3.67 \(\int \frac {\log ^3(e (\frac {a+b x}{c+d x})^n) \log (h (f+g x)^m)}{(a+b x) (c+d x)} \, dx\)

Optimal. Leaf size=620 \[ \frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_4\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{4 n (b c-a d)}+\frac {6 m n^2 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {m \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac {3 m n \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {m \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}+\frac {6 m n^3 \text {Li}_5\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {6 m n^3 \text {Li}_5\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d} \]

[Out]

1/4*m*ln(e*((b*x+a)/(d*x+c))^n)^4*ln((-a*d+b*c)/b/(d*x+c))/(-a*d+b*c)/n+1/4*ln(e*((b*x+a)/(d*x+c))^n)^4*ln(h*(
g*x+f)^m)/(-a*d+b*c)/n-1/4*m*ln(e*((b*x+a)/(d*x+c))^n)^4*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c
)/n+m*ln(e*((b*x+a)/(d*x+c))^n)^3*polylog(2,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)-m*ln(e*((b*x+a)/(d*x+c))^n)^3*poly
log(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)-3*m*n*ln(e*((b*x+a)/(d*x+c))^n)^2*polylog(3,d*(b*x+a)/
b/(d*x+c))/(-a*d+b*c)+3*m*n*ln(e*((b*x+a)/(d*x+c))^n)^2*polylog(3,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d
+b*c)+6*m*n^2*ln(e*((b*x+a)/(d*x+c))^n)*polylog(4,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)-6*m*n^2*ln(e*((b*x+a)/(d*x+c
))^n)*polylog(4,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)-6*m*n^3*polylog(5,d*(b*x+a)/b/(d*x+c))/(-a*d
+b*c)+6*m*n^3*polylog(5,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)

________________________________________________________________________________________

Rubi [A]  time = 1.13, antiderivative size = 649, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2507, 2489, 2488, 2506, 2508, 6610, 2503} \[ -\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {PolyLog}\left (4,1-\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {PolyLog}\left (2,1-\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {PolyLog}\left (3,1-\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac {6 m n^2 \text {PolyLog}\left (4,1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {m \text {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac {3 m n \text {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {6 m n^3 \text {PolyLog}\left (5,1-\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac {6 m n^3 \text {PolyLog}\left (5,1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{4 n (b c-a d)}+\frac {m \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 2507

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]

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]

Rubi steps

\begin {align*} \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx &=\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-\frac {(g m) \int \frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{4 (b c-a d) n}\\ &=\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-\frac {(d m) \int \frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4 (b c-a d) n}+\frac {((d f-c g) m) \int \frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x) (f+g x)} \, dx}{4 (b c-a d) n}\\ &=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-m \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (-\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+m \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (-\frac {(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-(3 m n) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+(3 m n) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\left (6 m n^2\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx-\left (6 m n^2\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1+\frac {(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_4\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\left (6 m n^3\right ) \int \frac {\text {Li}_4\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+\left (6 m n^3\right ) \int \frac {\text {Li}_4\left (1+\frac {(-b c+a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}+\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_4\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {6 m n^3 \text {Li}_5\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac {6 m n^3 \text {Li}_5\left (1-\frac {(b c-a d) (f+g x)}{(b f-a g) (c+d x)}\right )}{b c-a d}\\ \end {align*}

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Mathematica [B]  time = 22.15, size = 18164, normalized size = 29.30 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{3}}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

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maple [F]  time = 12.09, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3} \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

-1/4*(n^3*log(b*x + a)^4 + n^3*log(d*x + c)^4 - 4*n^2*log(b*x + a)^3*log(e) + 6*n*log(b*x + a)^2*log(e)^2 - 4*
(n^3*log(b*x + a) - n^2*log(e))*log(d*x + c)^3 - 4*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n)^3 + 4*(log(b
*x + a) - log(d*x + c))*log((d*x + c)^n)^3 - 4*log(b*x + a)*log(e)^3 + 6*(n^3*log(b*x + a)^2 - 2*n^2*log(b*x +
a)*log(e) + n*log(e)^2)*log(d*x + c)^2 + 6*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))
*log(d*x + c) - 2*log(b*x + a)*log(e))*log((b*x + a)^n)^2 + 6*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(
b*x + a) - log(e))*log(d*x + c) - 2*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n) - 2*log(b*x + a)*log(e))*lo
g((d*x + c)^n)^2 - 4*(n^3*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(e) + 3*n*log(b*x + a)*log(e)^2 - log(e)^3)
*log(d*x + c) - 4*(n^2*log(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) + 3*(n^2*log(b*x + a) -
n*log(e))*log(d*x + c)^2 + 3*log(b*x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n*log(b*x + a)*log(e) + log(e)
^2)*log(d*x + c))*log((b*x + a)^n) + 4*(n^2*log(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) +
3*(n^2*log(b*x + a) - n*log(e))*log(d*x + c)^2 + 3*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n)^2 + 3*log(b*
x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n*log(b*x + a)*log(e) + log(e)^2)*log(d*x + c) - 3*(n*log(b*x + a)
^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))*log(d*x + c) - 2*log(b*x + a)*log(e))*log((b*x + a)^n))*lo
g((d*x + c)^n))*log((g*x + f)^m)/(b*c - a*d) + integrate(1/4*(4*b*c*f*log(e)^3*log(h) - 4*a*d*f*log(e)^3*log(h
) + (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(b*x + a)^4 + (b*d*g*m*n^3*x^2 + a*c*g*
m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(d*x + c)^4 - 4*(b*d*g*m*n^2*x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c*
g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a)^3 + 4*(b*d*g*m*n^2*x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c
*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x - (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(
b*x + a))*log(d*x + c)^3 + 4*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a
*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*l
og((b*x + a)^n)^3 - 4*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g*m
+ (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((d*x
+ c)^n)^3 + 6*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*log(e)^2)*x)*log
(b*x + a)^2 + 6*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 +
a*d*g*m*n^3)*x)*log(b*x + a)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*log(e)^2)*x - 2*(b*d*g*m*n^2*x^2*log(e) + a*
c*g*m*n^2*log(e) + (b*c*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a))*log(d*x + c)^2 + 6*(2*b*c*f*log(
e)*log(h) - 2*a*d*f*log(e)*log(h) + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (
b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*
log(h))*x - 2*(b*d*g*m*x^2*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*
d*g*m*x^2*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*
m*n + a*d*g*m*n)*x)*log(b*x + a))*log(d*x + c))*log((b*x + a)^n)^2 + 6*(2*b*c*f*log(e)*log(h) - 2*a*d*f*log(e)
*log(h) + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (b*d*g*m*n*x^2 + a*c*g*m*n
+ (b*c*g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*log(h))*x - 2*(b*d*g*m*x^2
*log(e) + a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*d*g*m*x^2*log(e) + a*c*g*m
*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*
x + a))*log(d*x + c) + 2*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g
*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((
b*x + a)^n))*log((d*x + c)^n)^2 + 4*(b*c*g*log(e)^3*log(h) - a*d*g*log(e)^3*log(h))*x - 4*(b*d*g*m*x^2*log(e)^
3 + a*c*g*m*log(e)^3 + (b*c*g*m*log(e)^3 + a*d*g*m*log(e)^3)*x)*log(b*x + a) + 4*(b*d*g*m*x^2*log(e)^3 + a*c*g
*m*log(e)^3 - (b*d*g*m*n^3*x^2 + a*c*g*m*n^3 + (b*c*g*m*n^3 + a*d*g*m*n^3)*x)*log(b*x + a)^3 + 3*(b*d*g*m*n^2*
x^2*log(e) + a*c*g*m*n^2*log(e) + (b*c*g*m*n^2*log(e) + a*d*g*m*n^2*log(e))*x)*log(b*x + a)^2 + (b*c*g*m*log(e
)^3 + a*d*g*m*log(e)^3)*x - 3*(b*d*g*m*n*x^2*log(e)^2 + a*c*g*m*n*log(e)^2 + (b*c*g*m*n*log(e)^2 + a*d*g*m*n*l
og(e)^2)*x)*log(b*x + a))*log(d*x + c) + 4*(3*b*c*f*log(e)^2*log(h) - 3*a*d*f*log(e)^2*log(h) - (b*d*g*m*n^2*x
^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a)^3 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n
^2 + a*d*g*m*n^2)*x)*log(d*x + c)^3 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m
*n*log(e))*x)*log(b*x + a)^2 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(
e))*x - (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a))*log(d*x + c)^2 + 3*(b*c*
g*log(e)^2*log(h) - a*d*g*log(e)^2*log(h))*x - 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*c*g*m*log(e)^2
+ a*d*g*m*log(e)^2)*x)*log(b*x + a) + 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*d*g*m*n^2*x^2 + a*c*g*m*
n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x - 2*(b*d*g*m*n*x
^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a))*log(d*x + c))*log((b*x +
a)^n) - 4*(3*b*c*f*log(e)^2*log(h) - 3*a*d*f*log(e)^2*log(h) - (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2
+ a*d*g*m*n^2)*x)*log(b*x + a)^3 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(d*x + c
)^3 + 3*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a)^2 + 3
*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m*n*log(e) + a*d*g*m*n*log(e))*x - (b*d*g*m*n^2*x^2 + a*c*g
*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*log(b*x + a))*log(d*x + c)^2 + 3*(b*c*f*log(h) - a*d*f*log(h) + (b*c*g
*log(h) - a*d*g*log(h))*x - (b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + (b*d*g*m*x^2 + a*c*
g*m + (b*c*g*m + a*d*g*m)*x)*log(d*x + c))*log((b*x + a)^n)^2 + 3*(b*c*g*log(e)^2*log(h) - a*d*g*log(e)^2*log(
h))*x - 3*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x)*log(b*x + a) + 3
*(b*d*g*m*x^2*log(e)^2 + a*c*g*m*log(e)^2 + (b*d*g*m*n^2*x^2 + a*c*g*m*n^2 + (b*c*g*m*n^2 + a*d*g*m*n^2)*x)*lo
g(b*x + a)^2 + (b*c*g*m*log(e)^2 + a*d*g*m*log(e)^2)*x - 2*(b*d*g*m*n*x^2*log(e) + a*c*g*m*n*log(e) + (b*c*g*m
*n*log(e) + a*d*g*m*n*log(e))*x)*log(b*x + a))*log(d*x + c) + 3*(2*b*c*f*log(e)*log(h) - 2*a*d*f*log(e)*log(h)
+ (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a)^2 + (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*
g*m*n + a*d*g*m*n)*x)*log(d*x + c)^2 + 2*(b*c*g*log(e)*log(h) - a*d*g*log(e)*log(h))*x - 2*(b*d*g*m*x^2*log(e)
+ a*c*g*m*log(e) + (b*c*g*m*log(e) + a*d*g*m*log(e))*x)*log(b*x + a) + 2*(b*d*g*m*x^2*log(e) + a*c*g*m*log(e)
+ (b*c*g*m*log(e) + a*d*g*m*log(e))*x - (b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*x)*log(b*x + a))
*log(d*x + c))*log((b*x + a)^n))*log((d*x + c)^n))/(a*b*c^2*f - a^2*c*d*f + (b^2*c*d*g - a*b*d^2*g)*x^3 - (a*b
*d^2*f + a^2*d^2*g - (c*d*f + c^2*g)*b^2)*x^2 + (b^2*c^2*f + a*b*c^2*g - (d^2*f + c*d*g)*a^2)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(h*(f + g*x)^m)*log(e*((a + b*x)/(c + d*x))^n)^3)/((a + b*x)*(c + d*x)),x)

[Out]

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

[Out]

Timed out

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