3.107 \(\int \frac{\log ^2(\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)})}{e+f x} \, dx\)

Optimal. Leaf size=322 \[ -\frac{2 \text{PolyLog}\left (3,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}-\frac{2 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac{2 \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text{PolyLog}\left (2,\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac{2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )}{f}-\frac{\log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac{\log \left (1-\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f} \]

[Out]

-((Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2)/f) + (Log[((b*e -
a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[1 - ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f - (2*L
og[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/f + (2*Log[((b*e
- a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f + (2
*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))])/f - (2*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/
f

________________________________________________________________________________________

Rubi [A]  time = 0.507152, antiderivative size = 334, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2489, 2488, 2506, 6610, 2503} \[ -\frac{2 \text{PolyLog}\left (3,1-\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{f}+\frac{2 \text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}-\frac{2 \log \left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text{PolyLog}\left (2,1-\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{f}+\frac{2 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right )}{f}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac{\log \left (\frac{(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right ) \log ^2\left (\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x} \, dx &=\frac{d \int \frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{c+d x} \, dx}{f}-\frac{(d e-c f) \int \frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(c+d x) (e+f x)} \, dx}{f}\\ &=-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}+\frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}-\frac{(2 (b c-a d)) \int \frac{\log \left (-\frac{-b c+a d}{b (c+d x)}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{f}+\frac{(2 (b c-a d)) \int \frac{\log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (-\frac{(-b c+a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{f}\\ &=-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}+\frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}+\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{f}-\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}+\frac{(2 (b c-a d)) \int \frac{\text{Li}_2\left (1+\frac{-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{f}-\frac{(2 (b c-a d)) \int \frac{\text{Li}_2\left (1+\frac{(-b c+a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{f}\\ &=-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}+\frac{\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}+\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{f}-\frac{2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{Li}_2\left (1-\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}+\frac{2 \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{f}-\frac{2 \text{Li}_3\left (1-\frac{(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{f}\\ \end{align*}

Mathematica [B]  time = 0.234267, size = 1080, normalized size = 3.35 \[ \frac{\log (e+f x) \log ^2\left (\frac{a}{b}+x\right )-\log \left (\frac{b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac{a}{b}+x\right )-2 \log \left (\frac{c}{d}+x\right ) \log (e+f x) \log \left (\frac{a}{b}+x\right )+2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x) \log \left (\frac{a}{b}+x\right )+2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right ) \log \left (\frac{a}{b}+x\right )-2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right ) \log \left (\frac{a}{b}+x\right )+2 \log \left (\frac{c}{d}+x\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right ) \log \left (\frac{a}{b}+x\right )-2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right ) \log \left (\frac{a}{b}+x\right )-\log \left (\frac{a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+\log ^2\left (\frac{c}{d}+x\right ) \log (e+f x)+\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x)-2 \log \left (\frac{c}{d}+x\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x)-\log ^2\left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )-\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )+2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{b (e+f x)}{b e-a f}\right )-\log ^2\left (\frac{c}{d}+x\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )+\log ^2\left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )+2 \log \left (\frac{c}{d}+x\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )-2 \log \left (\frac{f (c+d x)}{c f-d e}\right ) \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{d (e+f x)}{d e-c f}\right )+\log ^2\left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac{(a d-b c) (e+f x)}{(d e-c f) (a+b x)}\right )-2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )+2 \log \left (\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text{PolyLog}\left (2,\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+2 \text{PolyLog}\left (3,\frac{b (c+d x)}{d (a+b x)}\right )-2 \text{PolyLog}\left (3,\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2) + Log[a/b + x]^2*
Log[e + f*x] - 2*Log[a/b + x]*Log[c/d + x]*Log[e + f*x] + Log[c/d + x]^2*Log[e + f*x] + 2*Log[a/b + x]*Log[((b
*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[e + f*x] - 2*Log[c/d + x]*Log[((b*e - a*f)*(c + d*x))/((d*e
- c*f)*(a + b*x))]*Log[e + f*x] + Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[e + f*x] - Log[a/
b + x]^2*Log[(b*(e + f*x))/(b*e - a*f)] + 2*Log[a/b + x]*Log[(f*(c + d*x))/(-(d*e) + c*f)]*Log[(b*(e + f*x))/(
b*e - a*f)] - Log[(f*(c + d*x))/(-(d*e) + c*f)]^2*Log[(b*(e + f*x))/(b*e - a*f)] - 2*Log[a/b + x]*Log[((b*e -
a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(b*(e + f*x))/(b*e - a*f)] + 2*Log[(f*(c + d*x))/(-(d*e) + c*f)]*
Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(b*(e + f*x))/(b*e - a*f)] - Log[((b*e - a*f)*(c + d*
x))/((d*e - c*f)*(a + b*x))]^2*Log[(b*(e + f*x))/(b*e - a*f)] + 2*Log[a/b + x]*Log[c/d + x]*Log[(d*(e + f*x))/
(d*e - c*f)] - Log[c/d + x]^2*Log[(d*(e + f*x))/(d*e - c*f)] - 2*Log[a/b + x]*Log[(f*(c + d*x))/(-(d*e) + c*f)
]*Log[(d*(e + f*x))/(d*e - c*f)] + Log[(f*(c + d*x))/(-(d*e) + c*f)]^2*Log[(d*(e + f*x))/(d*e - c*f)] + 2*Log[
c/d + x]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(d*(e + f*x))/(d*e - c*f)] - 2*Log[(f*(c + d
*x))/(-(d*e) + c*f)]*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Log[(d*(e + f*x))/(d*e - c*f)] + Log
[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]^2*Log[((-(b*c) + a*d)*(e + f*x))/((d*e - c*f)*(a + b*x))] -
2*Log[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + 2*Log[((b*e -
a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*PolyLog[2, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))] + 2*Poly
Log[3, (b*(c + d*x))/(d*(a + b*x))] - 2*PolyLog[3, ((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/f

________________________________________________________________________________________

Maple [B]  time = 0.072, size = 4733, normalized size = 14.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{b c e - a c f +{\left (b d e - a d f\right )} x}{a d e - a c f +{\left (b d e - b c f\right )} x}\right )^{2}}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))**2/(f*x+e),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{{\left (b e - a f\right )}{\left (d x + c\right )}}{{\left (d e - c f\right )}{\left (b x + a\right )}}\right )^{2}}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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