3.2.89 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{c i+d i x} \, dx\) [189]

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

[Out]

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

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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 i}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d i}-\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 i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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}{189 c+189 d x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d x)}{(a+b x) (c+d x)} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\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 (189 c+189 d 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 (189 c+189 d x)}{(b c-a d) (a+b x)}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \frac {\left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (189 c+189 d x)}{c+d x} \, dx-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 B n) \int \left (\frac {A \log (189 c+189 d x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x}\right ) \, dx-\frac {(2 b B n) \int \left (\frac {A \log (189 c+189 d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{a+b x}\right ) \, dx}{189 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}-\frac {1}{189} (2 A B n) \int \frac {\log (189 c+189 d x)}{-c-d x} \, dx-\frac {1}{189} \left (2 B^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (189 c+189 d x)}{-c-d x} \, dx-\frac {(2 A b B n) \int \frac {\log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+(2 A B n) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx-\frac {(2 A B n) \text {Subst}\left (\int -\frac {189 \log (x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}-\frac {\left (2 b B^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (189 c+189 d x)}{a+b x} \, dx}{189 d}+\frac {1}{189} \left (B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{c+d x} \, dx-\frac {\left (b B^2 n^2\right ) \int \frac {\log ^2(189 c+189 d x)}{a+b x} \, dx}{189 d}-\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 (189 c+189 d x)}{a+b x} \, dx}{189 d}\\ &=-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\frac {(2 A B n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {189 b c-189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}-\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 {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\left (2 B^2 n^2\right ) \int \frac {\log \left (\frac {189 d (a+b x)}{-189 b c+189 a d}\right ) \log (189 c+189 d x)}{189 c+189 d x} \, dx+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int \frac {189 \log ^2(x)}{x} \, dx,x,189 c+189 d x\right )}{35721 d}+\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 {189 d (a+b x)}{-189 b c+189 a d}\right )}{189 c+189 d x} \, dx\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\frac {B^2 \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {189 b c-189 a d}{b}+\frac {189 d 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 )}{189 b}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-189 b c+189 a d}{b}+\frac {189 d x}{b}\right )}{-\frac {-189 b c+189 a d}{b}+\frac {189 d 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,189 c+189 d x\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {189 d \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}+\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}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}+\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 )}{189 d}-\frac {\left (2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}+\frac {\left (B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (189 (c+d x))\right )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}+\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 c+189 d x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d^2}+\frac {\left (b B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-189 b c+189 a d}{189 d}+\frac {b x}{189 d}} \, dx,x,189 (c+d x)\right )}{35721 d^2}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {189 d x}{189 b c-189 a d}\right )}{x} \, dx,x,a+b x\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \log (189 (c+d x)) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}+\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 )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}-\frac {\left (2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-189 b c+189 a d}\right )}{x} \, dx,x,189 (c+d x)\right )}{189 d}\\ &=-\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log (189 (c+d x))}{189 d}+\frac {A B n \log ^2(189 (c+d x))}{189 d}-\frac {B^2 n^2 \log (a+b x) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 (c+d x))}{189 d}+\frac {B^2 n^2 \log ^3(189 (c+d x))}{567 d}+\frac {B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n \log (a+b x) \log (189 (c+d x)) \log \left ((c+d x)^{-n}\right )}{189 d}-\frac {B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{189 d}+\frac {B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{189 d}-\frac {2 A B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (189 c+189 d x)}{189 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (189 c+189 d x)}{189 d}+\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 (189 c+189 d x)}{189 d}-\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(189 c+189 d x)}{189 d}+\frac {B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(189 c+189 d x)}{189 d}+\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 )}{189 d}-\frac {2 A B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}-\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 )}{189 d}+\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 )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{189 d}-\frac {2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{189 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(306\) vs. \(2(137)=274\).
time = 0.12, size = 306, normalized size = 2.23 \begin {gather*} \frac {\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)-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 ^2\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 \log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )}{d i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{d i x +c i}\, dx\]

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*i*x+c*i),x)

[Out]

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

[Out]

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

[Out]

integral((-I*B^2*n^2*log((b*x + a)/(d*x + c))^2 - 2*(I*A*B + I*B^2)*n*log((b*x + a)/(d*x + c)) - I*A^2 - 2*I*A
*B - I*B^2)/(d*x + c), 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}{i} \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*i*x+c*i),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))/i

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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*i*x+c*i),x, algorithm="giac")

[Out]

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

[Out]

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

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