3.69 \(\int \frac{\log (e (\frac{a+b x}{c+d x})^n) \log (h (f+g x)^m)}{(a+b x) (c+d x)} \, dx\)
Optimal. Leaf size=371 \[ -\frac{m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{m n \text{PolyLog}\left (3,\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{m n \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac{(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{2 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)} \]
[Out]
(m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[(b*c - a*d)/(b*(c + d*x))])/(2*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c +
d*x))^n]^2*Log[h*(f + g*x)^m])/(2*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[1 - ((d*f - c*g)*(
a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, (d*(a + b
*x))/(b*(c + d*x))])/(b*c - a*d) - (m*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f
- a*g)*(c + d*x))])/(b*c - a*d) - (m*n*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/(b*c - a*d) + (m*n*PolyLog[3,
((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)
________________________________________________________________________________________
Rubi [A] time = 0.557658, antiderivative size = 384, normalized size of antiderivative =
1.04, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules
used = {2507, 2489, 2488, 2506, 6610, 2503} \[ -\frac{m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \text{PolyLog}\left (2,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac{m \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac{m n \text{PolyLog}\left (3,1-\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{b c-a d}-\frac{m n \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac{m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (\frac{(f+g x) (b c-a d)}{(c+d x) (b f-a g)}\right )}{2 n (b c-a d)}+\frac{m \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[(Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m])/((a + b*x)*(c + d*x)),x]
[Out]
(m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[(b*c - a*d)/(b*(c + d*x))])/(2*(b*c - a*d)*n) - (m*Log[e*((a + b*x)/(c
+ d*x))^n]^2*Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])/(2*(b*c - a*d)*n) + (Log[e*((a + b*x)/(c +
d*x))^n]^2*Log[h*(f + g*x)^m])/(2*(b*c - a*d)*n) + (m*Log[e*((a + b*x)/(c + d*x))^n]*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]*PolyLog[2, 1 - ((b*c - a*d)*(f + g*x))/((b*
f - a*g)*(c + d*x))])/(b*c - a*d) - (m*n*PolyLog[3, 1 - (b*c - a*d)/(b*(c + d*x))])/(b*c - a*d) + (m*n*PolyLog
[3, 1 - ((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))])/(b*c - a*d)
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 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]
Rubi steps
\begin{align*} \int \frac{\log \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 ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}-\frac{(g m) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{2 (b c-a d) n}\\ &=\frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}-\frac{(d m) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{2 (b c-a d) n}+\frac{((d f-c g) m) \int \frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x) (f+g x)} \, dx}{2 (b c-a d) n}\\ &=\frac{m \log ^2\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 )}{2 (b c-a d) n}-\frac{m \log ^2\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 )}{2 (b c-a d) n}+\frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}-m \int \frac{\log \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 \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 ^2\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 )}{2 (b c-a d) n}-\frac{m \log ^2\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 )}{2 (b c-a d) n}+\frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}+\frac{m \log \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 \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}-(m n) \int \frac{\text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx+(m n) \int \frac{\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 ^2\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 )}{2 (b c-a d) n}-\frac{m \log ^2\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 )}{2 (b c-a d) n}+\frac{\log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{2 (b c-a d) n}+\frac{m \log \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 \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{m n \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{b c-a d}+\frac{m n \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}\\ \end{align*}
Mathematica [B] time = 6.02764, size = 1408, normalized size = 3.8 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[(Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m])/((a + b*x)*(c + d*x)),x]
[Out]
(m*n*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] - m*n*Log[a/b + x]^2*Log[f + g*x] - m*n*Log[c/d
+ x]^2*Log[f + g*x] + 2*m*n*Log[a/b + x]*Log[a + b*x]*Log[f + g*x] - 2*m*n*Log[c/d + x]*Log[a + b*x]*Log[f +
g*x] + 2*m*n*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[f + g*x] + 2*m*Log[a/b + x]*Log[e*((a + b*x)/(
c + d*x))^n]*Log[f + g*x] - 2*m*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*m*Log[a + b*x]*Lo
g[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*m*n*Log[a/b + x]*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + 2*m*n*L
og[c/d + x]*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + m*n*Log[(a + b*x)/(c + d*x)]^2*Log[f + g*x] - 2*m*n*Log[a/
b + x]*Log[c + d*x]*Log[f + g*x] + 2*m*n*Log[c/d + x]*Log[c + d*x]*Log[f + g*x] + 2*m*Log[e*((a + b*x)/(c + d*
x))^n]*Log[c + d*x]*Log[f + g*x] + 2*m*n*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[f + g*x] - 2*m*Log[a/
b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(b*(f + g*x))/(b*f - a*g)] + 2*m*n*Log[a/b + x]*Log[(a + b*x)/(c + d
*x)]*Log[(b*(f + g*x))/(b*f - a*g)] + 2*m*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(d*(f + g*x))/(d*f -
c*g)] - 2*m*n*Log[c/d + x]*Log[(a + b*x)/(c + d*x)]*Log[(d*(f + g*x))/(d*f - c*g)] - m*n*Log[(a + b*x)/(c + d
*x)]^2*Log[((b*c - a*d)*(f + g*x))/((b*f - a*g)*(c + d*x))] + n*Log[a/b + x]^2*Log[h*(f + g*x)^m] + n*Log[c/d
+ x]^2*Log[h*(f + g*x)^m] - 2*n*Log[a/b + x]*Log[a + b*x]*Log[h*(f + g*x)^m] + 2*n*Log[c/d + x]*Log[a + b*x]*L
og[h*(f + g*x)^m] - 2*n*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[h*(f + g*x)^m] + 2*Log[a + b*x]*Log
[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m] + 2*n*Log[a/b + x]*Log[c + d*x]*Log[h*(f + g*x)^m] - 2*n*Log[c/
d + x]*Log[c + d*x]*Log[h*(f + g*x)^m] - 2*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]*Log[h*(f + g*x)^m] - 2*
n*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[h*(f + g*x)^m] + 2*n*(m*Log[f + g*x] - Log[h*(f + g*x)^m])*P
olyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 2*m*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])*Po
lyLog[2, (g*(a + b*x))/(-(b*f) + a*g)] + 2*m*n*Log[(a + b*x)/(c + d*x)]*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))
] - 2*m*n*Log[(a + b*x)/(c + d*x)]*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))] + 2*m*n*Log[f +
g*x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*n*Log[h*(f + g*x)^m]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 2
*m*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[2, (g*(c + d*x))/(-(d*f) + c*g)] - 2*m*n*Log[(a + b*x)/(c + d*x)]*Po
lyLog[2, (g*(c + d*x))/(-(d*f) + c*g)] - 2*m*n*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))] + 2*m*n*PolyLog[3, ((d*
f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*b*c - 2*a*d)
________________________________________________________________________________________
Maple [F] time = 4.115, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{ \left ( bx+a \right ) \left ( dx+c \right ) }\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(e*((b*x+a)/(d*x+c))^n)*ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x)
[Out]
int(ln(e*((b*x+a)/(d*x+c))^n)*ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*((b*x+a)/(d*x+c))^n)*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="maxima")
[Out]
-1/2*(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(d*x + c))*log((d*x + c)^n) - 2*log(b*x + a)*log(e))*log((g*x
+ f)^m)/(b*c - a*d) + integrate(1/2*(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*lo
g(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*(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))/(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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 )}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*((b*x+a)/(d*x+c))^n)*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)/(b*d*x^2 + a*c + (b*c + a*d)*x), 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(ln(e*((b*x+a)/(d*x+c))**n)*ln(h*(g*x+f)**m)/(b*x+a)/(d*x+c),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 )}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*((b*x+a)/(d*x+c))^n)*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)/((b*x + a)*(d*x + c)), x)