3.1.42 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{c g+d g x} \, dx\) [42]

Optimal. Leaf size=137 \[ -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d g}-\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d g} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2551, 2354, 2421, 6724} \begin {gather*} -\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 )}{d g}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d 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}{d g} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*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))])/(d*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))])/(d*g) + (2*B^2*n^2*PolyLog[3, (d*(a + b*x))/(b*(
c + d*x))])/(d*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 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 2551

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/d)^m, Subst[Int[(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] && IntegersQ[m, p] &&
EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

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}{c g+d g x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d 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 (c g+d g x)}{a+b x} \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d 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 (c g+d g x)}{(a+b x) (c+d x)} \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d 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 (c g+d g x)}{(a+b x) (c+d x)} \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac {(2 B (b c-a d) n) \int \left (\frac {d \left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{(b c-a d) (a+b x)}\right ) \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac {(2 B n) \int \frac {\left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c g+d g x)}{c+d x} \, dx}{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 (c g+d g x)}{a+b x} \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac {(2 B n) \int \left (\frac {A \log (c g+d g x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{-c-d x}\right ) \, dx}{g}-\frac {(2 b B n) \int \left (\frac {A \log (c g+d g x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{a+b x}\right ) \, dx}{d g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}-\frac {(2 A B n) \int \frac {\log (c g+d g x)}{-c-d x} \, dx}{g}-\frac {\left (2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c g+d g x)}{-c-d x} \, dx}{g}-\frac {(2 A b B n) \int \frac {\log (c g+d g x)}{a+b x} \, dx}{d 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 (c g+d g x)}{a+b x} \, dx}{d g}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+(2 A B n) \int \frac {\log \left (\frac {d g (a+b x)}{-b c g+a d g}\right )}{c g+d g x} \, dx+\frac {(2 A B n) \text {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,c g+d g x\right )}{d g^2}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (c g+d g x)}{a+b x} \, dx}{d g}+\frac {\left (B^2 n^2\right ) \int \frac {\log ^2(c g+d g x)}{c+d x} \, dx}{g}-\frac {\left (b B^2 n^2\right ) \int \frac {\log ^2(c g+d g x)}{a+b x} \, dx}{d 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 (c g+d g x)}{a+b x} \, dx}{d g}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c g+d g x\right )}{d g}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c g-a d g}{b}+\frac {d g x}{b}\right )}{x} \, dx,x,a+b x\right )}{d 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 c g+a d g}{b}+\frac {d g x}{b}\right )}{x} \, dx,x,a+b x\right )}{d g}+\left (2 B^2 n^2\right ) \int \frac {\log \left (\frac {d g (a+b x)}{-b c g+a d g}\right ) \log (c g+d g x)}{c g+d g x} \, dx+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,c g+d g x\right )}{d g^2}+\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 {d g (a+b x)}{-b c g+a d g}\right )}{c g+d g x} \, dx\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c g-a d g}{b}+\frac {d g x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{b g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c g+a d g}{b}+\frac {d g x}{b}\right )}{-\frac {-b c g+a d g}{b}+\frac {d g x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c g+d g x\right )}{d g}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {d g \left (\frac {-b c g+a d g}{d g}+\frac {b x}{d g}\right )}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d 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 c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \log (g (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d 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 (c+d x)}{b c-a d}\right )}{d g}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d g x}{b c g-a d g}\right )}{x} \, dx,x,a+b x\right )}{d g}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (g (c+d x))\right )}{d g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{d g}+\frac {b x}{d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \log (g (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d 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 (c+d x)}{b c-a d}\right )}{d g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}+\frac {\left (b B^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{d^2 g}+\frac {\left (b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c g+a d g}{d g}+\frac {b x}{d g}} \, dx,x,c g+d g x\right )}{d^2 g^2}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d g x}{b c g-a d g}\right )}{x} \, dx,x,a+b x\right )}{d g}\\ &=\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \log (g (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d 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 (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}+\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d 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 (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d g}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c g+a d g}\right )}{x} \, dx,x,c g+d g x\right )}{d g}\\ &=\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (g (c+d x))}{d g}+\frac {A B n \log ^2(g (c+d x))}{d g}-\frac {B^2 n^2 \log (a+b x) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(g (c+d x))}{d g}+\frac {B^2 n^2 \log ^3(g (c+d x))}{3 d g}-\frac {2 B^2 n \log (a+b x) \log (g (c+d x)) \log \left ((c+d x)^{-n}\right )}{d g}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{d g}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{d g}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c g+d g x)}{d g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c g+d g x)}{d g}+\frac {2 B^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\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 (c g+d g x)}{d g}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c g+d g x)}{d g}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c g+d g x)}{d g}+\frac {2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d 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 (c+d x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{d g}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{d g}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(537\) vs. \(2(137)=274\).
time = 0.19, size = 537, normalized size = 3.92 \begin {gather*} \frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2 \log (c+d x)-3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {c}{d}+x\right )+2 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \log (c+d x)-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )+B^2 n^2 \left (\log ^3\left (\frac {c}{d}+x\right )+3 \log ^2\left (\frac {c}{d}+x\right ) \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )+3 \left (-\log \left (\frac {a}{b}+x\right )+\log \left (\frac {c}{d}+x\right )+\log \left (\frac {a+b x}{c+d x}\right )\right )^2 \log (c+d x)+3 \log ^2\left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+6 \log \left (\frac {a}{b}+x\right ) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )+3 \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {c}{d}+x\right )-2 \left (\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )\right )+6 \log \left (\frac {c}{d}+x\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-6 \text {Li}_3\left (\frac {d (a+b x)}{-b c+a d}\right )-6 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{3 d g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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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/(d*g*x+c*g),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{c\,g+d\,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/(c*g + d*g*x),x)

[Out]

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

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