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3.1.68 \(\int \frac {\log ^2(e (\frac {a+b x}{c+d x})^n) \log (h (f+g x)^m)}{(a+b x) (c+d x)} \, dx\) [68]

Optimal. Leaf size=496 \[ \frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {2 m n^2 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {2 m n^2 \text {Li}_4\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.39, antiderivative size = 496, normalized size of antiderivative = 1.00, 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 = {2589, 2553, 2404, 2354, 2421, 2430, 6724} \begin {gather*} -\frac {m \log ^2\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 {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \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 \text {PolyLog}\left (2,\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 {2 m n \text {PolyLog}\left (3,\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 {2 m n^2 \text {PolyLog}\left (4,\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{b c-a d}+\frac {2 m n^2 \text {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {\log \left (h (f+g x)^m\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)}-\frac {m \log ^3\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 )}{3 n (b c-a d)}+\frac {m \log \left (\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 )}{3 n (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q), x] - Dist[b*n*(p/q), Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(
p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m +
 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c - a*d, 0] && Inte
gerQ[m] && IGtQ[p, 0]

Rule 2589

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3463\) vs. \(2(496)=992\).
time = 6.76, size = 3463, normalized size = 6.98 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*m*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] - 2*m*n^2*Log[
(a + b*x)/(c + d*x)]^3*Log[(b*c - a*d)/(b*c + b*d*x)] - 3*m*n*Log[a/b + x]^2*Log[e*((a + b*x)/(c + d*x))^n]*Lo
g[f + g*x] - 3*m*n*Log[c/d + x]^2*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] + 6*m*n*Log[a/b + x]*Log[a + b*x
]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 6*m*n*Log[c/d + x]*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n]
*Log[f + g*x] + 6*m*n*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*
x] + 3*m*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[f + g*x] - 3*m*Log[c/d + x]*Log[e*((a + b*x)/(c + d
*x))^n]^2*Log[f + g*x] - 3*m*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[f + g*x] + 3*m*n^2*Log[a/b + x]
^2*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + 3*m*n^2*Log[c/d + x]^2*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] - 6*m*
n^2*Log[a/b + x]*Log[a + b*x]*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + 6*m*n^2*Log[c/d + x]*Log[a + b*x]*Log[(a
 + b*x)/(c + d*x)]*Log[f + g*x] - 6*m*n^2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[(a + b*x)/(c + d*
x)]*Log[f + g*x] - 6*m*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + 6
*m*n*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]*Log[f + g*x] + 3*m*n^2*Log[a/b + x]*
Log[(a + b*x)/(c + d*x)]^2*Log[f + g*x] - 3*m*n^2*Log[c/d + x]*Log[(a + b*x)/(c + d*x)]^2*Log[f + g*x] + 3*m*n
^2*Log[a + b*x]*Log[(a + b*x)/(c + d*x)]^2*Log[f + g*x] + 3*m*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(
c + d*x)]^2*Log[f + g*x] - 3*m*n^2*Log[(a + b*x)/(c + d*x)]^3*Log[f + g*x] - 6*m*n*Log[a/b + x]*Log[e*((a + b*
x)/(c + d*x))^n]*Log[c + d*x]*Log[f + g*x] + 6*m*n*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]*Lo
g[f + g*x] + 3*m*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[c + d*x]*Log[f + g*x] + 6*m*n^2*Log[a/b + x]*Log[(a + b*
x)/(c + d*x)]*Log[c + d*x]*Log[f + g*x] - 6*m*n^2*Log[c/d + x]*Log[(a + b*x)/(c + d*x)]*Log[c + d*x]*Log[f + g
*x] - 3*m*n^2*Log[(a + b*x)/(c + d*x)]^2*Log[c + d*x]*Log[f + g*x] + 6*m*n*Log[a/b + x]*Log[e*((a + b*x)/(c +
d*x))^n]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[f + g*x] - 6*m*n^2*Log[a/b + x]*Log[(a + b*x)/(c + d*x)]*Log[(b*(c
 + d*x))/(b*c - a*d)]*Log[f + g*x] - 3*m*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[(b*(f + g*x))/(b*f
- a*g)] + 6*m*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]*Log[(b*(f + g*x))/(b*f -
a*g)] - 3*m*n^2*Log[a/b + x]*Log[(a + b*x)/(c + d*x)]^2*Log[(b*(f + g*x))/(b*f - a*g)] + 3*m*Log[c/d + x]*Log[
e*((a + b*x)/(c + d*x))^n]^2*Log[(d*(f + g*x))/(d*f - c*g)] - 6*m*n*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n
]*Log[(a + b*x)/(c + d*x)]*Log[(d*(f + g*x))/(d*f - c*g)] + 3*m*n^2*Log[c/d + x]*Log[(a + b*x)/(c + d*x)]^2*Lo
g[(d*(f + g*x))/(d*f - c*g)] - 3*m*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]^2*Log[((b*c - a*d
)*(f + g*x))/((b*f - a*g)*(c + d*x))] + 2*m*n^2*Log[(a + b*x)/(c + d*x)]^3*Log[((b*c - a*d)*(f + g*x))/((b*f -
 a*g)*(c + d*x))] + 3*n*Log[a/b + x]^2*Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m] + 3*n*Log[c/d + x]^2*
Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m] - 6*n*Log[a/b + x]*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^
n]*Log[h*(f + g*x)^m] + 6*n*Log[c/d + x]*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m] - 6*n*
Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[e*((a + b*x)/(c + d*x))^n]*Log[h*(f + g*x)^m] + 3*Log[a + b
*x]*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[h*(f + g*x)^m] - 3*n^2*Log[a/b + x]^2*Log[(a + b*x)/(c + d*x)]*Log[h*
(f + g*x)^m] - 3*n^2*Log[c/d + x]^2*Log[(a + b*x)/(c + d*x)]*Log[h*(f + g*x)^m] + 6*n^2*Log[a/b + x]*Log[a + b
*x]*Log[(a + b*x)/(c + d*x)]*Log[h*(f + g*x)^m] - 6*n^2*Log[c/d + x]*Log[a + b*x]*Log[(a + b*x)/(c + d*x)]*Log
[h*(f + g*x)^m] + 6*n^2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[(a + b*x)/(c + d*x)]*Log[h*(f + g*x
)^m] - 3*n^2*Log[a + b*x]*Log[(a + b*x)/(c + d*x)]^2*Log[h*(f + g*x)^m] + n^2*Log[(a + b*x)/(c + d*x)]^3*Log[h
*(f + g*x)^m] + 6*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]*Log[h*(f + g*x)^m] - 6*n*Log[c/d
+ x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x]*Log[h*(f + g*x)^m] - 3*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[c
 + d*x]*Log[h*(f + g*x)^m] - 6*n^2*Log[a/b + x]*Log[(a + b*x)/(c + d*x)]*Log[c + d*x]*Log[h*(f + g*x)^m] + 6*n
^2*Log[c/d + x]*Log[(a + b*x)/(c + d*x)]*Log[c + d*x]*Log[h*(f + g*x)^m] + 3*n^2*Log[(a + b*x)/(c + d*x)]^2*Lo
g[c + d*x]*Log[h*(f + g*x)^m] - 6*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(b*(c + d*x))/(b*c - a*d)]
*Log[h*(f + g*x)^m] + 6*n^2*Log[a/b + x]*Log[(a + b*x)/(c + d*x)]*Log[(b*(c + d*x))/(b*c - a*d)]*Log[h*(f + g*
x)^m] - 6*n*(-Log[e*((a + b*x)/(c + d*x))^n] + n*Log[(a + b*x)/(c + d*x)])*(m*Log[f + g*x] - Log[h*(f + g*x)^m
])*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 3*m*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]
)^2*PolyLog[2, (g*(a + b*x))/(-(b*f) + a*g)] + 6*m*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*x)/(c + d*x)]*P
olyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 3*m*n^...

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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(e*((b*x+a)/(d*x+c))^n)^2*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^2*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

1/3*(n^2*log(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2 + 3*(n^2*log(b*x + a) - n)*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(d*x + c))*log((d*x + c)^n)^2 - 3*(n
^2*log(b*x + a)^2 - 2*n*log(b*x + a) + 1)*log(d*x + c) - 3*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(b*x
 + a) - 1)*log(d*x + c) - 2*log(b*x + a))*log((b*x + a)^n) + 3*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log
(b*x + a) - 1)*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)
^n) + 3*log(b*x + a))*log((g*x + f)^m)/(b*c - a*d) - integrate(1/3*((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(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*c*f*log(h) + 3*a*d*f*log(h) - 3*(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 - 3*(b*d*g*m*n*x^2 + a*c*g*m*n + (b*c*g*m*n + a*d*g*m*n)*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*lo
g(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*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)^2 - 3*(b*c*g*log(h) - a*d*g*log(h))*x + 3*(b*d*g*m*x^2 + a*c*g*m + (b*c*g*m
 + a*d*g*m)*x)*log(b*x + a) - 3*(b*d*g*m*x^2 + a*c*g*m + (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 + a*d*g*m)*x - 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(b*x + a))*log(d*x + c) - 3*(2*b*c*f*log(h) - 2*a*d*f*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(h) - a*d*g*log(h))*x - 2*(b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + 2*(b*d*g*m*x^2 +
a*c*g*m + (b*c*g*m + a*d*g*m)*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) + 3*(2*b*c*f*log(h) - 2*a*d*f*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*l
og(h) - a*d*g*log(h))*x - 2*(b*d*g*m*x^2 + a*c*g*m + (b*c*g*m + a*d*g*m)*x)*log(b*x + a) + 2*(b*d*g*m*x^2 + a*
c*g*m + (b*c*g*m + a*d*g*m)*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))/(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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^2*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*((b*x+a)/(d*x+c))**n)**2*ln(h*(g*x+f)**m)/(b*x+a)/(d*x+c),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)^2*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^2}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \end {gather*}

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)^2)/((a + b*x)*(c + d*x)),x)

[Out]

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

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