\(\int \frac {\log ^2(e (\frac {a+b x}{c+d x})^n) \log (\frac {b c-a d}{b (c+d x)})}{(c+d x) (a g+b g x)} \, dx\) [92]

Optimal result

Integrand size = 58, antiderivative size = 160 \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=-\frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac {2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac {2 n^2 \operatorname {PolyLog}\left (4,1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g} \]

[Out]

-ln(e*((b*x+a)/(d*x+c))^n)^2*polylog(2,1+(a*d-b*c)/b/(d*x+c))/(-a*d+b*c)/g+2*n*ln(e*((b*x+a)/(d*x+c))^n)*polyl
og(3,1+(a*d-b*c)/b/(d*x+c))/(-a*d+b*c)/g-2*n^2*polylog(4,1+(a*d-b*c)/b/(d*x+c))/(-a*d+b*c)/g

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {2588, 2590, 6745} \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=-\frac {\operatorname {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}+\frac {2 n \operatorname {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {2 n^2 \operatorname {PolyLog}\left (4,1-\frac {b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \]

[In]

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

[Out]

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

Rule 2588

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)*PolyL
og[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]*(L
og[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 2590

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*PolyLo
g[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 6745

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*} \text {integral}& = -\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 )}{(b c-a d) g}+\frac {(2 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}{g} \\ & = -\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 )}{(b c-a d) g}+\frac {2 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) g}-\frac {\left (2 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}{g} \\ & = -\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 )}{(b c-a d) g}+\frac {2 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) g}-\frac {2 n^2 \text {Li}_4\left (1-\frac {b c-a d}{b (c+d x)}\right )}{(b c-a d) g} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(785\) vs. \(2(160)=320\).

Time = 0.31 (sec) , antiderivative size = 785, normalized size of antiderivative = 4.91 \[ \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 )}{(c+d x) (a g+b g x)} \, dx=\frac {\log \left (\frac {a+b x}{c+d x}\right ) \left (3 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {a+b x}{c+d x}\right )+n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+\frac {3}{2} \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2 \left (-\log ^2\left (\frac {c}{d}+x\right )-2 \log \left (\frac {a}{b}+x\right ) \log (c+d x)+2 \log \left (\frac {c}{d}+x\right ) \log (c+d x)+2 \log \left (\frac {a+b x}{c+d x}\right ) \log (c+d x)+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+n \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^3\left (\frac {c}{d}+x\right )+3 \log ^2\left (\frac {c}{d}+x\right ) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )+3 \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {a+b x}{c+d x}\right )\right )^2 \log (c+d x)+3 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+6 \log \left (\frac {a}{b}+x\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {c}{d}+x\right )-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )+6 \log \left (\frac {c}{d}+x\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )-6 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{-b c+a d}\right )-6 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right )\right )-n^2 \left (\log ^3\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+3 \log ^2\left (\frac {a+b x}{c+d x}\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )-6 \log \left (\frac {a+b x}{c+d x}\right ) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )+6 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )\right )}{3 (b c-a d) g} \]

[In]

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

[Out]

(Log[(a + b*x)/(c + d*x)]*(3*Log[e*((a + b*x)/(c + d*x))^n]^2 - 3*n*Log[e*((a + b*x)/(c + d*x))^n]*Log[(a + b*
x)/(c + d*x)] + n^2*Log[(a + b*x)/(c + d*x)]^2)*Log[(b*c - a*d)/(b*c + b*d*x)] + (3*(Log[e*((a + b*x)/(c + d*x
))^n] - n*Log[(a + b*x)/(c + d*x)])^2*(-Log[c/d + x]^2 - 2*Log[a/b + x]*Log[c + d*x] + 2*Log[c/d + x]*Log[c +
d*x] + 2*Log[(a + b*x)/(c + d*x)]*Log[c + d*x] + 2*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + 2*PolyLog[2,
(d*(a + b*x))/(-(b*c) + a*d)]))/2 + n*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])*(Log[c/d +
x]^3 + 3*Log[c/d + x]^2*(-Log[a/b + x] + Log[(d*(a + b*x))/(-(b*c) + a*d)]) + 3*(-Log[a/b + x] + Log[c/d + x]
+ Log[(a + b*x)/(c + d*x)])^2*Log[c + d*x] + 3*Log[a/b + x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 6*Log[a/b + x]
*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 3*(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)])*(Log[c/
d + x]^2 - 2*(Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])) + 6*Log
[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 6*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)] - 6*PolyLog[3, (b
*(c + d*x))/(b*c - a*d)]) - n^2*(Log[(a + b*x)/(c + d*x)]^3*Log[(b*c - a*d)/(b*c + b*d*x)] + 3*Log[(a + b*x)/(
c + d*x)]^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 6*Log[(a + b*x)/(c + d*x)]*PolyLog[3, (d*(a + b*x))/(b*(
c + d*x))] + 6*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))]))/(3*(b*c - a*d)*g)

Maple [F]

\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} \ln \left (\frac {-a d +c b}{b \left (d x +c \right )}\right )}{\left (d x +c \right ) \left (b g x +a g \right )}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=\int { \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )} {\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=\frac {\int \frac {\log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2} \log {\left (- \frac {a d}{b c + b d x} + \frac {b c}{b c + b d x} \right )}}{a c + a d x + b c x + b d x^{2}}\, dx}{g} \]

[In]

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

[Out]

Integral(log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2*log(-a*d/(b*c + b*d*x) + b*c/(b*c + b*d*x))/(a*c + a*d*x +
b*c*x + b*d*x**2), x)/g

Maxima [F]

\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=\int { \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )} {\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=\int { \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} \log \left (\frac {b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )} {\left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{(c+d x) (a g+b g x)} \, dx=\int \frac {\ln \left (-\frac {a\,d-b\,c}{b\,\left (c+d\,x\right )}\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{\left (a\,g+b\,g\,x\right )\,\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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