2*d^2 + a*b
*d*(-23*c + 147*d*x) + b^2*(4*c^2 - 15*c*d*x + 66*d^2*x^2)))*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 18*B^2*(6*A*(b
*c - a*d)^2 + B*n*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)))*Log[(e*(a + b*x)^n
)/(c + d*x)^n]^2 + 36*B^3*(b*c - a*d)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 6*B*d^3*n*(a + b*x)^3*Log[a + b*
x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*n^2*Log[c + d*x]^2 + 6*B*(6*A + 11*B*n)*Log[(e*(a + b*x)^n)/(c + d
*x)^n] + 18*B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 6*B*n*Log[c + d*x]*(6*A + 11*B*n + 6*B*Log[(e*(a + b*x)^n
)/(c + d*x)^n])))/(108*b*(b*c - a*d)^3*(a + b*x)^3)

________________________________________________________________________________________

Maple [C]  time = 27.216, size = 175812, normalized size = 287.7 \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)^4,x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [B]  time = 2.60012, size = 4901, normalized size = 8.02 \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)^4,x, algorithm="maxima")

[Out]

-1/3*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/108*(18*(6*d^3*e*n
*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) - 6*d^3*e*n*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) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n
- 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*
b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)
*x))*log((b*x + a)^n*e/(d*x + c)^n)^2/e + (6*(4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2
*n^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2
*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2
*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*e^2*n^2 - 54*a*b^2
*c*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2
*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^2 + 33*a^2*b*d^3*e
^2*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3
*d^3*e^2*n^2)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n*e/(d*x + c)^n)/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a
^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^
2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^
2*d^3)*x)*e) + (8*b^3*c^3*e^3*n^3 - 81*a*b^2*c^2*d*e^3*n^3 + 648*a^2*b*c*d^2*e^3*n^3 - 575*a^3*d^3*e^3*n^3 + 3
6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(b*x + a)^3 - 3
6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(d*x + c)^3 + 5
10*(b^3*c*d^2*e^3*n^3 - a*b^2*d^3*e^3*n^3)*x^2 - 198*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*
d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*log(b*x + a)^2 - 18*(11*b^3*d^3*e^3*n^3*x^3 + 33*a*b^2*d^3*e^3*n^3*x^2 + 33*a
^2*b*d^3*e^3*n^3*x + 11*a^3*d^3*e^3*n^3 - 6*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n
^3*x + a^3*d^3*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 - 3*(19*b^3*c^2*d*e^3*n^3 - 378*a*b^2*c*d^2*e^3*n^3 + 359
*a^2*b*d^3*e^3*n^3)*x + 510*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e
^3*n^3)*log(b*x + a) - 6*(85*b^3*d^3*e^3*n^3*x^3 + 255*a*b^2*d^3*e^3*n^3*x^2 + 255*a^2*b*d^3*e^3*n^3*x + 85*a^
3*d^3*e^3*n^3 + 18*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*l
og(b*x + a)^2 - 66*(b^3*d^3*e^3*n^3*x^3 + 3*a*b^2*d^3*e^3*n^3*x^2 + 3*a^2*b*d^3*e^3*n^3*x + a^3*d^3*e^3*n^3)*l
og(b*x + a))*log(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*
c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x
^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)*e^2))/e)*B^3 - 1/18*A*B^2*(6*(6*d^3*
e*n*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) - 6*d^3*e*n*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) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*
e*n - 3*(b^2*c*d*e*n - 5*a*b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a
^2*b^4*d^2)*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d
^2)*x))*log((b*x + a)^n*e/(d*x + c)^n)/e + (4*b^3*c^3*e^2*n^2 - 27*a*b^2*c^2*d*e^2*n^2 + 108*a^2*b*c*d^2*e^2*n
^2 - 85*a^3*d^3*e^2*n^2 + 66*(b^3*c*d^2*e^2*n^2 - a*b^2*d^3*e^2*n^2)*x^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d
^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(b*x + a)^2 - 18*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d
^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d^3*e^2*n^2)*log(d*x + c)^2 - 3*(5*b^3*c^2*d*e^2*n^2 - 54*a*b^2*c
*d^2*e^2*n^2 + 49*a^2*b*d^3*e^2*n^2)*x + 66*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n
^2*x + a^3*d^3*e^2*n^2)*log(b*x + a) - 6*(11*b^3*d^3*e^2*n^2*x^3 + 33*a*b^2*d^3*e^2*n^2*x^2 + 33*a^2*b*d^3*e^2
*n^2*x + 11*a^3*d^3*e^2*n^2 - 6*(b^3*d^3*e^2*n^2*x^3 + 3*a*b^2*d^3*e^2*n^2*x^2 + 3*a^2*b*d^3*e^2*n^2*x + a^3*d
^3*e^2*n^2)*log(b*x + a))*log(d*x + c))/((a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c
^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a
^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*x)*e^2)) - A*B^2*log((b*x
+ a)^n*e/(d*x + c)^n)^2/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/6*(6*d^3*e*n*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) - 6*d^3*e*n*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) + (6*b^2*d^2*e*n*x^2 + 2*b^2*c^2*e*n - 7*a*b*c*d*e*n + 11*a^2*d^2*e*n - 3*(b^2*c*d*e*n - 5*a*
b*d^2*e*n)*x)/(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2 + (b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*x^3 + 3*(a*b^5*
c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*x))*A^2*B/e - A^2*B*log
((b*x + a)^n*e/(d*x + c)^n)/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b) - 1/3*A^3/(b^4*x^3 + 3*a*b^3*x^2 + 3
*a^2*b^2*x + a^3*b)

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Fricas [B]  time = 1.86161, size = 8181, normalized size = 13.39 \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)^4,x, algorithm="fricas")

[Out]

-1/108*(36*A^3*b^3*c^3 - 108*A^3*a*b^2*c^2*d + 108*A^3*a^2*b*c*d^2 - 36*A^3*a^3*d^3 + (8*B^3*b^3*c^3 - 81*B^3*
a*b^2*c^2*d + 648*B^3*a^2*b*c*d^2 - 575*B^3*a^3*d^3)*n^3 + 36*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 +
3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^3)*log(b*x + a)^3 - 36*(B^3*b
^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^
2*b*c*d^2)*n^3)*log(d*x + c)^3 + 36*(B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2 - B^3*a^3*d^3)*log(e)
^3 + 6*(4*A*B^2*b^3*c^3 - 27*A*B^2*a*b^2*c^2*d + 108*A*B^2*a^2*b*c*d^2 - 85*A*B^2*a^3*d^3)*n^2 + 6*(85*(B^3*b^
3*c*d^2 - B^3*a*b^2*d^3)*n^3 + 66*(A*B^2*b^3*c*d^2 - A*B^2*a*b^2*d^3)*n^2 + 18*(A^2*B*b^3*c*d^2 - A^2*B*a*b^2*
d^3)*n)*x^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2
*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n
^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*
d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3*a^2*b*d^3*n^2*x + (B^
3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(b*x + a)^2 + 18*((2*B^3*b^3*c^3 - 9*B^3*a*
b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*
B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^
3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*
d^3*n^3*x^3 + 3*B^3*a*b^2*d^3*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b
*c*d^2)*n^3)*log(b*x + a) + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3*a^2*b*d^3*n^2*x + (B^3*b^
3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(d*x + c)^2 + 18*(6*A*B^2*b^3*c^3 - 18*A*B^2*a*
b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2 - 6*A*B^2*a^3*d^3 + 6*(B^3*b^3*c*d^2 - B^3*a*b^2*d^3)*n*x^2 - 3*(B^3*b^3*c^2*
d - 6*B^3*a*b^2*c*d^2 + 5*B^3*a^2*b*d^3)*n*x + (2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2 - 11*B^
3*a^3*d^3)*n)*log(e)^2 + 18*(2*A^2*B*b^3*c^3 - 9*A^2*B*a*b^2*c^2*d + 18*A^2*B*a^2*b*c*d^2 - 11*A^2*B*a^3*d^3)*
n - 3*((19*B^3*b^3*c^2*d - 378*B^3*a*b^2*c*d^2 + 359*B^3*a^2*b*d^3)*n^3 + 6*(5*A*B^2*b^3*c^2*d - 54*A*B^2*a*b^
2*c*d^2 + 49*A*B^2*a^2*b*d^3)*n^2 + 18*(A^2*B*b^3*c^2*d - 6*A^2*B*a*b^2*c*d^2 + 5*A^2*B*a^2*b*d^3)*n)*x + 6*((
4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108*B^3*a^2*b*c*d^2)*n^3 + (85*B^3*b^3*d^3*n^3 + 66*A*B^2*b^3*d^3*n^2 + 1
8*A^2*B*b^3*d^3*n)*x^3 + 6*(2*A*B^2*b^3*c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2)*n^2 + 3*(18*A^2*B*a*
b^2*d^3*n + (22*B^3*b^3*c*d^2 + 63*B^3*a*b^2*d^3)*n^3 + 6*(2*A*B^2*b^3*c*d^2 + 9*A*B^2*a*b^2*d^3)*n^2)*x^2 + 1
8*(B^3*b^3*d^3*n*x^3 + 3*B^3*a*b^2*d^3*n*x^2 + 3*B^3*a^2*b*d^3*n*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*
a^2*b*c*d^2)*n)*log(e)^2 + 18*(A^2*B*b^3*c^3 - 3*A^2*B*a*b^2*c^2*d + 3*A^2*B*a^2*b*c*d^2)*n + 3*(18*A^2*B*a^2*
b*d^3*n - (5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 - 36*B^3*a^2*b*d^3)*n^3 - 6*(A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2*c
*d^2 - 6*A*B^2*a^2*b*d^3)*n^2)*x + 6*((11*B^3*b^3*d^3*n^2 + 6*A*B^2*b^3*d^3*n)*x^3 + (2*B^3*b^3*c^3 - 9*B^3*a*
b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^2 + 3*(6*A*B^2*a*b^2*d^3*n + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^2)*x^2 +
6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n + 3*(6*A*B^2*a^2*b*d^3*n - (B^3*b^3*c^2*d - 6*
B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^2)*x)*log(e))*log(b*x + a) - 6*((4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108
*B^3*a^2*b*c*d^2)*n^3 + (85*B^3*b^3*d^3*n^3 + 66*A*B^2*b^3*d^3*n^2 + 18*A^2*B*b^3*d^3*n)*x^3 + 6*(2*A*B^2*b^3*
c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*B^2*a^2*b*c*d^2)*n^2 + 3*(18*A^2*B*a*b^2*d^3*n + (22*B^3*b^3*c*d^2 + 63*B^3*a
*b^2*d^3)*n^3 + 6*(2*A*B^2*b^3*c*d^2 + 9*A*B^2*a*b^2*d^3)*n^2)*x^2 + 18*(B^3*b^3*d^3*n^3*x^3 + 3*B^3*a*b^2*d^3
*n^3*x^2 + 3*B^3*a^2*b*d^3*n^3*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^3)*log(b*x + a)^2 +
18*(B^3*b^3*d^3*n*x^3 + 3*B^3*a*b^2*d^3*n*x^2 + 3*B^3*a^2*b*d^3*n*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^
3*a^2*b*c*d^2)*n)*log(e)^2 + 18*(A^2*B*b^3*c^3 - 3*A^2*B*a*b^2*c^2*d + 3*A^2*B*a^2*b*c*d^2)*n + 3*(18*A^2*B*a^
2*b*d^3*n - (5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 - 36*B^3*a^2*b*d^3)*n^3 - 6*(A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2
*c*d^2 - 6*A*B^2*a^2*b*d^3)*n^2)*x + 6*((2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^3 + (11*B^3
*b^3*d^3*n^3 + 6*A*B^2*b^3*d^3*n^2)*x^3 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A*B^2*a^2*b*c*d^2)*n^2 +
3*(6*A*B^2*a*b^2*d^3*n^2 + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^3)*x^2 + 3*(6*A*B^2*a^2*b*d^3*n^2 - (B^3*b^3*
c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^3)*x + 6*(B^3*b^3*d^3*n^2*x^3 + 3*B^3*a*b^2*d^3*n^2*x^2 + 3*B^3
*a^2*b*d^3*n^2*x + (B^3*b^3*c^3 - 3*B^3*a*b^2*c^2*d + 3*B^3*a^2*b*c*d^2)*n^2)*log(e))*log(b*x + a) + 6*((11*B^
3*b^3*d^3*n^2 + 6*A*B^2*b^3*d^3*n)*x^3 + (2*B^3*b^3*c^3 - 9*B^3*a*b^2*c^2*d + 18*B^3*a^2*b*c*d^2)*n^2 + 3*(6*A
*B^2*a*b^2*d^3*n + (2*B^3*b^3*c*d^2 + 9*B^3*a*b^2*d^3)*n^2)*x^2 + 6*(A*B^2*b^3*c^3 - 3*A*B^2*a*b^2*c^2*d + 3*A
*B^2*a^2*b*c*d^2)*n + 3*(6*A*B^2*a^2*b*d^3*n - (B^3*b^3*c^2*d - 6*B^3*a*b^2*c*d^2 - 6*B^3*a^2*b*d^3)*n^2)*x)*l
og(e))*log(d*x + c) + 6*(18*A^2*B*b^3*c^3 - 54*A^2*B*a*b^2*c^2*d + 54*A^2*B*a^2*b*c*d^2 - 18*A^2*B*a^3*d^3 + (
4*B^3*b^3*c^3 - 27*B^3*a*b^2*c^2*d + 108*B^3*a^2*b*c*d^2 - 85*B^3*a^3*d^3)*n^2 + 6*(11*(B^3*b^3*c*d^2 - B^3*a*
b^2*d^3)*n^2 + 6*(A*B^2*b^3*c*d^2 - A*B^2*a*b^2*d^3)*n)*x^2 + 6*(2*A*B^2*b^3*c^3 - 9*A*B^2*a*b^2*c^2*d + 18*A*
B^2*a^2*b*c*d^2 - 11*A*B^2*a^3*d^3)*n - 3*((5*B^3*b^3*c^2*d - 54*B^3*a*b^2*c*d^2 + 49*B^3*a^2*b*d^3)*n^2 + 6*(
A*B^2*b^3*c^2*d - 6*A*B^2*a*b^2*c*d^2 + 5*A*B^2*a^2*b*d^3)*n)*x)*log(e))/(a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^
5*b^2*c*d^2 - a^6*b*d^3 + (b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*x^3 + 3*(a*b^6*c^3 - 3*a^2
*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2
*d^3)*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)**4,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 )}^{4}}\,{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)^4,x, algorithm="giac")

[Out]

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