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 \[ \frac {\log \left (h (f+g x)^m\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}-\frac {m \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{2 n (b c-a d)}+\frac {m \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {m \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 n (b c-a d)}+\frac {m n \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m n \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 384, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, 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 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 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 \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 = 3.17, size = 1408, normalized size = 3.80 \[ \text {result too large to display} \]

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)

________________________________________________________________________________________

fricas [F]  time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)*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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)*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)

________________________________________________________________________________________

maple [F]  time = 6.68, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*((b*x+a)/(d*x+c))^n)*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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (n \log \left (b x + a\right )^{2} + n \log \left (d x + c\right )^{2} - 2 \, {\left (n \log \left (b x + a\right ) - \log \relax (e)\right )} \log \left (d x + c\right ) - 2 \, {\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) + 2 \, {\left (\log \left (b x + a\right ) - \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right ) - 2 \, \log \left (b x + a\right ) \log \relax (e)\right )} \log \left ({\left (g x + f\right )}^{m}\right )}{2 \, {\left (b c - a d\right )}} + \int \frac {2 \, b c f \log \relax (e) \log \relax (h) - 2 \, a d f \log \relax (e) \log \relax (h) + {\left (b d g m n x^{2} + a c g m n + {\left (b c g m n + a d g m n\right )} x\right )} \log \left (b x + a\right )^{2} + {\left (b d g m n x^{2} + a c g m n + {\left (b c g m n + a d g m n\right )} x\right )} \log \left (d x + c\right )^{2} + 2 \, {\left (b c g \log \relax (e) \log \relax (h) - a d g \log \relax (e) \log \relax (h)\right )} x - 2 \, {\left (b d g m x^{2} \log \relax (e) + a c g m \log \relax (e) + {\left (b c g m \log \relax (e) + a d g m \log \relax (e)\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (b d g m x^{2} \log \relax (e) + a c g m \log \relax (e) + {\left (b c g m \log \relax (e) + a d g m \log \relax (e)\right )} x - {\left (b d g m n x^{2} + a c g m n + {\left (b c g m n + a d g m n\right )} x\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right ) + 2 \, {\left (b c f \log \relax (h) - a d f \log \relax (h) + {\left (b c g \log \relax (h) - a d g \log \relax (h)\right )} x - {\left (b d g m x^{2} + a c g m + {\left (b c g m + a d g m\right )} x\right )} \log \left (b x + a\right ) + {\left (b d g m x^{2} + a c g m + {\left (b c g m + a d g m\right )} x\right )} \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (b c f \log \relax (h) - a d f \log \relax (h) + {\left (b c g \log \relax (h) - a d g \log \relax (h)\right )} x - {\left (b d g m x^{2} + a c g m + {\left (b c g m + a d g m\right )} x\right )} \log \left (b x + a\right ) + {\left (b d g m x^{2} + a c g m + {\left (b c g m + a d g m\right )} x\right )} \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, {\left (a b c^{2} f - a^{2} c d f + {\left (b^{2} c d g - a b d^{2} g\right )} x^{3} - {\left (a b d^{2} f + a^{2} d^{2} g - {\left (c d f + c^{2} g\right )} b^{2}\right )} x^{2} + {\left (b^{2} c^{2} f + a b c^{2} g - {\left (d^{2} f + c d g\right )} a^{2}\right )} x\right )}}\,{d x} \]

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)

________________________________________________________________________________________

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

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))/((a + b*x)*(c + d*x)),x)

[Out]

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

________________________________________________________________________________________

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)*ln(h*(g*x+f)**m)/(b*x+a)/(d*x+c),x)

[Out]

Timed out

________________________________________________________________________________________