3.2.99 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) [199]

Optimal. Leaf size=392 \[ -\frac {2 A B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}+\frac {2 B^2 d^2 n^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B^2 n^2 (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g^2 i^2 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.26, antiderivative size = 392, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {2561, 2395, 2333, 2332, 2342, 2341, 2339, 30} \begin {gather*} -\frac {b^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac {2 b^2 B n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^3}+\frac {d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac {2 A B d^2 n (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac {2 b d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g^2 i^2 n (b c-a d)^3}-\frac {2 b^2 B^2 n^2 (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g^2 i^2 (c+d x) (b c-a d)^3}+\frac {2 B^2 d^2 n^2 (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[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 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]

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}{(199 c+199 d x)^2 (a g+b g x)^2} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)^2}-\frac {2 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)^2}+\frac {2 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (2 b^2 d\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b d^2\right ) \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}{39601 (b c-a d)^3 g^2}+\frac {b^2 \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2} \, dx}{39601 (b c-a d)^2 g^2}+\frac {d^2 \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}{39601 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d n) \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) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {(4 b B d 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}{39601 (b c-a d)^3 g^2}+\frac {(2 b 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 )}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {(2 B d 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}{39601 (b c-a d)^2 g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d n) \int \frac {(b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^3 g^2}-\frac {(4 b B d 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}{39601 (b c-a d)^3 g^2}+\frac {(2 b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d) g^2}+\frac {(2 B d n) \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}{39601 (b c-a d) g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 b B d n) \int \frac {\log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}-\frac {(4 b B d 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}{39601 (b c-a d)^2 g^2}+\frac {(2 b B n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b 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 (a+b x)}+\frac {d^2 \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}{39601 (b c-a d) g^2}+\frac {(2 B d n) \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}{39601 (b c-a d) g^2}\\ &=-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{39601 (b c-a d)^2 g^2}+\frac {(4 b B d n) \int \left (\frac {A \log (a+b x)}{(a+b x) (c+d x)}+\frac {B \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)}\right ) \, dx}{39601 (b c-a d)^2 g^2}-\frac {(4 b B d n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (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}{39601 (b c-a d)^2 g^2}-\frac {\left (2 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{39601 (b c-a d)^2 g^2}\\ &=-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B d 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}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B d^2 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)}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {\left (4 b B^2 d n\right ) \int \frac {\log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}-\frac {\left (2 B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{39601 (b c-a d)^2 g^2}\\ &=\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B^2 d\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B d 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}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B d^2 n\right ) \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}{39601 (b c-a d)^3 g^2}+\frac {(4 A B d n) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{39601 (b c-a d) g^2}-\frac {\left (2 B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{39601 (b c-a d) g^2}\\ &=\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 A b^2 B d n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B^2 d 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}{39601 (b c-a d)^3 g^2}-\frac {\left (4 A B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 A b B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d^2 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}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d n\right ) \int \frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}+\frac {\left (2 b B^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{39601 (b c-a d) g^2}-\frac {\left (2 B^2 d 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}{39601 (b c-a d) g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b^2 B^2 d n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 A b B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{39601 (b c-a d)^2 g^2}-\frac {\left (4 b^2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {(4 A b B d n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d^2 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}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d^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 (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 B^2 d^2\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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 B^2 d^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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b B^2 d n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 B^2 d^2 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b^2 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 )}{39601 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B^2 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}+\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-2 \frac {\left (4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}\\ &=-\frac {2 b B^2 n^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {2 B^2 d n^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {4 b B^2 d n^2 \log (a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(a+b x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (a+b x)}+\frac {2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^2 g^2 (c+d x)}-\frac {2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 A b B d n \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d n^2 \log ^3(c+d x)}{118803 (b c-a d)^3 g^2}-\frac {4 A b B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}-\frac {2 b B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {2 b B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d 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 )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 A b B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}+\frac {4 b B^2 d n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{39601 (b c-a d)^3 g^2}-\frac {4 b B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{39601 (b c-a d)^3 g^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(870\) vs. \(2(392)=784\).
time = 0.79, size = 870, normalized size = 2.22 \begin {gather*} -\frac {2 b B^2 d n^2 (a+b x) (c+d x) \log ^3\left (\frac {a+b x}{c+d x}\right )+3 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (2 a A b c d+b^2 B c^2 n-a^2 B d^2 n+2 A b^2 c d x+2 a A b d^2 x+2 b^2 B c d n x-2 a b B d^2 n x+2 A b^2 d^2 x^2+2 b B d (a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 b B d n (a+b x) (c+d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (A b c+a A d+b B c n-a B d n+2 A b d x+B (a d+b (c+2 d x)) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n (b c+a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 b d (a+b x) (c+d x) \log (a+b x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right )+3 b (b c-a d) (c+d x) \left (A^2+2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n (A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (A+B n-B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+3 d (b c-a d) (a+b x) \left (A^2-2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-A+B n+B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b d (a+b x) (c+d x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right ) \log (c+d x)}{3 (b c-a d)^3 g^2 i^2 (a+b x) (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(2*b*B^2*d*n^2*(a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^3 + 3*B*n*Log[(a + b*x)/(c + d*x)]^2*(2*a*A*b
*c*d + b^2*B*c^2*n - a^2*B*d^2*n + 2*A*b^2*c*d*x + 2*a*A*b*d^2*x + 2*b^2*B*c*d*n*x - 2*a*b*B*d^2*n*x + 2*A*b^2
*d^2*x^2 + 2*b*B*d*(a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*b*B*d*n*(a + b*x)*(c + d*x)*Log[(a +
b*x)/(c + d*x)]) + 6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(A*b*c + a*A*d + b*B*c*n - a*B*d*n + 2*A*b*d*x
+ B*(a*d + b*(c + 2*d*x))*Log[e*((a + b*x)/(c + d*x))^n] - B*n*(b*c + a*d + 2*b*d*x)*Log[(a + b*x)/(c + d*x)])
+ 6*b*d*(a + b*x)*(c + d*x)*Log[a + b*x]*(A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a
+ b*x)/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2) + 3*b*(b*c - a*d)*(c
+ d*x)*(A^2 + 2*A*B*n + 2*B^2*n^2 + B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(A + B*n)*Log[(a + b*x)/(c +
d*x)] + B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(A + B*n - B*n*Log[(a + b*x)/
(c + d*x)])) + 3*d*(b*c - a*d)*(a + b*x)*(A^2 - 2*A*B*n + 2*B^2*n^2 + B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2
*B*n*(-A + B*n)*Log[(a + b*x)/(c + d*x)] + B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x)
)^n]*(-A + B*n + B*n*Log[(a + b*x)/(c + d*x)])) - 6*b*d*(a + b*x)*(c + d*x)*(A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((
a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(
c + d*x)])^2)*Log[c + d*x])/((b*c - a*d)^3*g^2*i^2*(a + b*x)*(c + d*x))

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1854 vs. \(2 (371) = 742\).
time = 0.43, size = 1854, normalized size = 4.73 \begin {gather*} B^{2} {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )^{2} + 2 \, A B {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {2}{3} \, {\left (\frac {{\left (3 \, b^{2} c^{2} - 3 \, a^{2} d^{2} + {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{3} + 3 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{3} + 6 \, {\left (b^{2} c d - a b d^{2}\right )} x + 6 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, b^{2} d^{2} x^{2} + 2 \, a b c d + {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )\right )} n^{2}}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x} + \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} n \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right )}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x}\right )} B^{2} + \frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} A B n}{a b^{3} c^{4} g^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} - a^{4} c d^{3} g^{2} + {\left (b^{4} c^{3} d g^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} - a^{3} b d^{4} g^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} - 2 \, a b^{3} c^{3} d g^{2} + 2 \, a^{3} b c d^{3} g^{2} - a^{4} d^{4} g^{2}\right )} x} + A^{2} {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2}}\right )} \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/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

B^2*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*g^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c
*d^2 + a^3*d^3)*g^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*
c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2) - 2*b*d*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3
)*g^2))*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2 + 2*A*B*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 +
a^2*b*d^3)*g^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*g^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*
c*d^2)*g^2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2) - 2*b*d*log(d*x + c
)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 2/3*((3*
b^2*c^2 - 3*a^2*d^2 + (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^3 + 3*(b^2*d^2*x^2 + a*b*c*
d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*log(d*x + c)^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(d
*x + c)^3 + 6*(b^2*c*d - a*b*d^2)*x + 6*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a) - 3*(2*b^
2*d^2*x^2 + 2*a*b*c*d + (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^2 + 2*(b^2*c*d + a*b*d^2)
*x)*log(d*x + c))*n^2/(a*b^3*c^4*g^2 - 3*a^2*b^2*c^3*d*g^2 + 3*a^3*b*c^2*d^2*g^2 - a^4*c*d^3*g^2 + (b^4*c^3*d*
g^2 - 3*a*b^3*c^2*d^2*g^2 + 3*a^2*b^2*c*d^3*g^2 - a^3*b*d^4*g^2)*x^2 + (b^4*c^4*g^2 - 2*a*b^3*c^3*d*g^2 + 2*a^
3*b*c*d^3*g^2 - a^4*d^4*g^2)*x) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d
^2)*x)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*log(d*x + c) - (b^2*d^2
*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(d*x + c)^2)*n*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(a*b^3*c^4*g^
2 - 3*a^2*b^2*c^3*d*g^2 + 3*a^3*b*c^2*d^2*g^2 - a^4*c*d^3*g^2 + (b^4*c^3*d*g^2 - 3*a*b^3*c^2*d^2*g^2 + 3*a^2*b
^2*c*d^3*g^2 - a^3*b*d^4*g^2)*x^2 + (b^4*c^4*g^2 - 2*a*b^3*c^3*d*g^2 + 2*a^3*b*c*d^3*g^2 - a^4*d^4*g^2)*x))*B^
2 + 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)^2 + 2*(b^2
*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*log(d*x + c) - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*
b*d^2)*x)*log(d*x + c)^2)*A*B*n/(a*b^3*c^4*g^2 - 3*a^2*b^2*c^3*d*g^2 + 3*a^3*b*c^2*d^2*g^2 - a^4*c*d^3*g^2 + (
b^4*c^3*d*g^2 - 3*a*b^3*c^2*d^2*g^2 + 3*a^2*b^2*c*d^3*g^2 - a^3*b*d^4*g^2)*x^2 + (b^4*c^4*g^2 - 2*a*b^3*c^3*d*
g^2 + 2*a^3*b*c*d^3*g^2 - a^4*d^4*g^2)*x) + A^2*((2*b*d*x + b*c + a*d)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3
)*g^2*x^2 + (b^3*c^3 - a*b^2*c^2*d - a^2*b*c*d^2 + a^3*d^3)*g^2*x + (a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2)*g^
2) + 2*b*d*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2) - 2*b*d*log(d*x + c)/((b^3*c
^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^2))

________________________________________________________________________________________

Fricas [A]
time = 0.41, size = 740, normalized size = 1.89 \begin {gather*} \frac {3 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c^{2} - 3 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a^{2} d^{2} + 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + B^{2} a b c d n^{2} + {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 6 \, {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 3 \, {\left (2 \, {\left (A B + B^{2}\right )} b^{2} d^{2} n x^{2} + 2 \, {\left (A B + B^{2}\right )} a b c d n + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} + {\left ({\left (A B + B^{2}\right )} b^{2} c d + {\left (A B + B^{2}\right )} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, {\left ({\left (A B + B^{2}\right )} b^{2} c^{2} - 2 \, {\left (A B + B^{2}\right )} a b c d + {\left (A B + B^{2}\right )} a^{2} d^{2}\right )} n + 6 \, {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d - {\left (A^{2} + 2 \, A B + B^{2}\right )} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2}\right )} x + 6 \, {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} a b c d + {\left (B^{2} b^{2} c^{2} + B^{2} a^{2} d^{2}\right )} n^{2} + {\left (2 \, B^{2} b^{2} d^{2} n^{2} + {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} d^{2}\right )} x^{2} + {\left ({\left (A B + B^{2}\right )} b^{2} c^{2} - {\left (A B + B^{2}\right )} a^{2} d^{2}\right )} n + {\left ({\left (A^{2} + 2 \, A B + B^{2}\right )} b^{2} c d + {\left (A^{2} + 2 \, A B + B^{2}\right )} a b d^{2} + 2 \, {\left (B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left ({\left (A B + B^{2}\right )} b^{2} c d - {\left (A B + B^{2}\right )} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{3 \, {\left ({\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2}\right )}} \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/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

1/3*(3*(A^2 + 2*A*B + B^2)*b^2*c^2 - 3*(A^2 + 2*A*B + B^2)*a^2*d^2 + 2*(B^2*b^2*d^2*n^2*x^2 + B^2*a*b*c*d*n^2
+ (B^2*b^2*c*d + B^2*a*b*d^2)*n^2*x)*log((b*x + a)/(d*x + c))^3 + 6*(B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 3*(2*(A*
B + B^2)*b^2*d^2*n*x^2 + 2*(A*B + B^2)*a*b*c*d*n + (B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 2*((B^2*b^2*c*d - B^2*a*b
*d^2)*n^2 + ((A*B + B^2)*b^2*c*d + (A*B + B^2)*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c))^2 + 6*((A*B + B^2)*b^2*
c^2 - 2*(A*B + B^2)*a*b*c*d + (A*B + B^2)*a^2*d^2)*n + 6*((A^2 + 2*A*B + B^2)*b^2*c*d - (A^2 + 2*A*B + B^2)*a*
b*d^2 + 2*(B^2*b^2*c*d - B^2*a*b*d^2)*n^2)*x + 6*((A^2 + 2*A*B + B^2)*a*b*c*d + (B^2*b^2*c^2 + B^2*a^2*d^2)*n^
2 + (2*B^2*b^2*d^2*n^2 + (A^2 + 2*A*B + B^2)*b^2*d^2)*x^2 + ((A*B + B^2)*b^2*c^2 - (A*B + B^2)*a^2*d^2)*n + ((
A^2 + 2*A*B + B^2)*b^2*c*d + (A^2 + 2*A*B + B^2)*a*b*d^2 + 2*(B^2*b^2*c*d + B^2*a*b*d^2)*n^2 + 2*((A*B + B^2)*
b^2*c*d - (A*B + B^2)*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^4*c^3*d - 3*a*b^3*c^2*d^2 + 3*a^2*b^2*c*d^3
- a^3*b*d^4)*g^2*x^2 + (b^4*c^4 - 2*a*b^3*c^3*d + 2*a^3*b*c*d^3 - a^4*d^4)*g^2*x + (a*b^3*c^4 - 3*a^2*b^2*c^3
*d + 3*a^3*b*c^2*d^2 - a^4*c*d^3)*g^2)

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

[Out]

Timed out

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Giac [A]
time = 257.14, size = 164, normalized size = 0.42 \begin {gather*} {\left (\frac {{\left (d x + c\right )} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )} g^{2}} + \frac {2 \, {\left (B^{2} n^{2} + A B n + B^{2} n\right )} {\left (d x + c\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2}} + \frac {{\left (2 \, B^{2} n^{2} + 2 \, A B n + 2 \, B^{2} n + A^{2} + 2 \, A B + B^{2}\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \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/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

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

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

[Out]

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

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