3.2.96 \(\int \frac {(a g+b g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^2} \, dx\) [196]

Optimal. Leaf size=282 \[ \frac {2 A B g n (a+b x)}{d i^2 (c+d x)}-\frac {2 B^2 g n^2 (a+b x)}{d i^2 (c+d x)}+\frac {2 B^2 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d i^2 (c+d x)}-\frac {g (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d i^2 (c+d x)}-\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^2 i^2}-\frac {2 b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}+\frac {2 b B^2 g n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.19, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2561, 2395, 2333, 2332, 2354, 2421, 6724} \begin {gather*} -\frac {2 b B g 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^2 i^2}+\frac {2 b B^2 g n^2 \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i^2}-\frac {b g \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^2 i^2}-\frac {g (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)}+\frac {2 A B g n (a+b x)}{d i^2 (c+d x)}+\frac {2 B^2 g n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d i^2 (c+d x)}-\frac {2 B^2 g n^2 (a+b x)}{d i^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

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 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

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 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

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 {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(196 c+196 d x)^2} \, dx &=\int \left (\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d (c+d x)^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{38416 d}-\frac {((b c-a d) g) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{38416 d}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B g 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+d x)}{a+b x} \, dx}{19208 d^2}-\frac {(B (b c-a d) g 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 )}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B g 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+d x)}{(a+b x) (c+d x)} \, dx}{19208 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {(b B (b c-a d) g n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{19208 d^2}-\frac {\left (B (b c-a d)^2 g n\right ) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19208 d^2}\\ &=\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{19208 d^2}+\frac {(b B g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{19208 d}-\frac {(b B (b c-a d) g n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d 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 (c+d x)}{(b c-a d) (c+d x)}\right ) \, dx}{19208 d^2}+\frac {(B (b c-a d) g n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{19208 d}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}+\frac {(b B g n) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}+\frac {\left (B^2 (b c-a d) g n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (b^2 B g n\right ) \int \left (\frac {A \log (c+d x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x}\right ) \, dx}{19208 d^2}+\frac {(b B g n) \int \left (\frac {A \log (c+d x)}{c+d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x}\right ) \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{19208 d^2}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{19208 d^2}\\ &=-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}-\frac {\left (A b^2 B g n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}+\frac {(A b B g n) \int \frac {\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {(A b B g n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b^2 B^2 g n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{19208 d^2}-\frac {\left (b^2 B^2 g n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{19208 d^2}+\frac {(A b B g n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{38416 d^2}+\frac {\left (b^2 B^2 g n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{38416 d}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}-\frac {\left (b^2 B^2 g 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+d x)}{a+b x} \, dx}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}+\frac {(A b B g n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (b B^2 g n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )}{x} \, dx,x,a+b x\right )}{19208 d^2}-\frac {\left (b B^2 g n\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right ) \log \left (\left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )^{-n}\right )}{x} \, dx,x,a+b x\right )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{38416 d^2}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}+\frac {\left (b B^2 g n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)}{c+d x} \, dx}{19208 d}+\frac {\left (b B^2 g 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 (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{19208 d}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (B^2 g\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^n\right )}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{38416 d}+\frac {\left (B^2 g 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 )}{19208 d}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{38416 d^2}+\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {d \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}-\frac {\left (B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-b c+a d}{b}+\frac {d x}{b}\right )}{-\frac {-b c+a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{19208 d}+\frac {\left (b B^2 g 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+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g n\right ) \text {Subst}\left (\int \frac {\log \left (x^n\right ) \log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (b^2 B^2 g\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 )}{38416 d^3}+\frac {\left (b^2 B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c+a d}{d}+\frac {b x}{d}} \, dx,x,c+d x\right )}{38416 d^3}-\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {\left (b B^2 g 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 )}{19208 d^2}-\frac {\left (b B^2 g n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-2 \frac {\left (b B^2 g 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 )}{19208 d^2}\\ &=\frac {B^2 (b c-a d) g n^2}{19208 d^2 (c+d x)}+\frac {b B^2 g n^2 \log (a+b x)}{19208 d^2}+\frac {b B^2 g n^2 \log ^2(a+b x)}{38416 d^2}-\frac {B (b c-a d) g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2 (c+d x)}-\frac {b B g n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{19208 d^2}+\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{38416 d^2 (c+d x)}-\frac {b B^2 g n^2 \log (c+d x)}{19208 d^2}-\frac {A b B g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{19208 d^2}-\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{38416 d^2}+\frac {b B g n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{19208 d^2}+\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{38416 d^2}+\frac {A b B g n \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^2(c+d x)}{38416 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{38416 d^2}+\frac {b B^2 g n^2 \log ^3(c+d x)}{115248 d^2}-\frac {b B^2 g n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38416 d^2}-\frac {b B^2 g n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{19208 d^2}-\frac {b B^2 g \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{38416 d^2}+\frac {b B^2 g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {A b B g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}+\frac {b B^2 g 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 )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{19208 d^2}-\frac {b B^2 g n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{19208 d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1261\) vs. \(2(282)=564\).
time = 1.21, size = 1261, normalized size = 4.47 \begin {gather*} \frac {g \left (\frac {(b c-a d) \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}{c+d x}+b \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)+\frac {2 a B d 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 (b c-a d+b (c+d x) \log \left (\frac {a}{b}+x\right )+(-b c+a d) \log \left (\frac {a+b x}{c+d x}\right )-b c \log \left (\frac {b (c+d x)}{b c-a d}\right )-b d x \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )}{(-b c+a d) (c+d x)}+b 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 \log \left (\frac {c}{d}+x\right ) \log (c+d x)+2 \left (-\frac {c}{c+d x}+\frac {b c \log (a+b x)}{-b c+a d}+\frac {b c \log (c+d x)}{b c-a d}-\log \left (\frac {a}{b}+x\right ) \log (c+d x)+\log \left (\frac {a+b x}{c+d x}\right ) \left (\frac {c}{c+d x}+\log (c+d x)\right )+\log \left (\frac {a}{b}+x\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )+2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-\frac {a B^2 d n^2 \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {a+b x}{c+d x}\right )+(b c-a d) \log ^2\left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+b (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}+b B^2 n^2 \left (\frac {c \log ^2\left (\frac {a+b x}{c+d x}\right )}{c+d x}-\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 )+\frac {c \left (2 b c-2 a d+2 b (c+d x) \log (a+b x)-2 (b c-a d) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (a+b x) \log \left (\frac {a+b x}{c+d x}\right )-2 b (c+d x) \log (c+d x)-2 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+b (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+b (c+d x) \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d) (c+d x)}+2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{d^2 i^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-2*A*B*a*g*n*(b*log(b*x + a)/(b*c*d - a*d^2) - b*log(d*x + c)/(b*c*d - a*d^2) + 1/(d^2*x + c*d)) - A^2*b*g*(c/
(d^3*x + c*d^2) + log(d*x + c)/d^2) + 2*A*B*a*g*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(d^2*x + c*d) + A^2*a*g
/(d^2*x + c*d) - ((b*c*g - a*d*g)*B^2 + (B^2*b*d*g*x + B^2*b*c*g)*log(d*x + c))*log((d*x + c)^n)^2/(d^3*x + c*
d^2) + integrate(-(B^2*a*d*g + (B^2*b*d*g*x + B^2*a*d*g)*log((b*x + a)^n)^2 + (2*A*B*b*d*g + B^2*b*d*g)*x + 2*
(B^2*a*d*g + (A*B*b*d*g + B^2*b*d*g)*x)*log((b*x + a)^n) + 2*((a*d*g*(n - 1) - b*c*g*n)*B^2 - (A*B*b*d*g + B^2
*b*d*g)*x - (B^2*b*d*g*n*x + B^2*b*c*g*n)*log(d*x + c) - (B^2*b*d*g*x + B^2*a*d*g)*log((b*x + a)^n))*log((d*x
+ c)^n))/(d^3*x^2 + 2*c*d^2*x + c^2*d), 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((b*g*x+a*g)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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