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

Optimal. Leaf size=425 \[ -\frac{6 B^2 n^2 \text{PolyLog}\left (3,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}+\frac{6 B^2 n^2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h}+\frac{3 B n \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h}-\frac{3 B n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{h}-\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{d (a+b x)}{b (c+d x)}\right )}{h}+\frac{\log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.64183, antiderivative size = 921, normalized size of antiderivative = 2.17, number of steps used = 25, number of rules used = 11, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6742, 2494, 2394, 2393, 2391, 2489, 2488, 2506, 6610, 2503, 2508} \[ \frac{\log (g+h x) A^3}{h}-\frac{3 B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x) A^2}{h}+\frac{3 B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x) A^2}{h}+\frac{3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x) A^2}{h}-\frac{3 B n \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right ) A^2}{h}+\frac{3 B n \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right ) A^2}{h}-\frac{3 B^2 \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 ) A}{h}+\frac{3 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) A}{h}-\frac{6 B^2 n \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 ) A}{h}+\frac{6 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) A}{h}+\frac{6 B^2 n^2 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) A}{h}-\frac{6 B^2 n^2 \text{PolyLog}\left (3,1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) A}{h}-\frac{B^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{3 B^3 n \log ^2\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 )}{h}+\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right )}{h}-\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (3,1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{6 B^3 n^3 \text{PolyLog}\left (4,1-\frac{b c-a d}{b (c+d x)}\right )}{h}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6742

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

Rule 2494

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]/((g_.) + (h_.)*(x_)), x_Sym
bol] :> Simp[(Log[g + h*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/h, x] + (-Dist[(b*p*r)/h, Int[Log[g + h*x]/(a
+ b*x), x], x] - Dist[(d*q*r)/h, Int[Log[g + h*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, p, q,
r}, x] && NeQ[b*c - a*d, 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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

Rule 2489

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_)/((g_.) + (h_.)*(x_)),
x_Symbol] :> Dist[d/h, Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[
Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/((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] && NeQ[d*g - c*h, 0] && IGtQ[s, 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 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 2503

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), 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)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; 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] && LinearQ[Simplify[1/
(u*(a + b*x))], 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}{g+h x} \, dx &=\int \left (\frac{A^3}{g+h x}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x}\right ) \, dx\\ &=\frac{A^3 \log (g+h x)}{h}+\left (3 A^2 B\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx+\left (3 A B^2\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx+B^3 \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{g+h x} \, dx\\ &=\frac{A^3 \log (g+h x)}{h}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{\left (3 A B^2 d\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{h}+\frac{\left (B^3 d\right ) \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{c+d x} \, dx}{h}-\frac{\left (3 A B^2 (d g-c h)\right ) \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{h}-\frac{\left (B^3 (d g-c h)\right ) \int \frac{\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{h}-\frac{\left (3 A^2 b B n\right ) \int \frac{\log (g+h x)}{a+b x} \, dx}{h}+\frac{\left (3 A^2 B d n\right ) \int \frac{\log (g+h x)}{c+d x} \, dx}{h}\\ &=-\frac{3 A B^2 \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 )}{h}-\frac{B^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A^3 \log (g+h x)}{h}-\frac{3 A^2 B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\left (3 A^2 B n\right ) \int \frac{\log \left (\frac{h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx-\left (3 A^2 B n\right ) \int \frac{\log \left (\frac{h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx+\frac{\left (6 A B^2 (b c-a d) n\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}{h}-\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 ) \log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\frac{\left (3 B^3 (b c-a d) n\right ) \int \frac{\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 )}{(a+b x) (c+d x)} \, dx}{h}-\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 ) \log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac{3 A B^2 \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 )}{h}-\frac{B^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A^3 \log (g+h x)}{h}-\frac{3 A^2 B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\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}{b (c+d x)}\right )}{h}-\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}{b (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{\left (3 A^2 B n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}-\frac{\left (3 A^2 B n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h}+\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}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}-\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\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}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}-\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac{3 A B^2 \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 )}{h}-\frac{B^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A^3 \log (g+h x)}{h}-\frac{3 A^2 B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{3 A^2 B n \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{3 A^2 B n \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\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}{b (c+d x)}\right )}{h}-\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}{b (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{6 A B^2 n^2 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h}+\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}{b (c+d x)}\right )}{h}-\frac{6 A B^2 n^2 \text{Li}_3\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{\left (6 B^3 (b c-a d) n^3\right ) \int \frac{\text{Li}_3\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}+\frac{\left (6 B^3 (b c-a d) n^3\right ) \int \frac{\text{Li}_3\left (1+\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{h}\\ &=-\frac{3 A B^2 \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 )}{h}-\frac{B^3 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h}+\frac{A^3 \log (g+h x)}{h}-\frac{3 A^2 B n \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B n \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{h}+\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{h}+\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{3 A^2 B n \text{Li}_2\left (\frac{b (g+h x)}{b g-a h}\right )}{h}+\frac{3 A^2 B n \text{Li}_2\left (\frac{d (g+h x)}{d g-c h}\right )}{h}-\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}{b (c+d x)}\right )}{h}-\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}{b (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}+\frac{6 A B^2 n^2 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h}+\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}{b (c+d x)}\right )}{h}-\frac{6 A B^2 n^2 \text{Li}_3\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\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) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}-\frac{6 B^3 n^3 \text{Li}_4\left (1-\frac{b c-a d}{b (c+d x)}\right )}{h}+\frac{6 B^3 n^3 \text{Li}_4\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{h}\\ \end{align*}

Mathematica [F]  time = 1.45157, size = 0, normalized size = 0. \[ \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{g+h x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 3.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{hx+g} \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/(h*x+g),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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}}{h x + g}, 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/(h*x+g),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)/(h*x + g), 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((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**3/(h*x+g),x)

[Out]

Timed out

________________________________________________________________________________________

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}}{h x + g}\,{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/(h*x+g),x, algorithm="giac")

[Out]

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