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

Optimal. Leaf size=203 \[ \frac{6 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b d}-\frac{6 B^3 n^3 (b c-a d) \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{b d}+\frac{3 B n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b d}+\frac{(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 0.590136, antiderivative size = 408, normalized size of antiderivative = 2.01, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6742, 2486, 31, 2488, 2411, 2343, 2333, 2315, 2506, 6610} \[ \frac{6 A B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d}+\frac{6 B^3 n^2 (b c-a d) \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}-\frac{6 B^3 n^3 (b c-a d) \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right )}{b d}+\frac{3 A^2 B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{3 A^2 B n (b c-a d) \log (c+d x)}{b d}+\frac{3 A B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 A B^2 n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+\frac{B^3 (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{3 B^3 n (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b d}+A^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

A^3*x - (3*A^2*B*(b*c - a*d)*n*Log[c + d*x])/(b*d) + (3*A^2*B*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/b +
(6*A*B^2*(b*c - a*d)*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(b*d) + (3*A*B^2*(a +
b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/b + (3*B^3*(b*c - a*d)*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b
*x)^n)/(c + d*x)^n]^2)/(b*d) + (B^3*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3)/b + (6*A*B^2*(b*c - a*d)*n^2
*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b*d) + (6*B^3*(b*c - a*d)*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Poly
Log[2, 1 - (b*c - a*d)/(b*(c + d*x))])/(b*d) - (6*B^3*(b*c - a*d)*n^3*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))
])/(b*d)

Rule 6742

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

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

Mathematica [A]  time = 0.279521, size = 378, normalized size = 1.86 \[ \frac{3 A B^2 n (b c-a d) \left (2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )-\log \left (\frac{b c-a d}{b c+b d x}\right ) \left (-2 \log \left (e (a+b x)^n (c+d x)^{-n}\right )+2 n \log \left (\frac{d (a+b x)}{a d-b c}\right )+n \log \left (\frac{b c-a d}{b c+b d x}\right )\right )\right )+3 B^3 n (b c-a d) \left (2 n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-2 n^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )+\log \left (\frac{b c-a d}{b c+b d x}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 A^2 B d (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )-3 A^2 B n (b c-a d) \log (c+d x)+3 A B^2 d (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 d (a+b x) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )+A^3 b d x}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 2.737, size = 0, normalized size = 0. \begin{align*} \int \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((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3,x)

[Out]

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

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

[Out]

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

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^3,x, algorithm="giac")

[Out]

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