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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.51, 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 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 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 ^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 = 0.28, 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 {Li}_2\left (\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 {Li}_2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 \text {Li}_3\left (\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

________________________________________________________________________________________

fricas [F]  time = 1.34, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right )^{2}}{f x + e}, x\right ) \]

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 4733, normalized size = 14.70 \[ \text {output too large to display} \]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Memory limit reached. Please jump to an outer pointer, quit progra
m and enlarge thememory limits before executing the program again.

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}^2}{e+f\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (- \frac {a c f}{- a c f + a d e - b c f x + b d e x} - \frac {a d f x}{- a c f + a d e - b c f x + b d e x} + \frac {b c e}{- a c f + a d e - b c f x + b d e x} + \frac {b d e x}{- a c f + a d e - b c f x + b d e x} \right )}^{2}}{e + f x}\, dx \]

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]

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

________________________________________________________________________________________