3.259 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^3}{a f h+b g h x^2+h (b f x+a g x)} \, dx\)

Optimal. Leaf size=282 \[ \frac{6 B^2 n^2 \text{PolyLog}\left (3,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac{3 B n \text{PolyLog}\left (2,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac{\log \left (1-\frac{(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h (b f-a g)} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.31199, antiderivative size = 656, normalized size of antiderivative = 2.33, number of steps used = 17, number of rules used = 11, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used = {6688, 12, 6742, 36, 31, 2503, 2502, 2315, 2506, 6610, 2508} \[ \frac{3 A^2 B n \text{PolyLog}\left (2,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{6 A B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{6 A B^2 n^2 \text{PolyLog}\left (3,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{6 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (3,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{3 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text{PolyLog}\left (2,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}+\frac{6 B^3 n^3 \text{PolyLog}\left (4,\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}+1\right )}{h (b f-a g)}-\frac{3 A^2 B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}+\frac{A^3 \log (a+b x)}{h (b f-a g)}-\frac{A^3 \log (f+g x)}{h (b f-a g)}-\frac{3 A B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac{B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac{(f+g x) (b c-a d)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6688

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 2502

Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]

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

Mathematica [F]  time = 2.64478, size = 0, normalized size = 0. \[ \int \frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{a f h+b g h x^2+h (b f x+a g x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 3.54, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{afh+bgh{x}^{2}+h \left ( agx+bxf \right ) } \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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x)

[Out]

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

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

[Out]

A^3*(log(b*x + a)/((b*f - a*g)*h) - log(g*x + f)/((b*f - a*g)*h)) - integrate(-(B^3*log((b*x + a)^n)^3 - B^3*l
og((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))/(b*g*h*x^2 + a*f*h + (b*f*h + a*g*h)*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 g h x^{2} + a f h +{\left (b f + a g\right )} h x}, 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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),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*g*h*x^2 + a*f*h + (b*f + a*g)*h*x), 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/(a*f*h+b*g*h*x**2+h*(a*g*x+b*f*x)),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 g h x^{2} + a f h +{\left (b f x + a g x\right )} h}\,{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/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x, algorithm="giac")

[Out]

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