3.165 \(\int (a+b x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^3 \, dx\)

Optimal. Leaf size=614 \[ \frac{2 B^2 n^2 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^3}+\frac{4 B^3 n^3 (b c-a d)^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac{B^3 n^3 (b c-a d)^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b d^3}-\frac{2 B^3 n^3 (b c-a d)^3 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{b d^3}+\frac{4 B^2 n^2 (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^3}-\frac{B^2 n^2 (b c-a d)^3 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^3}+\frac{B^2 n^2 (a+b x) (b c-a d)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d^2}+\frac{B n (b c-a d)^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b d^3}+\frac{2 B n (a+b x) (b c-a d)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b d^2}-\frac{b B n (c+d x)^2 (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 d^3}+\frac{(a+b x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 b}-\frac{B^3 n^3 (b c-a d)^3 \log (c+d x)}{b d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.7369, antiderivative size = 915, normalized size of antiderivative = 1.49, number of steps used = 40, number of rules used = 13, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.394, Rules used = {6742, 2492, 43, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ \frac{(a+b x)^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{3 b}-\frac{(b c-a d) n (a+b x)^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{2 b d}+\frac{(b c-a d)^2 n (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{b d^2}+\frac{(b c-a d)^3 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{b d^3}-\frac{(b c-a d)^3 n^3 \log (c+d x) B^3}{b d^3}+\frac{(b c-a d)^2 n^2 (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{b d^2}+\frac{3 (b c-a d)^3 n^2 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B^3}{b d^3}+\frac{3 (b c-a d)^3 n^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) B^3}{b d^3}+\frac{2 (b c-a d)^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) B^3}{b d^3}-\frac{2 (b c-a d)^3 n^3 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) B^3}{b d^3}+\frac{A (a+b x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) B^2}{b}+\frac{A (b c-a d)^2 n^2 x B^2}{d^2}-\frac{3 A (b c-a d)^3 n^2 \log (c+d x) B^2}{b d^3}-\frac{A (b c-a d) n (a+b x)^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B^2}{b d}+\frac{2 A (b c-a d)^2 n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B^2}{b d^2}+\frac{2 A (b c-a d)^3 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B^2}{b d^3}+\frac{2 A (b c-a d)^3 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) B^2}{b d^3}-\frac{A^2 (b c-a d) n (a+b x)^2 B}{2 b d}+\frac{A^2 (b c-a d)^2 n x B}{d^2}-\frac{A^2 (b c-a d)^3 n \log (c+d x) B}{b d^3}+\frac{A^2 (a+b x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) B}{b}+\frac{A^3 (a+b x)^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A^2*B*(b*c - a*d)^2*n*x)/d^2 + (A*B^2*(b*c - a*d)^2*n^2*x)/d^2 - (A^2*B*(b*c - a*d)*n*(a + b*x)^2)/(2*b*d) +
(A^3*(a + b*x)^3)/(3*b) - (A^2*B*(b*c - a*d)^3*n*Log[c + d*x])/(b*d^3) - (3*A*B^2*(b*c - a*d)^3*n^2*Log[c + d*
x])/(b*d^3) - (B^3*(b*c - a*d)^3*n^3*Log[c + d*x])/(b*d^3) + (2*A*B^2*(b*c - a*d)^2*n*(a + b*x)*Log[(e*(a + b*
x)^n)/(c + d*x)^n])/(b*d^2) + (B^3*(b*c - a*d)^2*n^2*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*d^2) - (A*
B^2*(b*c - a*d)*n*(a + b*x)^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*d) + (A^2*B*(a + b*x)^3*Log[(e*(a + b*x)^n)
/(c + d*x)^n])/b + (2*A*B^2*(b*c - a*d)^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(
b*d^3) + (3*B^3*(b*c - a*d)^3*n^2*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*d^3) + (
B^3*(b*c - a*d)^2*n*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(b*d^2) - (B^3*(b*c - a*d)*n*(a + b*x)^2*Log
[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(2*b*d) + (A*B^2*(a + b*x)^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/b + (B^3*(b*
c - a*d)^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(b*d^3) + (B^3*(a + b*x)^3*Log
[(e*(a + b*x)^n)/(c + d*x)^n]^3)/(3*b) + (2*A*B^2*(b*c - a*d)^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(
b*d^3) + (3*B^3*(b*c - a*d)^3*n^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d^3) + (2*B^3*(b*c - a*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*d^3) - (2*B^3*(b*c - a*d)^3*n^3
*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/(b*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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2486

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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

Mathematica [B]  time = 4.11063, size = 5668, normalized size = 9.23 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [F]  time = 4.456, size = 0, normalized size = 0. \begin{align*} \int \left ( bx+a \right ) ^{2} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A^2*B*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^3*b^2*x^3 + 3*A^2*B*a*b*x^2*log((b*x + a)^n*e/(d*x + c)^n
) + A^3*a*b*x^2 + 3*A^2*B*a^2*x*log((b*x + a)^n*e/(d*x + c)^n) + A^3*a^2*x + 3*(a*e*n*log(b*x + a)/b - c*e*n*l
og(d*x + c)/d)*A^2*B*a^2/e - 3*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b
*d))*A^2*B*a*b/e + 1/2*(2*a^3*e*n*log(b*x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)
*x^2 - 2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A^2*B*b^2/e - 1/6*(2*(B^3*b^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2
+ 3*B^3*a^2*b*d^3*x)*log((d*x + c)^n)^3 - 3*(2*B^3*a^3*d^3*n*log(b*x + a) - 2*(b^3*c^3*n - 3*a*b^2*c^2*d*n +
3*a^2*b*c*d^2*n)*B^3*log(d*x + c) + 2*(B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + (6*A*B^2*a*b^2*d^3 + (a*b^2*d
^3*(n + 6*log(e)) - b^3*c*d^2*n)*B^3)*x^2 + 2*(3*A*B^2*a^2*b*d^3 + (a^2*b*d^3*(2*n + 3*log(e)) + b^3*c^2*d*n -
3*a*b^2*c*d^2*n)*B^3)*x + 2*(B^3*b^3*d^3*x^3 + 3*B^3*a*b^2*d^3*x^2 + 3*B^3*a^2*b*d^3*x)*log((b*x + a)^n))*log
((d*x + c)^n)^2)/(b*d^3) - integrate(-(B^3*a^2*b*c*d^2*log(e)^3 + 3*A*B^2*a^2*b*c*d^2*log(e)^2 + (B^3*b^3*d^3*
log(e)^3 + 3*A*B^2*b^3*d^3*log(e)^2)*x^3 + (B^3*b^3*d^3*x^3 + B^3*a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*B^3*
x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*B^3*x)*log((b*x + a)^n)^3 + (3*(b^3*c*d^2*log(e)^2 + 2*a*b^2*d^3*log(e)^2)*A
*B^2 + (b^3*c*d^2*log(e)^3 + 2*a*b^2*d^3*log(e)^3)*B^3)*x^2 + 3*(B^3*a^2*b*c*d^2*log(e) + A*B^2*a^2*b*c*d^2 +
(B^3*b^3*d^3*log(e) + A*B^2*b^3*d^3)*x^3 + ((b^3*c*d^2 + 2*a*b^2*d^3)*A*B^2 + (b^3*c*d^2*log(e) + 2*a*b^2*d^3*
log(e))*B^3)*x^2 + ((2*a*b^2*c*d^2 + a^2*b*d^3)*A*B^2 + (2*a*b^2*c*d^2*log(e) + a^2*b*d^3*log(e))*B^3)*x)*log(
(b*x + a)^n)^2 + (3*(2*a*b^2*c*d^2*log(e)^2 + a^2*b*d^3*log(e)^2)*A*B^2 + (2*a*b^2*c*d^2*log(e)^3 + a^2*b*d^3*
log(e)^3)*B^3)*x + 3*(B^3*a^2*b*c*d^2*log(e)^2 + 2*A*B^2*a^2*b*c*d^2*log(e) + (B^3*b^3*d^3*log(e)^2 + 2*A*B^2*
b^3*d^3*log(e))*x^3 + (2*(b^3*c*d^2*log(e) + 2*a*b^2*d^3*log(e))*A*B^2 + (b^3*c*d^2*log(e)^2 + 2*a*b^2*d^3*log
(e)^2)*B^3)*x^2 + (2*(2*a*b^2*c*d^2*log(e) + a^2*b*d^3*log(e))*A*B^2 + (2*a*b^2*c*d^2*log(e)^2 + a^2*b*d^3*log
(e)^2)*B^3)*x)*log((b*x + a)^n) - (2*B^3*a^3*d^3*n^2*log(b*x + a) + 3*B^3*a^2*b*c*d^2*log(e)^2 + 6*A*B^2*a^2*b
*c*d^2*log(e) - 2*(b^3*c^3*n^2 - 3*a*b^2*c^2*d*n^2 + 3*a^2*b*c*d^2*n^2)*B^3*log(d*x + c) + ((2*n*log(e) + 3*lo
g(e)^2)*B^3*b^3*d^3 + 2*A*B^2*b^3*d^3*(n + 3*log(e)))*x^3 + (6*(a*b^2*d^3*(n + 2*log(e)) + b^3*c*d^2*log(e))*A
*B^2 - ((n^2 - 3*log(e)^2)*b^3*c*d^2 - (n^2 + 6*n*log(e) + 6*log(e)^2)*a*b^2*d^3)*B^3)*x^2 + 3*(B^3*b^3*d^3*x^
3 + B^3*a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*B^3*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*B^3*x)*log((b*x + a)^n)^
2 + (6*(a^2*b*d^3*(n + log(e)) + 2*a*b^2*c*d^2*log(e))*A*B^2 + (2*b^3*c^2*d*n^2 - 6*(n^2 - log(e)^2)*a*b^2*c*d
^2 + (4*n^2 + 6*n*log(e) + 3*log(e)^2)*a^2*b*d^3)*B^3)*x + 2*(3*B^3*a^2*b*c*d^2*log(e) + 3*A*B^2*a^2*b*c*d^2 +
(B^3*b^3*d^3*(n + 3*log(e)) + 3*A*B^2*b^3*d^3)*x^3 + 3*((b^3*c*d^2 + 2*a*b^2*d^3)*A*B^2 + (a*b^2*d^3*(n + 2*l
og(e)) + b^3*c*d^2*log(e))*B^3)*x^2 + 3*((2*a*b^2*c*d^2 + a^2*b*d^3)*A*B^2 + (a^2*b*d^3*(n + log(e)) + 2*a*b^2
*c*d^2*log(e))*B^3)*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b*d^3*x + b*c*d^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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