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

Optimal. Leaf size=186 \[ \frac{6 B^2 n^2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b}+\frac{3 B n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{b}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{b (c+d x)}{d (a+b x)}\right )}{b}-\frac{\log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{b} \]

[Out]

-(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (3*B*n*(A + B*Log[(e*
(a + b*x)^n)/(c + d*x)^n])^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b + (6*B^2*n^2*(A + B*Log[(e*(a + b*x)^n
)/(c + d*x)^n])*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/b + (6*B^3*n^3*PolyLog[4, (b*(c + d*x))/(d*(a + b*x))
])/b

________________________________________________________________________________________

Rubi [B]  time = 0.850658, antiderivative size = 424, normalized size of antiderivative = 2.28, number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6742, 2488, 2411, 2343, 2333, 2315, 2506, 6610, 2508} \[ \frac{3 A^2 B n \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{b}+\frac{6 A B^2 n \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 A B^2 n^2 \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right )}{b}+\frac{6 B^3 n^2 \text{PolyLog}\left (3,\frac{b c-a d}{d (a+b x)}+1\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{3 B^3 n \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{b c-a d}{d (a+b x)}+1\right )}{b}-\frac{3 A^2 B \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{A^3 \log (a+b x)}{b}-\frac{3 A B^2 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A^3*Log[a + b*x])/b - (3*A^2*B*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n])/b - (3*A*B
^2*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/b - (B^3*Log[-((b*c - a*d)/(d*(a + b*
x)))]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3)/b + (3*A^2*B*n*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b + (6*A*B
^2*n*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b + (3*B^3*n*Log[(e*(a + b*x)
^n)/(c + d*x)^n]^2*PolyLog[2, 1 + (b*c - a*d)/(d*(a + b*x))])/b + (6*A*B^2*n^2*PolyLog[3, 1 + (b*c - a*d)/(d*(
a + b*x))])/b + (6*B^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/b + (6*
B^3*n^3*PolyLog[4, 1 + (b*c - a*d)/(d*(a + b*x))])/b

Rule 6742

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

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]

Rule 2508

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_)*PolyLog[n_, v_],
x_Symbol] :> With[{g = Simplify[(v*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*PolyL
og[n + 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[n + 1, v]*Lo
g[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, n, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]

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} \, dx &=\int \left (\frac{A^3}{a+b x}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x}\right ) \, dx\\ &=\frac{A^3 \log (a+b x)}{b}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, 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} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx\\ &=\frac{A^3 \log (a+b x)}{b}-\frac{3 A^2 B \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}-\frac{3 A B^2 \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}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \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{\log \left (-\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}+\frac{\left (6 A B^2 (b c-a d) n\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}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{\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 )}{(a+b x) (c+d x)} \, dx}{b}\\ &=\frac{A^3 \log (a+b x)}{b}-\frac{3 A^2 B \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}-\frac{3 A B^2 \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}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 A B^2 n \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}+\frac{3 B^3 n \log ^2\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}+\frac{\left (3 A^2 B (b c-a d) n\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}-\frac{\left (6 A B^2 (b c-a d) n^2\right ) \int \frac{\text{Li}_2\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}-\frac{\left (6 B^3 (b c-a d) n^2\right ) \int \frac{\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 )}{(a+b x) (c+d x)} \, dx}{b}\\ &=\frac{A^3 \log (a+b x)}{b}-\frac{3 A^2 B \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}-\frac{3 A B^2 \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}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 A B^2 n \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}+\frac{3 B^3 n \log ^2\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}+\frac{6 A B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}-\frac{\left (3 A^2 B (b c-a d) n\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}-\frac{\left (6 B^3 (b c-a d) n^3\right ) \int \frac{\text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b}\\ &=\frac{A^3 \log (a+b x)}{b}-\frac{3 A^2 B \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}-\frac{3 A B^2 \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}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{6 A B^2 n \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}+\frac{3 B^3 n \log ^2\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}+\frac{6 A B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{6 B^3 n^3 \text{Li}_4\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}-\frac{\left (3 A^2 B (b c-a d) n\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}\\ &=\frac{A^3 \log (a+b x)}{b}-\frac{3 A^2 B \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}-\frac{3 A B^2 \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}-\frac{B^3 \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{b}+\frac{3 A^2 B n \text{Li}_2\left (\frac{b (c+d x)}{d (a+b x)}\right )}{b}+\frac{6 A B^2 n \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}+\frac{3 B^3 n \log ^2\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}+\frac{6 A B^2 n^2 \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{Li}_3\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}+\frac{6 B^3 n^3 \text{Li}_4\left (1+\frac{b c-a d}{d (a+b x)}\right )}{b}\\ \end{align*}

Mathematica [B]  time = 1.02327, size = 2513, normalized size = 13.51 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*A^3*Log[a + b*x] - 6*A^2*B*n*Log[a + b*x]^2 + 4*A*B^2*n^2*Log[a + b*x]^3 - B^3*n^3*Log[a + b*x]^4 + B^3*n^3
*Log[(d*(a + b*x))/(-(b*c) + a*d)]^4 - 4*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]^3*Log[-((d*(a + b*x))/(b*(c
+ d*x)))] + 6*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]^2*Log[-((d*(a + b*x))/(b*(c + d*x)))]^2 - 4*B^3*n^3*L
og[(d*(a + b*x))/(-(b*c) + a*d)]*Log[-((d*(a + b*x))/(b*(c + d*x)))]^3 + B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d
*x)))]^4 - 12*A*B^2*n^2*Log[a + b*x]*Log[c + d*x]^2 + 12*B^3*n^3*Log[a + b*x]^2*Log[c + d*x]^2 + 12*A*B^2*n^2*
Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - 12*B^3*n^3*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*L
og[c + d*x]^2 - 8*B^3*n^3*Log[a + b*x]*Log[c + d*x]^3 + 8*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*
x]^3 + 12*A^2*B*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 12*A*B^2*n^2*Log[a + b*x]^2*Log[(b*(c + d*x))/
(b*c - a*d)] + 4*B^3*n^3*Log[a + b*x]^3*Log[(b*(c + d*x))/(b*c - a*d)] + 8*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) +
a*d)]^3*Log[(b*(c + d*x))/(b*c - a*d)] - 12*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]^2*Log[-((d*(a + b*x))/(
b*(c + d*x)))]*Log[(b*(c + d*x))/(b*c - a*d)] + 24*A*B^2*n^2*Log[a + b*x]*Log[c + d*x]*Log[(b*(c + d*x))/(b*c
- a*d)] - 24*B^3*n^3*Log[a + b*x]^2*Log[c + d*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 12*B^3*n^3*Log[a + b*x]*Log[
c + d*x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 6*B^3*n^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)]^2 + 12*B^3
*n^3*Log[a + b*x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[(b*(c + d*x))/(b*c - a*d)]^2 - 18*B^3*n^3*Log[(d*(a +
b*x))/(-(b*c) + a*d)]^2*Log[(b*(c + d*x))/(b*c - a*d)]^2 + 12*A^2*B*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)
^n] - 12*A*B^2*n*Log[a + b*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 4*B^3*n^2*Log[a + b*x]^3*Log[(e*(a + b*x)^n
)/(c + d*x)^n] - 12*B^3*n^2*Log[a + b*x]*Log[c + d*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 12*B^3*n^2*Log[(d*(
a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*A*B^2*n*Log[a + b*x]*Log[(b*(c
+ d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 12*B^3*n^2*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d
)]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 24*B^3*n^2*Log[a + b*x]*Log[c + d*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[
(e*(a + b*x)^n)/(c + d*x)^n] + 12*A*B^2*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 6*B^3*n*Log[a + b*x]
^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 12*B^3*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[(e*(a + b*x)^
n)/(c + d*x)^n]^2 + 4*B^3*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3 - 4*B^3*n^3*Log[-((d*(a + b*x))/(b*(
c + d*x)))]^3*Log[(b*c - a*d)/(b*c + b*d*x)] + 12*B*n*(A^2 + B^2*n^2*Log[(d*(a + b*x))/(-(b*c) + a*d)]^2 + B^2
*n^2*Log[c + d*x]^2 + 2*B^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 2*B^2*n^2*Log[(d*(a + b*x))/(-(b
*c) + a*d)]*(Log[-((d*(a + b*x))/(b*(c + d*x)))] + Log[(b*(c + d*x))/(b*c - a*d)]) + 2*A*B*Log[(e*(a + b*x)^n)
/(c + d*x)^n] + B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*B*n*Log[c + d*x]*(A - B*n*Log[a + b*x] + B*Log[(e*(
a + b*x)^n)/(c + d*x)^n]))*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 12*B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d
*x)))]^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] + 12*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]^2*PolyLog[2, (
b*(c + d*x))/(b*c - a*d)] - 24*B^3*n^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[-((d*(a + b*x))/(b*(c + d*x)))]*P
olyLog[2, (b*(c + d*x))/(b*c - a*d)] + 12*B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d*x)))]^2*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)] + 24*A*B^2*n^2*Log[c + d*x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 24*B^3*n^3*Log[a + b*x]*Lo
g[c + d*x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 12*B^3*n^3*Log[c + d*x]^2*PolyLog[2, (b*(c + d*x))/(b*c - a
*d)] + 24*B^3*n^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 24*B^3*n
^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[(b*(c + d*x))/(b*c - a*d)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 24
*B^3*n^2*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 24*A*B^2*n^2*Po
lyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] + 24*B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d*x)))]*PolyLog[3, (d*(a + b*x
))/(-(b*c) + a*d)] - 24*B^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] + 24
*B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d*x)))]*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] - 24*A*B^2*n^2*PolyLog[3,
(b*(c + d*x))/(b*c - a*d)] + 24*B^3*n^3*Log[-((d*(a + b*x))/(b*(c + d*x)))]*PolyLog[3, (b*(c + d*x))/(b*c - a
*d)] - 24*B^3*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] - 24*B^3*n^3*PolyLog[
4, (d*(a + b*x))/(b*(c + d*x))])/(4*b)

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Maple [F]  time = 2.592, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a} \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/(b*x+a),x)

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{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}}{b x + a}, 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/(b*x+a),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)/(b*x + a), 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),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}}{b x + a}\,{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),x, algorithm="giac")

[Out]

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