3.249 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^4}{(f+g x) (a h+b h x)} \, dx\)

Optimal. Leaf size=361 \[ \frac {24 B^3 n^3 \text {Li}_4\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac {12 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)}+\frac {4 B n \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{h (b f-a g)}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^4}{h (b f-a g)}+\frac {24 B^4 n^4 \text {Li}_5\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{h (b f-a g)} \]

[Out]

-(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^4*ln(1-(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+4*B*n*(A+B*ln(e*
(b*x+a)^n/((d*x+c)^n)))^3*polylog(2,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+12*B^2*n^2*(A+B*ln(e*(
b*x+a)^n/((d*x+c)^n)))^2*polylog(3,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+24*B^3*n^3*(A+B*ln(e*(b
*x+a)^n/((d*x+c)^n)))*polylog(4,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+24*B^4*n^4*polylog(5,(-a*g
+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h

________________________________________________________________________________________

Rubi [B]  time = 1.93, antiderivative size = 1021, normalized size of antiderivative = 2.83, number of steps used = 20, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {6742, 36, 31, 2503, 2502, 2315, 2506, 6610, 2508} \[ \frac {\log (a+b x) A^4}{(b f-a g) h}-\frac {\log (f+g x) A^4}{(b f-a g) h}-\frac {4 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 ) A^3}{(b f-a g) h}+\frac {4 B n \text {PolyLog}\left (2,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^3}{(b f-a g) h}-\frac {6 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 ) A^2}{(b f-a g) h}+\frac {12 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^2}{(b f-a g) h}+\frac {12 B^2 n^2 \text {PolyLog}\left (3,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A^2}{(b f-a g) h}-\frac {4 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 ) A}{(b f-a g) h}+\frac {12 B^3 n \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}+\frac {24 B^3 n^2 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (3,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}+\frac {24 B^3 n^3 \text {PolyLog}\left (4,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right ) A}{(b f-a g) h}-\frac {B^4 \log ^4\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 {4 B^4 n \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (2,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac {12 B^4 n^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (3,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac {24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {PolyLog}\left (4,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h}+\frac {24 B^4 n^4 \text {PolyLog}\left (5,\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}+1\right )}{(b f-a g) h} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^4/((f + g*x)*(a*h + b*h*x)),x]

[Out]

(A^4*Log[a + b*x])/((b*f - a*g)*h) - (A^4*Log[f + g*x])/((b*f - a*g)*h) - (4*A^3*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) - (6*A^2*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) - (4*A*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) - (B^4*Log[
(e*(a + b*x)^n)/(c + d*x)^n]^4*Log[-(((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x)))])/((b*f - a*g)*h) + (4*A
^3*B*n*PolyLog[2, 1 + ((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (12*A^2*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) +
(12*A*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) + (4*B^4*n*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3*PolyLog[2, 1 + ((b*c - a*d)*(f + g*x))/((d*f -
c*g)*(a + b*x))])/((b*f - a*g)*h) + (12*A^2*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) + (24*A*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) + (12*B^4*n^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2*PolyLog[3, 1 + ((
b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (24*A*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) + (24*B^4*n^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]*PolyLog[4, 1
+ ((b*c - a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h) + (24*B^4*n^4*PolyLog[5, 1 + ((b*c - a*d)
*(f + g*x))/((d*f - c*g)*(a + b*x))])/((b*f - a*g)*h)

Rule 31

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

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 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 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 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 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 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]

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 6742

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

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx &=\int \left (\frac {A^4}{h (a+b x) (f+g x)}+\frac {4 A^3 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {6 A^2 B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {4 A B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}+\frac {B^4 \log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{h (a+b x) (f+g x)}\right ) \, dx\\ &=\frac {A^4 \int \frac {1}{(a+b x) (f+g x)} \, dx}{h}+\frac {\left (4 A^3 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 (6 A^2 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 {\left (4 A B^3\right ) \int \frac {\log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac {B^4 \int \frac {\log ^4\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}\\ &=-\frac {4 A^3 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 {6 A^2 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 {4 A 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 {B^4 \log ^4\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^4 b\right ) \int \frac {1}{a+b x} \, dx}{(b f-a g) h}-\frac {\left (A^4 g\right ) \int \frac {1}{f+g x} \, dx}{(b f-a g) h}+\frac {\left (4 A^3 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 (12 A^2 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 (12 A 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 {\left (4 B^4 (b c-a d) n\right ) \int \frac {\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 )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 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 {6 A^2 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 {4 A 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 {B^4 \log ^4\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 {12 A^2 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 {12 A 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 {4 B^4 n \log ^3\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 (4 A^3 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 (12 A^2 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 (24 A 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 {\left (12 B^4 (b c-a d) n^2\right ) \int \frac {\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 )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 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 {6 A^2 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 {4 A 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 {B^4 \log ^4\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 {4 A^3 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 {12 A^2 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 {12 A 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 {4 B^4 n \log ^3\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 {12 A^2 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 {24 A 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 {12 B^4 n^2 \log ^2\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 (24 A 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 {\left (24 B^4 (b c-a d) n^3\right ) \int \frac {\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 )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 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 {6 A^2 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 {4 A 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 {B^4 \log ^4\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 {4 A^3 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 {12 A^2 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 {12 A 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 {4 B^4 n \log ^3\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 {12 A^2 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 {24 A 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 {12 B^4 n^2 \log ^2\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 {24 A 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}+\frac {24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \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}-\frac {\left (24 B^4 (b c-a d) n^4\right ) \int \frac {\text {Li}_4\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^4 \log (a+b x)}{(b f-a g) h}-\frac {A^4 \log (f+g x)}{(b f-a g) h}-\frac {4 A^3 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 {6 A^2 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 {4 A 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 {B^4 \log ^4\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 {4 A^3 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 {12 A^2 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 {12 A 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 {4 B^4 n \log ^3\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 {12 A^2 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 {24 A 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 {12 B^4 n^2 \log ^2\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 {24 A 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}+\frac {24 B^4 n^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \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}+\frac {24 B^4 n^4 \text {Li}_5\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*}

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Mathematica [F]  time = 4.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^4}{(f+g x) (a h+b h x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^4/((f + g*x)*(a*h + b*h*x)),x]

[Out]

Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^4/((f + g*x)*(a*h + b*h*x)), x]

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fricas [F]  time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B^{4} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{4} + 4 \, A B^{3} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{3} + 6 \, A^{2} B^{2} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 4 \, A^{3} B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{4}}{b g h x^{2} + a f h + {\left (b f + a g\right )} h x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^4/(g*x+f)/(b*h*x+a*h),x, algorithm="fricas")

[Out]

integral((B^4*log((b*x + a)^n*e/(d*x + c)^n)^4 + 4*A*B^3*log((b*x + a)^n*e/(d*x + c)^n)^3 + 6*A^2*B^2*log((b*x
+ a)^n*e/(d*x + c)^n)^2 + 4*A^3*B*log((b*x + a)^n*e/(d*x + c)^n) + A^4)/(b*g*h*x^2 + a*f*h + (b*f + a*g)*h*x)
, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{4}}{{\left (b h x + a h\right )} {\left (g x + f\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^4/(g*x+f)/(b*h*x+a*h),x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^4/((b*h*x + a*h)*(g*x + f)), x)

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maple [F]  time = 5.97, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{4}}{\left (g x +f \right ) \left (b h x +a h \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^4/(g*x+f)/(b*h*x+a*h),x)

[Out]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^4/(g*x+f)/(b*h*x+a*h),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ A^{4} {\left (\frac {\log \left (b x + a\right )}{{\left (b f - a g\right )} h} - \frac {\log \left (g x + f\right )}{{\left (b f - a g\right )} h}\right )} + \int \frac {B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{4} + B^{4} \log \left ({\left (d x + c\right )}^{n}\right )^{4} + B^{4} \log \relax (e)^{4} + 4 \, A B^{3} \log \relax (e)^{3} + 6 \, A^{2} B^{2} \log \relax (e)^{2} + 4 \, A^{3} B \log \relax (e) + 4 \, {\left (B^{4} \log \relax (e) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{3} - 4 \, {\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right ) + B^{4} \log \relax (e) + A B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{3} + 6 \, {\left (B^{4} \log \relax (e)^{2} + 2 \, A B^{3} \log \relax (e) + A^{2} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 6 \, {\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + B^{4} \log \relax (e)^{2} + 2 \, A B^{3} \log \relax (e) + A^{2} B^{2} + 2 \, {\left (B^{4} \log \relax (e) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )^{2} + 4 \, {\left (B^{4} \log \relax (e)^{3} + 3 \, A B^{3} \log \relax (e)^{2} + 3 \, A^{2} B^{2} \log \relax (e) + A^{3} B\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 4 \, {\left (B^{4} \log \left ({\left (b x + a\right )}^{n}\right )^{3} + B^{4} \log \relax (e)^{3} + 3 \, A B^{3} \log \relax (e)^{2} + 3 \, A^{2} B^{2} \log \relax (e) + A^{3} B + 3 \, {\left (B^{4} \log \relax (e) + A B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 3 \, {\left (B^{4} \log \relax (e)^{2} + 2 \, A B^{3} \log \relax (e) + A^{2} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b g h x^{2} + a f h + {\left (b f h + a g h\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^4/(g*x+f)/(b*h*x+a*h),x, algorithm="maxima")

[Out]

A^4*(log(b*x + a)/((b*f - a*g)*h) - log(g*x + f)/((b*f - a*g)*h)) + integrate((B^4*log((b*x + a)^n)^4 + B^4*lo
g((d*x + c)^n)^4 + B^4*log(e)^4 + 4*A*B^3*log(e)^3 + 6*A^2*B^2*log(e)^2 + 4*A^3*B*log(e) + 4*(B^4*log(e) + A*B
^3)*log((b*x + a)^n)^3 - 4*(B^4*log((b*x + a)^n) + B^4*log(e) + A*B^3)*log((d*x + c)^n)^3 + 6*(B^4*log(e)^2 +
2*A*B^3*log(e) + A^2*B^2)*log((b*x + a)^n)^2 + 6*(B^4*log((b*x + a)^n)^2 + B^4*log(e)^2 + 2*A*B^3*log(e) + A^2
*B^2 + 2*(B^4*log(e) + A*B^3)*log((b*x + a)^n))*log((d*x + c)^n)^2 + 4*(B^4*log(e)^3 + 3*A*B^3*log(e)^2 + 3*A^
2*B^2*log(e) + A^3*B)*log((b*x + a)^n) - 4*(B^4*log((b*x + a)^n)^3 + B^4*log(e)^3 + 3*A*B^3*log(e)^2 + 3*A^2*B
^2*log(e) + A^3*B + 3*(B^4*log(e) + A*B^3)*log((b*x + a)^n)^2 + 3*(B^4*log(e)^2 + 2*A*B^3*log(e) + A^2*B^2)*lo
g((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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^4}{\left (f+g\,x\right )\,\left (a\,h+b\,h\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^4/((f + g*x)*(a*h + b*h*x)),x)

[Out]

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^4/((f + g*x)*(a*h + b*h*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**4/(g*x+f)/(b*h*x+a*h),x)

[Out]

Timed out

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