3.1.71 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{f+g x} \, dx\) [71]

Optimal. Leaf size=297 \[ -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2553, 2404, 2354, 2421, 6724} \begin {gather*} \frac {2 B n \text {PolyLog}\left (2,\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac {2 B n \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{g}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{g}+\frac {\log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1441\) vs. \(2(297)=594\).
time = 0.26, size = 1441, normalized size = 4.85 \begin {gather*} \frac {-B^2 n^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+A^2 \log (f+g x)-2 A B n \log \left (\frac {a}{b}+x\right ) \log (f+g x)+B^2 n^2 \log ^2\left (\frac {a}{b}+x\right ) \log (f+g x)+2 A B n \log \left (\frac {c}{d}+x\right ) \log (f+g x)-2 B^2 n^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log (f+g x)+B^2 n^2 \log ^2\left (\frac {c}{d}+x\right ) \log (f+g x)+2 A B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)-2 B^2 n \log \left (\frac {a}{b}+x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)+2 B^2 n \log \left (\frac {c}{d}+x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (f+g x)+2 A B n \log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 n^2 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 n \log \left (\frac {a}{b}+x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 n^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 n^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )+2 B^2 n^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-B^2 n^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {b (f+g x)}{b f-a g}\right )-2 A B n \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+2 B^2 n^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-B^2 n^2 \log ^2\left (\frac {c}{d}+x\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 n \log \left (\frac {c}{d}+x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 n^2 \log \left (\frac {a}{b}+x\right ) \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+B^2 n^2 \log ^2\left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )-2 B^2 n^2 \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {d (f+g x)}{d f-c g}\right )+B^2 n^2 \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \log \left (\frac {(-b c+a d) (f+g x)}{(d f-c g) (a+b x)}\right )+2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {g (a+b x)}{-b f+a g}\right )-2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {g (c+d x)}{-d f+c g}\right )-2 B^2 n^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )+2 B^2 n^2 \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )+2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(B^2*n^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]^2) + A^2*Log
[f + g*x] - 2*A*B*n*Log[a/b + x]*Log[f + g*x] + B^2*n^2*Log[a/b + x]^2*Log[f + g*x] + 2*A*B*n*Log[c/d + x]*Log
[f + g*x] - 2*B^2*n^2*Log[a/b + x]*Log[c/d + x]*Log[f + g*x] + B^2*n^2*Log[c/d + x]^2*Log[f + g*x] + 2*A*B*Log
[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] - 2*B^2*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] +
2*B^2*n*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[f + g*x] + B^2*Log[e*((a + b*x)/(c + d*x))^n]^2*Log[f
+ g*x] + 2*A*B*n*Log[a/b + x]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*n^2*Log[a/b + x]^2*Log[(b*(f + g*x))/(b*f -
a*g)] + 2*B^2*n*Log[a/b + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(b*(f + g*x))/(b*f - a*g)] + 2*B^2*n^2*Log[a/
b + x]*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*n^2*Log[(g*(c + d*x))/(-(d*f) +
c*g)]^2*Log[(b*(f + g*x))/(b*f - a*g)] + 2*B^2*n^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[((b*f - a*g)*(c + d*x
))/((d*f - c*g)*(a + b*x))]*Log[(b*(f + g*x))/(b*f - a*g)] - B^2*n^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*
(a + b*x))]^2*Log[(b*(f + g*x))/(b*f - a*g)] - 2*A*B*n*Log[c/d + x]*Log[(d*(f + g*x))/(d*f - c*g)] + 2*B^2*n^2
*Log[a/b + x]*Log[c/d + x]*Log[(d*(f + g*x))/(d*f - c*g)] - B^2*n^2*Log[c/d + x]^2*Log[(d*(f + g*x))/(d*f - c*
g)] - 2*B^2*n*Log[c/d + x]*Log[e*((a + b*x)/(c + d*x))^n]*Log[(d*(f + g*x))/(d*f - c*g)] - 2*B^2*n^2*Log[a/b +
x]*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[(d*(f + g*x))/(d*f - c*g)] + B^2*n^2*Log[(g*(c + d*x))/(-(d*f) + c*g
)]^2*Log[(d*(f + g*x))/(d*f - c*g)] - 2*B^2*n^2*Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[((b*f - a*g)*(c + d*x))/
((d*f - c*g)*(a + b*x))]*Log[(d*(f + g*x))/(d*f - c*g)] + B^2*n^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a
+ b*x))]^2*Log[((-(b*c) + a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))] + 2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^
n] + B*n*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*PolyLog[2, (g*(a + b*x))/(-(b*f) + a*g)] - 2*B*
n*(A + B*Log[e*((a + b*x)/(c + d*x))^n] + B*n*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*PolyLog[2,
(g*(c + d*x))/(-(d*f) + c*g)] - 2*B^2*n^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*PolyLog[2, (b*
(c + d*x))/(d*(a + b*x))] + 2*B^2*n^2*Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*PolyLog[2, ((b*f -
a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))] + 2*B^2*n^2*PolyLog[3, (b*(c + d*x))/(d*(a + b*x))] - 2*B^2*n^2*PolyL
og[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/g

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{g x +f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f),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((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f),x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\right )^{2}}{f + g x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{f+g\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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