\(\int \frac {\log ^2(\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)})}{e+f x} \, dx\) [107]

Optimal result

Integrand size = 42, antiderivative size = 322 \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (1-\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}-\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{f}+\frac {2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}+\frac {2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{f}-\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f} \]

[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] (verified)

Time = 0.30 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2554, 2404, 2354, 2421, 6724} \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x} \, dx=-\frac {2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f}-\frac {2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac {2 \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{f}+\frac {2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{f} \]

[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)/(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

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 2554

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1080\) vs. \(2(322)=644\).

Time = 0.40 (sec) , antiderivative size = 1080, normalized size of antiderivative = 3.35 \[ \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 {-\log \left (\frac {-b c+a d}{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 {a}{b}+x\right ) \log (e+f x)-2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (e+f x)+\log ^2\left (\frac {c}{d}+x\right ) \log (e+f x)+2 \log \left (\frac {a}{b}+x\right ) \log \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 {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log (e+f x)-\log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )-\log ^2\left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )-2 \log \left (\frac {a}{b}+x\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 )+2 \log \left (\frac {f (c+d x)}{-d e+c f}\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 {(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 {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-\log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {f (c+d x)}{-d e+c f}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )+\log ^2\left (\frac {f (c+d x)}{-d e+c f}\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)}{-d e+c f}\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 {(-b c+a d) (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 ) \operatorname {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 ) \operatorname {PolyLog}\left (2,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )-2 \operatorname {PolyLog}\left (3,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(812\) vs. \(2(321)=642\).

Time = 3.18 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.52

method	result	size
derivativedivides	\(\left (a f -b e \right ) \left (a d -c b \right ) \left (\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{\left (a f -b e \right ) f \left (a d -c b \right )}-\frac {b \left (c f -d e \right ) \left (\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (1+\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )\right )}{\left (b c f -b d e \right ) \left (a f -b e \right ) f \left (a d -c b \right )}\right )\)	\(813\)
default	\(\left (a f -b e \right ) \left (a d -c b \right ) \left (\frac {\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}-\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}+1\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{\left (a f -b e \right ) f \left (a d -c b \right )}-\frac {b \left (c f -d e \right ) \left (\ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )^{2} \ln \left (1+\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )+2 \ln \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right ) \operatorname {Li}_{2}\left (-\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {\left (b c f -b d e \right ) \left (-\frac {\left (a f -b e \right ) \left (a d -c b \right )}{b \left (c f -d e \right ) \left (b x +a \right )}+\frac {d \left (a f -b e \right )}{\left (c f -d e \right ) b}\right )}{-a d f +b d e}\right )\right )}{\left (b c f -b d e \right ) \left (a f -b e \right ) f \left (a d -c b \right )}\right )\)	\(813\)
risch	\(\text {Expression too large to display}\)	\(4733\)

[In]

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

[Out]

(a*f-b*e)*(a*d-b*c)*(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((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+1)+2*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)-2*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*(c*f-d*e)/(b*c*f-b*d*e)/(
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+(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))+2*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,-(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))-2*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))))

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x} \, dx=\text {Exception raised: RuntimeError} \]

[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.

Giac [F]

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

[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)

Mupad [F(-1)]

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

[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)