3.3.7 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) [207]

Optimal. Leaf size=562 \[ -\frac {B^2 d^3 n^2 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {6 A b B d^2 n (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {6 b B^2 d^2 n^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {2 b^3 B^2 n^2 (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {6 b B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {B d^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {2 b^3 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{B (b c-a d)^4 g^2 i^3 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.32, antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 12, 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^3 (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^3 (a+b x) (b c-a d)^4}-\frac {2 b^3 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^3 (a+b x) (b c-a d)^4}-\frac {b^2 d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{B g^2 i^3 n (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {B d^3 n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b 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^3 (c+d x) (b c-a d)^4}-\frac {6 A b B d^2 n (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {2 b^3 B^2 n^2 (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {B^2 d^3 n^2 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {6 b 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^3 (c+d x) (b c-a d)^4}+\frac {6 b B^2 d^2 n^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \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)^3),x]

[Out]

-1/4*(B^2*d^3*n^2*(a + b*x)^2)/((b*c - a*d)^4*g^2*i^3*(c + d*x)^2) - (6*A*b*B*d^2*n*(a + b*x))/((b*c - a*d)^4*
g^2*i^3*(c + d*x)) + (6*b*B^2*d^2*n^2*(a + b*x))/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (2*b^3*B^2*n^2*(c + d*x))
/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (6*b*B^2*d^2*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)^4*g
^2*i^3*(c + d*x)) + (B*d^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^2*i^3*(c +
d*x)^2) - (2*b^3*B*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (d
^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^2*i^3*(c + d*x)^2) + (3*b*d^2*(a +
b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^2*i^3*(c + d*x)) - (b^3*(c + d*x)*(A + B*Log[
e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^2*i^3*(a + b*x)) - (b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^3)/(B*(b*c - a*d)^4*g^2*i^3*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}{(207 c+207 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (a+b x)^2}-\frac {b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 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}{8869743 (b c-a d)^2 g^2 (c+d x)^3}+\frac {2 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)^2}+\frac {b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^3 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^2 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}{2956581 (b c-a d)^4 g^2}+\frac {b^3 \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}{8869743 (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)^2} \, dx}{8869743 (b c-a d)^3 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)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\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) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B n\right ) \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}{8869743 (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 )}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}+\frac {(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)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^4 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 (c+d x)} \, dx}{8869743 (b c-a d)^2 g^2}+\frac {(4 b 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}{8869743 (b c-a d)^2 g^2}+\frac {(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)^3} \, dx}{8869743 (b c-a d) g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 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) (c+d x)} \, dx}{2956581 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B n\right ) \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}{8869743 (b c-a d)^2 g^2}+\frac {(4 b 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}{8869743 (b c-a d)^2 g^2}+\frac {(B d n) \int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (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)^3}-\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)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{8869743 (b c-a d) g^2}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^3 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^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} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^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} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^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} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 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}{8869743 (b c-a d)^3 g^2}+\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 b^2 B d n\right ) \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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b 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}{8869743 (b c-a d)^3 g^2}-\frac {\left (4 b 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}{8869743 (b c-a d)^3 g^2}-\frac {\left (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)^3} \, dx}{8869743 (b c-a d)^2 g^2}\\ &=-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{2956581 (b c-a d)^3 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (b B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (4 b B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^3 g^2}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{17739486 (b c-a d)^2 g^2}\\ &=\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^3 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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^4 g^2}+\frac {(2 A b 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 )}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d 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}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (b B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (4 b B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8869743 (b c-a d)^2 g^2}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{17739486 (b c-a d) g^2}\\ &=\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 A b^3 B d n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 A b 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d^2 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{8869743 (b c-a d)^2 g^2}-\frac {\left (b 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}{8869743 (b c-a d)^2 g^2}-\frac {\left (4 b 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}{8869743 (b c-a d)^2 g^2}-\frac {\left (B^2 d n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{17739486 (b c-a d) g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^3 B^2 d n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d^2 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^3 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^3 B^2 d n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d^2 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^3 g^2}-\frac {\left (2 b^3 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}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 A b^2 B d n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}-\frac {\left (b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}-\frac {\left (4 b^2 B^2 d 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 )}{8869743 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^2 B^2 d n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 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 ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 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 )}{2956581 (b c-a d)^4 g^2}+\frac {\left (b^3 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}+\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}-2 \frac {\left (2 b^2 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 )}{2956581 (b c-a d)^4 g^2}\\ &=-\frac {2 b^2 B^2 n^2}{8869743 (b c-a d)^3 g^2 (a+b x)}-\frac {B^2 d n^2}{35478972 (b c-a d)^2 g^2 (c+d x)^2}-\frac {11 b B^2 d n^2}{17739486 (b c-a d)^3 g^2 (c+d x)}-\frac {5 b^2 B^2 d n^2 \log (a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(a+b x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(a+b x)}{5913162 (b c-a d)^4 g^2}+\frac {b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (a+b x)}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17739486 (b c-a d)^2 g^2 (c+d x)^2}+\frac {5 b B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8869743 (b c-a d)^3 g^2 (c+d x)}+\frac {b^2 B d n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 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}{17739486 (b c-a d)^2 g^2 (c+d x)^2}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8869743 (b c-a d)^3 g^2 (c+d x)}-\frac {b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2956581 (b c-a d)^4 g^2}+\frac {5 b^2 B^2 d n^2 \log (c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {A b^2 B d n \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log ^2(c+d x)}{5913162 (b c-a d)^4 g^2}-\frac {b^2 B^2 d n^2 \log (a+b x) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log ^3(c+d x)}{8869743 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}-\frac {b^2 B^2 d \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 A b^2 B d n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}+\frac {b^2 B^2 d n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}+\frac {2 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 ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 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 )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{2956581 (b c-a d)^4 g^2}-\frac {2 b^2 B^2 d n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{2956581 (b c-a d)^4 g^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1334\) vs. \(2(562)=1124\).
time = 1.20, size = 1334, normalized size = 2.37 \begin {gather*} -\frac {4 b^2 B^2 d n^2 (a+b x) (c+d x)^2 \log ^3\left (\frac {a+b x}{c+d x}\right )+2 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (6 a A b^2 c^2 d+2 b^3 B c^3 n-6 a^2 b B c d^2 n+a^3 B d^3 n+6 A b^3 c^2 d x+12 a A b^2 c d^2 x+6 b^3 B c^2 d n x-12 a b^2 B c d^2 n x-3 a^2 b B d^3 n x+12 A b^3 c d^2 x^2+6 a A b^2 d^3 x^2-9 a b^2 B d^3 n x^2+6 A b^3 d^3 x^3-3 b^3 B d^3 n x^3+6 b^2 B d (a+b x) (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-6 b^2 B d n (a+b x) (c+d x)^2 \log \left (\frac {a+b x}{c+d x}\right )\right )+4 b^2 (b c-a d) (c+d x)^2 \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 )+2 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (2 b d (a+b x) (c+d x) \left (4 A-5 B n+4 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-4 B n \log \left (\frac {a+b x}{c+d x}\right )\right )+d (b c-a d) (a+b x) \left (2 A-B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \left (A+B n+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 )\right )+d (b c-a d)^2 (a+b x) \left (2 A^2-2 A B n+B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 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 (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+6 b^2 d (a+b x) (c+d x)^2 \log (a+b x) \left (2 A^2-2 A B n+5 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 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 (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+2 b d (b c-a d) (a+b x) (c+d x) \left (4 A^2-10 A B n+11 B^2 n^2+4 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-4 A+5 B n) \log \left (\frac {a+b x}{c+d x}\right )+4 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 (-4 A+5 B n+4 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b^2 d (a+b x) (c+d x)^2 \left (2 A^2-2 A B n+5 B^2 n^2+2 B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-2 A+B n) \log \left (\frac {a+b x}{c+d x}\right )+2 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 (-2 A+B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{4 (b c-a d)^4 g^2 i^3 (a+b x) (c+d x)^2} \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)^3),x]

[Out]

-1/4*(4*b^2*B^2*d*n^2*(a + b*x)*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]^3 + 2*B*n*Log[(a + b*x)/(c + d*x)]^2*(6*a
*A*b^2*c^2*d + 2*b^3*B*c^3*n - 6*a^2*b*B*c*d^2*n + a^3*B*d^3*n + 6*A*b^3*c^2*d*x + 12*a*A*b^2*c*d^2*x + 6*b^3*
B*c^2*d*n*x - 12*a*b^2*B*c*d^2*n*x - 3*a^2*b*B*d^3*n*x + 12*A*b^3*c*d^2*x^2 + 6*a*A*b^2*d^3*x^2 - 9*a*b^2*B*d^
3*n*x^2 + 6*A*b^3*d^3*x^3 - 3*b^3*B*d^3*n*x^3 + 6*b^2*B*d*(a + b*x)*(c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n]
- 6*b^2*B*d*n*(a + b*x)*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]) + 4*b^2*(b*c - a*d)*(c + d*x)^2*(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)])) + 2*B*(b*c
- a*d)*n*Log[(a + b*x)/(c + d*x)]*(2*b*d*(a + b*x)*(c + d*x)*(4*A - 5*B*n + 4*B*Log[e*((a + b*x)/(c + d*x))^n
] - 4*B*n*Log[(a + b*x)/(c + d*x)]) + d*(b*c - a*d)*(a + b*x)*(2*A - B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]
- 2*B*n*Log[(a + b*x)/(c + d*x)]) + 4*b^2*(c + d*x)^2*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a
+ b*x)/(c + d*x)])) + d*(b*c - a*d)^2*(a + b*x)*(2*A^2 - 2*A*B*n + B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x)
)^n]^2 + 2*B*n*(-2*A + B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a +
b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])) + 6*b^2*d*(a + b*x)*(c + d*x)^2*Log[a + b*x]
*(2*A^2 - 2*A*B*n + 5*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + B*n)*Log[(a + b*x)/(c +
d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*Log[(a
+ b*x)/(c + d*x)])) + 2*b*d*(b*c - a*d)*(a + b*x)*(c + d*x)*(4*A^2 - 10*A*B*n + 11*B^2*n^2 + 4*B^2*Log[e*((a +
b*x)/(c + d*x))^n]^2 + 2*B*n*(-4*A + 5*B*n)*Log[(a + b*x)/(c + d*x)] + 4*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 -
2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-4*A + 5*B*n + 4*B*n*Log[(a + b*x)/(c + d*x)])) - 6*b^2*d*(a + b*x)*(c +
d*x)^2*(2*A^2 - 2*A*B*n + 5*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + B*n)*Log[(a + b*x
)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*L
og[(a + b*x)/(c + d*x)]))*Log[c + d*x])/((b*c - a*d)^4*g^2*i^3*(a + b*x)*(c + d*x)^2)

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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3}}\, dx\]

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)^3,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)^3,x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4053 vs. \(2 (530) = 1060\).
time = 1.01, size = 4053, normalized size = 7.21 \begin {gather*} \text {Too large to display} \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)^3,x, algorithm="maxima")

[Out]

1/2*B^2*(6*b^2*d*log(b*x + a)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^
4)*g^2) - 6*b^2*d*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d
^4)*g^2) + (6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((I*b^4*c^3*d^2 - 3*I
*a*b^3*c^2*d^3 + 3*I*a^2*b^2*c*d^4 - I*a^3*b*d^5)*g^2*x^3 + (2*I*b^4*c^4*d - 5*I*a*b^3*c^3*d^2 + 3*I*a^2*b^2*c
^2*d^3 + I*a^3*b*c*d^4 - I*a^4*d^5)*g^2*x^2 + (I*b^4*c^5 - I*a*b^3*c^4*d - 3*I*a^2*b^2*c^3*d^2 + 5*I*a^3*b*c^2
*d^3 - 2*I*a^4*c*d^4)*g^2*x + (I*a*b^3*c^5 - 3*I*a^2*b^2*c^4*d + 3*I*a^3*b*c^3*d^2 - I*a^4*c^2*d^3)*g^2))*log(
(b*x/(d*x + c) + a/(d*x + c))^n*e)^2 + A*B*(6*b^2*d*log(b*x + a)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c
^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^2) - 6*b^2*d*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*
c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^2) + (6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d
+ a*b*d^2)*x)/((I*b^4*c^3*d^2 - 3*I*a*b^3*c^2*d^3 + 3*I*a^2*b^2*c*d^4 - I*a^3*b*d^5)*g^2*x^3 + (2*I*b^4*c^4*d
- 5*I*a*b^3*c^3*d^2 + 3*I*a^2*b^2*c^2*d^3 + I*a^3*b*c*d^4 - I*a^4*d^5)*g^2*x^2 + (I*b^4*c^5 - I*a*b^3*c^4*d -
3*I*a^2*b^2*c^3*d^2 + 5*I*a^3*b*c^2*d^3 - 2*I*a^4*c*d^4)*g^2*x + (I*a*b^3*c^5 - 3*I*a^2*b^2*c^4*d + 3*I*a^3*b*
c^3*d^2 - I*a^4*c^2*d^3)*g^2))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/4*((-8*I*b^3*c^3 - 15*I*a*b^2*c^2*d
+ 24*I*a^2*b*c*d^2 - I*a^3*d^3 - 4*(I*b^3*d^3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 + (I*b^3
*c^2*d + 2*I*a*b^2*c*d^2)*x)*log(b*x + a)^3 - 4*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*d^
3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(d*x + c)^3 - 30*(I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 - 6*(I*b^3*d^
3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x)*log(b*x + a)^2
- 6*(I*b^3*d^3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x + 2
*(I*b^3*d^3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x)*log(b
*x + a))*log(d*x + c)^2 - 3*(13*I*b^3*c^2*d - 6*I*a*b^2*c*d^2 - 7*I*a^2*b*d^3)*x - 30*(I*b^3*d^3*x^3 + I*a*b^2
*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x)*log(b*x + a) - 6*(-5*I*b^3*d^3
*x^3 - 5*I*a*b^2*c^2*d + 5*(-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 + 2*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*
c*d^2 - I*a*b^2*d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(b*x + a)^2 + 5*(-I*b^3*c^2*d - 2*I*a*b^2*c*
d^2)*x + 2*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*
d^2)*x)*log(b*x + a))*log(d*x + c))*n^2/(a*b^4*c^6*g^2 - 4*a^2*b^3*c^5*d*g^2 + 6*a^3*b^2*c^4*d^2*g^2 - 4*a^4*b
*c^3*d^3*g^2 + a^5*c^2*d^4*g^2 + (b^5*c^4*d^2*g^2 - 4*a*b^4*c^3*d^3*g^2 + 6*a^2*b^3*c^2*d^4*g^2 - 4*a^3*b^2*c*
d^5*g^2 + a^4*b*d^6*g^2)*x^3 + (2*b^5*c^5*d*g^2 - 7*a*b^4*c^4*d^2*g^2 + 8*a^2*b^3*c^3*d^3*g^2 - 2*a^3*b^2*c^2*
d^4*g^2 - 2*a^4*b*c*d^5*g^2 + a^5*d^6*g^2)*x^2 + (b^5*c^6*g^2 - 2*a*b^4*c^5*d*g^2 - 2*a^2*b^3*c^4*d^2*g^2 + 8*
a^3*b^2*c^3*d^3*g^2 - 7*a^4*b*c^2*d^4*g^2 + 2*a^5*c*d^5*g^2)*x) + 2*(-4*I*b^3*c^3 + 15*I*a*b^2*c^2*d - 12*I*a^
2*b*c*d^2 + I*a^3*d^3 - 6*(-I*b^3*c*d^2 + I*a*b^2*d^3)*x^2 - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d
^2 - I*a*b^2*d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d
+ (-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(-I*b^3*c^2*d -
2*I*a*b^2*c*d^2 + 3*I*a^2*b*d^3)*x - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 +
(-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(b*x + a) - 6*(I*b^3*d^3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^
2*d^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x + 2*(I*b^3*d^3*x^3 + I*a*b^2*c^2*d + (2*I*b^3*c*d^2 + I*a*b^2*d
^3)*x^2 + (I*b^3*c^2*d + 2*I*a*b^2*c*d^2)*x)*log(b*x + a))*log(d*x + c))*n*log((b*x/(d*x + c) + a/(d*x + c))^n
*e)/(a*b^4*c^6*g^2 - 4*a^2*b^3*c^5*d*g^2 + 6*a^3*b^2*c^4*d^2*g^2 - 4*a^4*b*c^3*d^3*g^2 + a^5*c^2*d^4*g^2 + (b^
5*c^4*d^2*g^2 - 4*a*b^4*c^3*d^3*g^2 + 6*a^2*b^3*c^2*d^4*g^2 - 4*a^3*b^2*c*d^5*g^2 + a^4*b*d^6*g^2)*x^3 + (2*b^
5*c^5*d*g^2 - 7*a*b^4*c^4*d^2*g^2 + 8*a^2*b^3*c^3*d^3*g^2 - 2*a^3*b^2*c^2*d^4*g^2 - 2*a^4*b*c*d^5*g^2 + a^5*d^
6*g^2)*x^2 + (b^5*c^6*g^2 - 2*a*b^4*c^5*d*g^2 - 2*a^2*b^3*c^4*d^2*g^2 + 8*a^3*b^2*c^3*d^3*g^2 - 7*a^4*b*c^2*d^
4*g^2 + 2*a^5*c*d^5*g^2)*x))*B^2 + 1/2*(-4*I*b^3*c^3 + 15*I*a*b^2*c^2*d - 12*I*a^2*b*c*d^2 + I*a^3*d^3 - 6*(-I
*b^3*c*d^2 + I*a*b^2*d^3)*x^2 - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 + (-I*b
^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*
d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(-I*b^3*c^2*d - 2*I*a*b^2*c*d^2 + 3*I*a^2*b*
d^3)*x - 6*(-I*b^3*d^3*x^3 - I*a*b^2*c^2*d + (-2*I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 + (-I*b^3*c^2*d - 2*I*a*b^2*c*
d^2)*x)*log(b*x + a) - 6*(I*b^3*d^3*x^3 + I*a*b...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1656 vs. \(2 (530) = 1060\).
time = 0.45, size = 1656, normalized size = 2.95 \begin {gather*} -\frac {4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 12 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a^{2} b c d^{2} + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a^{3} d^{3} + 4 \, {\left (i \, B^{2} b^{3} d^{3} n^{2} x^{3} + i \, B^{2} a b^{2} c^{2} d n^{2} + {\left (2 i \, B^{2} b^{3} c d^{2} + i \, B^{2} a b^{2} d^{3}\right )} n^{2} x^{2} + {\left (i \, B^{2} b^{3} c^{2} d + 2 i \, B^{2} a b^{2} c d^{2}\right )} n^{2} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} - {\left (-8 i \, B^{2} b^{3} c^{3} - 15 i \, B^{2} a b^{2} c^{2} d + 24 i \, B^{2} a^{2} b c d^{2} - i \, B^{2} a^{3} d^{3}\right )} n^{2} + 6 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + 2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b^{2} d^{3} + 5 \, {\left (i \, B^{2} b^{3} c d^{2} - i \, B^{2} a b^{2} d^{3}\right )} n^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{3} c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a b^{2} d^{3}\right )} n\right )} x^{2} + 2 \, {\left (6 \, {\left (i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d n + 3 \, {\left (-i \, B^{2} b^{3} d^{3} n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} d^{3} n\right )} x^{3} + {\left (2 i \, B^{2} b^{3} c^{3} - 6 i \, B^{2} a^{2} b c d^{2} + i \, B^{2} a^{3} d^{3}\right )} n^{2} + 3 \, {\left (-3 i \, B^{2} a b^{2} d^{3} n^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a b^{2} d^{3}\right )} n\right )} x^{2} + 3 \, {\left ({\left (2 i \, B^{2} b^{3} c^{2} d - 4 i \, B^{2} a b^{2} c d^{2} - i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left ({\left (i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 2 \, {\left (i \, A B + i \, B^{2}\right )} a b^{2} c d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (4 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 15 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c^{2} d + 12 \, {\left (i \, A B + i \, B^{2}\right )} a^{2} b c d^{2} + {\left (-i \, A B - i \, B^{2}\right )} a^{3} d^{3}\right )} n + 3 \, {\left (6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + 2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a^{2} b d^{3} + {\left (13 i \, B^{2} b^{3} c^{2} d - 6 i \, B^{2} a b^{2} c d^{2} - 7 i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{3} c^{2} d + 2 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + 3 \, {\left (i \, A B + i \, B^{2}\right )} a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c^{2} d + 3 \, {\left (5 i \, B^{2} b^{3} d^{3} n^{2} + 2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{3} d^{3} n + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} d^{3}\right )} x^{3} + {\left (4 i \, B^{2} b^{3} c^{3} + 12 i \, B^{2} a^{2} b c d^{2} - i \, B^{2} a^{3} d^{3}\right )} n^{2} + 3 \, {\left (6 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} d^{3} n + 4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c d^{2} + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} d^{3} + {\left (8 i \, B^{2} b^{3} c d^{2} + 7 i \, B^{2} a b^{2} d^{3}\right )} n^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{3} + 6 \, {\left (-i \, A B - i \, B^{2}\right )} a^{2} b c d^{2} + {\left (i \, A B + i \, B^{2}\right )} a^{3} d^{3}\right )} n + 3 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a b^{2} c d^{2} + {\left (4 i \, B^{2} b^{3} c^{2} d + 8 i \, B^{2} a b^{2} c d^{2} + 3 i \, B^{2} a^{2} b d^{3}\right )} n^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (-i \, A B - i \, B^{2}\right )} a b^{2} c d^{2} + {\left (-i \, A B - i \, B^{2}\right )} a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\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)^3,x, algorithm="fricas")

[Out]

-1/4*(4*(I*A^2 + 2*I*A*B + I*B^2)*b^3*c^3 + 6*(I*A^2 + 2*I*A*B + I*B^2)*a*b^2*c^2*d + 12*(-I*A^2 - 2*I*A*B - I
*B^2)*a^2*b*c*d^2 + 2*(I*A^2 + 2*I*A*B + I*B^2)*a^3*d^3 + 4*(I*B^2*b^3*d^3*n^2*x^3 + I*B^2*a*b^2*c^2*d*n^2 + (
2*I*B^2*b^3*c*d^2 + I*B^2*a*b^2*d^3)*n^2*x^2 + (I*B^2*b^3*c^2*d + 2*I*B^2*a*b^2*c*d^2)*n^2*x)*log((b*x + a)/(d
*x + c))^3 - (-8*I*B^2*b^3*c^3 - 15*I*B^2*a*b^2*c^2*d + 24*I*B^2*a^2*b*c*d^2 - I*B^2*a^3*d^3)*n^2 + 6*(2*(I*A^
2 + 2*I*A*B + I*B^2)*b^3*c*d^2 + 2*(-I*A^2 - 2*I*A*B - I*B^2)*a*b^2*d^3 + 5*(I*B^2*b^3*c*d^2 - I*B^2*a*b^2*d^3
)*n^2 + 2*((-I*A*B - I*B^2)*b^3*c*d^2 + (I*A*B + I*B^2)*a*b^2*d^3)*n)*x^2 + 2*(6*(I*A*B + I*B^2)*a*b^2*c^2*d*n
+ 3*(-I*B^2*b^3*d^3*n^2 + 2*(I*A*B + I*B^2)*b^3*d^3*n)*x^3 + (2*I*B^2*b^3*c^3 - 6*I*B^2*a^2*b*c*d^2 + I*B^2*a
^3*d^3)*n^2 + 3*(-3*I*B^2*a*b^2*d^3*n^2 + 2*(2*(I*A*B + I*B^2)*b^3*c*d^2 + (I*A*B + I*B^2)*a*b^2*d^3)*n)*x^2 +
3*((2*I*B^2*b^3*c^2*d - 4*I*B^2*a*b^2*c*d^2 - I*B^2*a^2*b*d^3)*n^2 + 2*((I*A*B + I*B^2)*b^3*c^2*d + 2*(I*A*B
+ I*B^2)*a*b^2*c*d^2)*n)*x)*log((b*x + a)/(d*x + c))^2 + 2*(4*(I*A*B + I*B^2)*b^3*c^3 + 15*(-I*A*B - I*B^2)*a*
b^2*c^2*d + 12*(I*A*B + I*B^2)*a^2*b*c*d^2 + (-I*A*B - I*B^2)*a^3*d^3)*n + 3*(6*(I*A^2 + 2*I*A*B + I*B^2)*b^3*
c^2*d + 4*(-I*A^2 - 2*I*A*B - I*B^2)*a*b^2*c*d^2 + 2*(-I*A^2 - 2*I*A*B - I*B^2)*a^2*b*d^3 + (13*I*B^2*b^3*c^2*
d - 6*I*B^2*a*b^2*c*d^2 - 7*I*B^2*a^2*b*d^3)*n^2 + 2*((-I*A*B - I*B^2)*b^3*c^2*d + 2*(-I*A*B - I*B^2)*a*b^2*c*
d^2 + 3*(I*A*B + I*B^2)*a^2*b*d^3)*n)*x + 2*(6*(I*A^2 + 2*I*A*B + I*B^2)*a*b^2*c^2*d + 3*(5*I*B^2*b^3*d^3*n^2
+ 2*(-I*A*B - I*B^2)*b^3*d^3*n + 2*(I*A^2 + 2*I*A*B + I*B^2)*b^3*d^3)*x^3 + (4*I*B^2*b^3*c^3 + 12*I*B^2*a^2*b*
c*d^2 - I*B^2*a^3*d^3)*n^2 + 3*(6*(-I*A*B - I*B^2)*a*b^2*d^3*n + 4*(I*A^2 + 2*I*A*B + I*B^2)*b^3*c*d^2 + 2*(I*
A^2 + 2*I*A*B + I*B^2)*a*b^2*d^3 + (8*I*B^2*b^3*c*d^2 + 7*I*B^2*a*b^2*d^3)*n^2)*x^2 + 2*(2*(I*A*B + I*B^2)*b^3
*c^3 + 6*(-I*A*B - I*B^2)*a^2*b*c*d^2 + (I*A*B + I*B^2)*a^3*d^3)*n + 3*(2*(I*A^2 + 2*I*A*B + I*B^2)*b^3*c^2*d
+ 4*(I*A^2 + 2*I*A*B + I*B^2)*a*b^2*c*d^2 + (4*I*B^2*b^3*c^2*d + 8*I*B^2*a*b^2*c*d^2 + 3*I*B^2*a^2*b*d^3)*n^2
+ 2*(2*(I*A*B + I*B^2)*b^3*c^2*d + 4*(-I*A*B - I*B^2)*a*b^2*c*d^2 + (-I*A*B - I*B^2)*a^2*b*d^3)*n)*x)*log((b*x
+ a)/(d*x + c)))/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*g^2*x^3 +
(2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*g^2*x^2 + (
b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*g^2*x + (a*b^
4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4)*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)**3,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((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

integrate((B*log(((b*x + a)/(d*x + c))^n*e) + A)^2/((b*g*x + a*g)^2*(I*d*x + I*c)^3), x)

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Mupad [B]
time = 10.14, size = 1785, normalized size = 3.18 \begin {gather*} \frac {\frac {-2\,A^2\,a^2\,d^2+10\,A^2\,a\,b\,c\,d+4\,A^2\,b^2\,c^2+2\,A\,B\,a^2\,d^2\,n-22\,A\,B\,a\,b\,c\,d\,n+8\,A\,B\,b^2\,c^2\,n-B^2\,a^2\,d^2\,n^2+23\,B^2\,a\,b\,c\,d\,n^2+8\,B^2\,b^2\,c^2\,n^2}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A^2\,b^2\,d^2-2\,A\,B\,b^2\,d^2\,n+5\,B^2\,b^2\,d^2\,n^2\right )}{a\,d-b\,c}+\frac {3\,x\,\left (6\,c\,A^2\,b^2\,d+2\,a\,A^2\,b\,d^2-2\,c\,A\,B\,b^2\,d\,n-6\,a\,A\,B\,b\,d^2\,n+13\,c\,B^2\,b^2\,d\,n^2+7\,a\,B^2\,b\,d^2\,n^2\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (4\,a^3\,c\,d^3\,g^2\,i^3-6\,a^2\,b\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (2\,a^3\,d^4\,g^2\,i^3-6\,a\,b^2\,c^2\,d^2\,g^2\,i^3+4\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (2\,a^2\,b\,d^4\,g^2\,i^3-4\,a\,b^2\,c\,d^3\,g^2\,i^3+2\,b^3\,c^2\,d^2\,g^2\,i^3\right )+2\,a^3\,c^2\,d^2\,g^2\,i^3+2\,a\,b^2\,c^4\,g^2\,i^3-4\,a^2\,b\,c^3\,d\,g^2\,i^3}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,\left (a\,d+2\,b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B^2\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3}+\frac {3\,B\,b^2\,d\,\left (2\,A-B\,n\right )}{2\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}-\frac {3\,B^2\,b^2\,d\,\left (d\,g^2\,i^3\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^2\,i^3\,n\,\left (a\,d-b\,c\right )}{b}+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3\right )}\right )-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {x\,\left (\frac {3\,b\,d\,n\,B^2}{2}+3\,A\,b\,d\,B\right )-\frac {B^2\,a\,d\,n}{2}+2\,B^2\,b\,c\,n+A\,B\,a\,d+2\,A\,B\,b\,c}{x\,\left (2\,a^3\,c\,d^3\,g^2\,i^3-3\,a^2\,b\,c^2\,d^2\,g^2\,i^3+b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (a^3\,d^4\,g^2\,i^3-3\,a\,b^2\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (a^2\,b\,d^4\,g^2\,i^3-2\,a\,b^2\,c\,d^3\,g^2\,i^3+b^3\,c^2\,d^2\,g^2\,i^3\right )+a^3\,c^2\,d^2\,g^2\,i^3+a\,b^2\,c^4\,g^2\,i^3-2\,a^2\,b\,c^3\,d\,g^2\,i^3}-\frac {3\,B\,b^2\,d\,\left (2\,A-B\,n\right )\,\left (d\,g^2\,i^3\,n\,x^2\,{\left (a\,d-b\,c\right )}^3+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,{\left (a\,d-b\,c\right )}^3}{b}+\frac {a\,c\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^3}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (2\,a^3\,c\,d^3\,g^2\,i^3-3\,a^2\,b\,c^2\,d^2\,g^2\,i^3+b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (a^3\,d^4\,g^2\,i^3-3\,a\,b^2\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (a^2\,b\,d^4\,g^2\,i^3-2\,a\,b^2\,c\,d^3\,g^2\,i^3+b^3\,c^2\,d^2\,g^2\,i^3\right )+a^3\,c^2\,d^2\,g^2\,i^3+a\,b^2\,c^4\,g^2\,i^3-2\,a^2\,b\,c^3\,d\,g^2\,i^3\right )}\right )-\frac {B^2\,b^2\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}+\frac {b^2\,d\,\mathrm {atan}\left (\frac {b^2\,d\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,\left (2\,a^4\,d^4\,g^2\,i^3-4\,a^3\,b\,c\,d^3\,g^2\,i^3+4\,a\,b^3\,c^3\,d\,g^2\,i^3-2\,b^4\,c^4\,g^2\,i^3\right )\,3{}\mathrm {i}}{2\,g^2\,i^3\,{\left (a\,d-b\,c\right )}^4\,\left (6\,d\,A^2\,b^2-6\,d\,A\,B\,b^2\,n+15\,d\,B^2\,b^2\,n^2\right )}+\frac {b^3\,d^2\,x\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,\left (a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3\right )\,6{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4\,\left (6\,d\,A^2\,b^2-6\,d\,A\,B\,b^2\,n+15\,d\,B^2\,b^2\,n^2\right )}\right )\,\left (2\,A^2-2\,A\,B\,n+5\,B^2\,n^2\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \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)^3),x)

[Out]

((4*A^2*b^2*c^2 - 2*A^2*a^2*d^2 - B^2*a^2*d^2*n^2 + 8*B^2*b^2*c^2*n^2 + 10*A^2*a*b*c*d + 2*A*B*a^2*d^2*n + 8*A
*B*b^2*c^2*n + 23*B^2*a*b*c*d*n^2 - 22*A*B*a*b*c*d*n)/(2*(a*d - b*c)) + (3*x^2*(2*A^2*b^2*d^2 + 5*B^2*b^2*d^2*
n^2 - 2*A*B*b^2*d^2*n))/(a*d - b*c) + (3*x*(2*A^2*a*b*d^2 + 6*A^2*b^2*c*d + 7*B^2*a*b*d^2*n^2 + 13*B^2*b^2*c*d
*n^2 - 6*A*B*a*b*d^2*n - 2*A*B*b^2*c*d*n))/(2*(a*d - b*c)))/(x*(2*b^3*c^4*g^2*i^3 + 4*a^3*c*d^3*g^2*i^3 - 6*a^
2*b*c^2*d^2*g^2*i^3) + x^2*(2*a^3*d^4*g^2*i^3 + 4*b^3*c^3*d*g^2*i^3 - 6*a*b^2*c^2*d^2*g^2*i^3) + x^3*(2*b^3*c^
2*d^2*g^2*i^3 + 2*a^2*b*d^4*g^2*i^3 - 4*a*b^2*c*d^3*g^2*i^3) + 2*a^3*c^2*d^2*g^2*i^3 + 2*a*b^2*c^4*g^2*i^3 - 4
*a^2*b*c^3*d*g^2*i^3) - log(e*((a + b*x)/(c + d*x))^n)^2*(((B^2*(a*d + 2*b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c
*d)) + (3*B^2*b*d*x)/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i^3) + x^2*(a*d^2*g^
2*i^3 + 2*b*c*d*g^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^3*x^3) + (3*B*b^2*d*(2*A - B*n))/(2*g^2*i^3*n*(a*d - b*
c)^4) - (3*B^2*b^2*d*(d*g^2*i^3*n*x^2*(a*d - b*c) + (a*c*g^2*i^3*n*(a*d - b*c))/b + (g^2*i^3*n*x*(a*d + b*c)*(
a*d - b*c))/b))/(g^2*i^3*n*(a*d - b*c)^4*(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i^3) + x^2*(a*d^2*g^2*i^3 + 2*b*c*d*g
^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^3*x^3))) - log(e*((a + b*x)/(c + d*x))^n)*((x*((3*B^2*b*d*n)/2 + 3*A*B*b
*d) - (B^2*a*d*n)/2 + 2*B^2*b*c*n + A*B*a*d + 2*A*B*b*c)/(x*(b^3*c^4*g^2*i^3 + 2*a^3*c*d^3*g^2*i^3 - 3*a^2*b*c
^2*d^2*g^2*i^3) + x^2*(a^3*d^4*g^2*i^3 + 2*b^3*c^3*d*g^2*i^3 - 3*a*b^2*c^2*d^2*g^2*i^3) + x^3*(b^3*c^2*d^2*g^2
*i^3 + a^2*b*d^4*g^2*i^3 - 2*a*b^2*c*d^3*g^2*i^3) + a^3*c^2*d^2*g^2*i^3 + a*b^2*c^4*g^2*i^3 - 2*a^2*b*c^3*d*g^
2*i^3) - (3*B*b^2*d*(2*A - B*n)*(d*g^2*i^3*n*x^2*(a*d - b*c)^3 + (g^2*i^3*n*x*(a*d + b*c)*(a*d - b*c)^3)/b + (
a*c*g^2*i^3*n*(a*d - b*c)^3)/b))/(g^2*i^3*n*(a*d - b*c)^4*(x*(b^3*c^4*g^2*i^3 + 2*a^3*c*d^3*g^2*i^3 - 3*a^2*b*
c^2*d^2*g^2*i^3) + x^2*(a^3*d^4*g^2*i^3 + 2*b^3*c^3*d*g^2*i^3 - 3*a*b^2*c^2*d^2*g^2*i^3) + x^3*(b^3*c^2*d^2*g^
2*i^3 + a^2*b*d^4*g^2*i^3 - 2*a*b^2*c*d^3*g^2*i^3) + a^3*c^2*d^2*g^2*i^3 + a*b^2*c^4*g^2*i^3 - 2*a^2*b*c^3*d*g
^2*i^3))) + (b^2*d*atan((b^2*d*(2*A^2 + 5*B^2*n^2 - 2*A*B*n)*(2*a^4*d^4*g^2*i^3 - 2*b^4*c^4*g^2*i^3 + 4*a*b^3*
c^3*d*g^2*i^3 - 4*a^3*b*c*d^3*g^2*i^3)*3i)/(2*g^2*i^3*(a*d - b*c)^4*(6*A^2*b^2*d + 15*B^2*b^2*d*n^2 - 6*A*B*b^
2*d*n)) + (b^3*d^2*x*(2*A^2 + 5*B^2*n^2 - 2*A*B*n)*(a^3*d^3*g^2*i^3 - b^3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3
- 3*a^2*b*c*d^2*g^2*i^3)*6i)/(g^2*i^3*(a*d - b*c)^4*(6*A^2*b^2*d + 15*B^2*b^2*d*n^2 - 6*A*B*b^2*d*n)))*(2*A^2
+ 5*B^2*n^2 - 2*A*B*n)*3i)/(g^2*i^3*(a*d - b*c)^4) - (B^2*b^2*d*log(e*((a + b*x)/(c + d*x))^n)^3)/(g^2*i^3*n*(
a*d - b*c)^4)

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