3.3.6 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x) (c i+d i x)^3} \, dx\) [206]

Optimal. Leaf size=402 \[ \frac {B^2 d^2 n^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d n (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d n^2 (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g i^3 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.31, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2561, 2388, 2339, 30, 2333, 2332, 2367, 2342, 2341} \begin {gather*} \frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B g i^3 n (b c-a d)^3}+\frac {d^2 (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 i^3 (c+d x)^2 (b c-a d)^3}-\frac {B d^2 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 i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g i^3 (c+d x) (b c-a d)^3}+\frac {4 A b B d n (a+b x)}{g i^3 (c+d x) (b c-a d)^3}+\frac {B^2 d^2 n^2 (a+b x)^2}{4 g i^3 (c+d x)^2 (b c-a d)^3}+\frac {4 b B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {4 b B^2 d n^2 (a+b x)}{g i^3 (c+d x) (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B^2*d^2*n^2*(a + b*x)^2)/(4*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (4*A*b*B*d*n*(a + b*x))/((b*c - a*d)^3*g*i^3*(
c + d*x)) - (4*b*B^2*d*n^2*(a + b*x))/((b*c - a*d)^3*g*i^3*(c + d*x)) + (4*b*B^2*d*n*(a + b*x)*Log[e*((a + b*x
)/(c + d*x))^n])/((b*c - a*d)^3*g*i^3*(c + d*x)) - (B*d^2*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
)/(2*(b*c - a*d)^3*g*i^3*(c + d*x)^2) + (d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a
*d)^3*g*i^3*(c + d*x)^2) - (2*b*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^3*g*i^3*(c
+ d*x)) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^3*g*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 2367

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

Rule 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

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}{(206 c+206 d x)^3 (a g+b g x)} \, 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}{8741816 (b c-a d)^3 g (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d) g (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 )^2}{8741816 (b c-a d)^2 g (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 )^2}{8741816 (b c-a d)^3 g (c+d x)}\right ) \, dx\\ &=\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} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 d\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{c+d x} \, dx}{8741816 (b c-a d)^3 g}-\frac {(b d) \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}{8741816 (b c-a d)^2 g}-\frac {d \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}{8741816 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}-\frac {(b B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}-\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^3} \, dx}{8741816 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {(B 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}{8741816 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^3 g}+\frac {\left (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 ) \log (c+d x)}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^3 g}-\frac {(b B 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}{4370908 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {(B 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}{8741816 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}+\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}-\frac {(b B 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}{4370908 (b c-a d) g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (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} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (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} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B d n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B 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}{4370908 (b c-a d)^2 g}+\frac {\left (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 ) \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}{4370908 (b c-a d)^2 g}+\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8741816 (b c-a d)^2 g}+\frac {(b B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}+\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8741816 (b c-a d) g}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B 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}{4370908 (b c-a d)^3 g}-\frac {\left (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)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \int \frac {\log (a+b x)}{(a+b x) (c+d x)} \, dx}{4370908 (b c-a d)^2 g}-\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^2 g}+\frac {\left (b^2 B^2 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}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8741816 (b c-a d)^2 g}+\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{17483632 (b c-a d) g}\\ &=-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2\right ) \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B 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}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B d 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}{4370908 (b c-a d)^3 g}-\frac {(A b B 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 )}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{17483632 g}+\frac {\left (b^2 B^2 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}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8741816 (b c-a d) g}+\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{4370908 (b c-a d) g}\\ &=-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (A b^3 B n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 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}{4370908 (b c-a d)^3 g}+\frac {(A b B d n) \text {Subst}\left (\int \frac {\log (x)}{\frac {b c-a d}{b}+\frac {d x}{b}} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B d n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^2 g}+\frac {\left (B^2 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}{17483632 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 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}{8741816 (b c-a d) g}+\frac {\left (b B^2 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}{4370908 (b c-a d) g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B d n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 d n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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}{4370908 (b c-a d)^2 g}+\frac {\left (b^3 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (A b^2 B 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{8741816 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d 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}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 d n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b B^2 d\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 )}{8741816 (b c-a d)^3 g}-\frac {\left (b B^2 d 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{8741816 (b c-a d)^3 g}-\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b B^2 d 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n \left (-\log \left ((a+b x)^n\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\log \left ((c+d x)^{-n}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\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 )}{8741816 d (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}-\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 )}{8741816 d (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}-\frac {\left (b^2 B^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}+\frac {\left (b^2 B^2 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 )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}+2 \frac {\left (b^2 B^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{4370908 (b c-a d)^3 g}\\ &=\frac {B^2 n^2}{34967264 (b c-a d) g (c+d x)^2}+\frac {7 b B^2 n^2}{17483632 (b c-a d)^2 g (c+d x)}+\frac {7 b^2 B^2 n^2 \log (a+b x)}{17483632 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(a+b x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(a+b x)}{17483632 (b c-a d)^3 g}-\frac {b^2 B^2 \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 )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{8741816 (b c-a d)^3 g}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{17483632 (b c-a d) g (c+d x)^2}-\frac {3 b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^2 g (c+d x)}-\frac {3 b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8741816 (b c-a d)^3 g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{17483632 (b c-a d) g (c+d x)^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^2 g (c+d x)}+\frac {b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8741816 (b c-a d)^3 g}-\frac {7 b^2 B^2 n^2 \log (c+d x)}{17483632 (b c-a d)^3 g}+\frac {A b^2 B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8741816 (b c-a d)^3 g}-\frac {A b^2 B n \log ^2(c+d x)}{8741816 (b c-a d)^3 g}+\frac {3 b^2 B^2 n^2 \log ^2(c+d x)}{17483632 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \log (a+b x) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n^2 \log ^3(c+d x)}{26225448 (b c-a d)^3 g}+\frac {A b^2 B n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}-\frac {b^2 B^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {A b^2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {3 b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8741816 (b c-a d)^3 g}+\frac {b^2 B^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}-\frac {b^2 B^2 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 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 )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4370908 (b c-a d)^3 g}+\frac {b^2 B^2 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{4370908 (b c-a d)^3 g}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(971\) vs. \(2(402)=804\).
time = 0.75, size = 971, normalized size = 2.42 \begin {gather*} \frac {4 b^2 B^2 n^2 \log ^3\left (\frac {a+b x}{c+d x}\right )-\frac {6 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (-2 A b^2 c^2+4 a b B c d n-a^2 B d^2 n-4 A b^2 c d x+4 b^2 B c d n x+2 a b B d^2 n x-2 A b^2 d^2 x^2+3 b^2 B d^2 n x^2-2 b^2 B (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 b^2 B n (c+d x)^2 \log \left (\frac {a+b x}{c+d x}\right )\right )}{(c+d x)^2}-\frac {6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (-6 A b c+2 a A d+7 b B c n-a B d n-4 A b d x+6 b B d n x+2 B (-3 b c+a d-2 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (3 b c-a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )}{(c+d x)^2}+\frac {3 (b c-a d)^2 \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 )}{(c+d x)^2}+\frac {6 b (b c-a d) \left (2 A^2-6 A B n+7 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+3 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+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )}{c+d x}+6 b^2 \log (a+b x) \left (2 A^2-6 A B n+7 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+3 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+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b^2 \left (2 A^2-6 A B n+7 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+3 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+3 B n+2 B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right ) \log (c+d x)}{12 (b c-a d)^3 g i^3} \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)*(c*i + d*i*x)^3),x]

[Out]

(4*b^2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^3 - (6*B*n*Log[(a + b*x)/(c + d*x)]^2*(-2*A*b^2*c^2 + 4*a*b*B*c*d*n -
a^2*B*d^2*n - 4*A*b^2*c*d*x + 4*b^2*B*c*d*n*x + 2*a*b*B*d^2*n*x - 2*A*b^2*d^2*x^2 + 3*b^2*B*d^2*n*x^2 - 2*b^2*
B*(c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 2*b^2*B*n*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]))/(c + d*x)^2 -
(6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(-6*A*b*c + 2*a*A*d + 7*b*B*c*n - a*B*d*n - 4*A*b*d*x + 6*b*B*d*n*
x + 2*B*(-3*b*c + a*d - 2*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n] + 2*B*n*(3*b*c - a*d + 2*b*d*x)*Log[(a + b*x)/
(c + d*x)]))/(c + d*x)^2 + (3*(b*c - a*d)^2*(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)])))/(c + d*x)^2 + (6*b*(b*c - a*d)*(2*A^2 - 6*A*B*n
+ 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*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 + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*
x)])))/(c + d*x) + 6*b^2*Log[a + b*x]*(2*A^2 - 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 +
2*B*n*(-2*A + 3*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 + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])) - 6*b^2*(2*A^2 - 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[
e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*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 + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)]))*Log[c + d*x])/(12*
(b*c - a*d)^3*g*i^3)

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

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)/(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)/(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. 2035 vs. \(2 (375) = 750\).
time = 0.68, size = 2035, normalized size = 5.06 \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)/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/2*B^2*(2*b^2*log(b*x + a)/((-I*b^3*c^3 + 3*I*a*b^2*c^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*g) - 2*b^2*log(d*x +
c)/((-I*b^3*c^3 + 3*I*a*b^2*c^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*g) + (2*b*d*x + 3*b*c - a*d)/((-I*b^2*c^2*d^
2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*g*x^2 + 2*(-I*b^2*c^3*d + 2*I*a*b*c^2*d^2 - I*a^2*c*d^3)*g*x + (-I*b^2*c^4 + 2*
I*a*b*c^3*d - I*a^2*c^2*d^2)*g))*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2 + A*B*(2*b^2*log(b*x + a)/((-I*b^3*c
^3 + 3*I*a*b^2*c^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*g) - 2*b^2*log(d*x + c)/((-I*b^3*c^3 + 3*I*a*b^2*c^2*d - 3
*I*a^2*b*c*d^2 + I*a^3*d^3)*g) + (2*b*d*x + 3*b*c - a*d)/((-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*g*x^2 +
2*(-I*b^2*c^3*d + 2*I*a*b*c^2*d^2 - I*a^2*c*d^3)*g*x + (-I*b^2*c^4 + 2*I*a*b*c^3*d - I*a^2*c^2*d^2)*g))*log((
b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/12*((-45*I*b^2*c^2 + 48*I*a*b*c*d - 3*I*a^2*d^2 - 4*(I*b^2*d^2*x^2 + 2*I
*b^2*c*d*x + I*b^2*c^2)*log(b*x + a)^3 - 4*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(d*x + c)^3 - 18*(I
*b^2*d^2*x^2 + 2*I*b^2*c*d*x + I*b^2*c^2)*log(b*x + a)^2 - 6*(3*I*b^2*d^2*x^2 + 6*I*b^2*c*d*x + 3*I*b^2*c^2 +
2*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + I*b^2*c^2)*log(b*x + a))*log(d*x + c)^2 - 42*(I*b^2*c*d - I*a*b*d^2)*x - 42
*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + I*b^2*c^2)*log(b*x + a) - 6*(-7*I*b^2*d^2*x^2 - 14*I*b^2*c*d*x - 7*I*b^2*c^2
+ 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(b*x + a)^2 + 6*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c
^2)*log(b*x + a))*log(d*x + c))*n^2/(b^3*c^5*g - 3*a*b^2*c^4*d*g + 3*a^2*b*c^3*d^2*g - a^3*c^2*d^3*g + (b^3*c^
3*d^2*g - 3*a*b^2*c^2*d^3*g + 3*a^2*b*c*d^4*g - a^3*d^5*g)*x^2 + 2*(b^3*c^4*d*g - 3*a*b^2*c^3*d^2*g + 3*a^2*b*
c^2*d^3*g - a^3*c*d^4*g)*x) + 6*(7*I*b^2*c^2 - 8*I*a*b*c*d + I*a^2*d^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I
*b^2*c^2)*log(b*x + a)^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(d*x + c)^2 - 6*(-I*b^2*c*d + I*a
*b*d^2)*x - 6*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(b*x + a) - 2*(3*I*b^2*d^2*x^2 + 6*I*b^2*c*d*x +
3*I*b^2*c^2 + 2*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + I*b^2*c^2)*log(b*x + a))*log(d*x + c))*n*log((b*x/(d*x + c)
+ a/(d*x + c))^n*e)/(b^3*c^5*g - 3*a*b^2*c^4*d*g + 3*a^2*b*c^3*d^2*g - a^3*c^2*d^3*g + (b^3*c^3*d^2*g - 3*a*b^
2*c^2*d^3*g + 3*a^2*b*c*d^4*g - a^3*d^5*g)*x^2 + 2*(b^3*c^4*d*g - 3*a*b^2*c^3*d^2*g + 3*a^2*b*c^2*d^3*g - a^3*
c*d^4*g)*x))*B^2 - 1/2*(7*I*b^2*c^2 - 8*I*a*b*c*d + I*a^2*d^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)
*log(b*x + a)^2 - 2*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(d*x + c)^2 - 6*(-I*b^2*c*d + I*a*b*d^2)*x
- 6*(-I*b^2*d^2*x^2 - 2*I*b^2*c*d*x - I*b^2*c^2)*log(b*x + a) - 2*(3*I*b^2*d^2*x^2 + 6*I*b^2*c*d*x + 3*I*b^2*
c^2 + 2*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x + I*b^2*c^2)*log(b*x + a))*log(d*x + c))*A*B*n/(b^3*c^5*g - 3*a*b^2*c^4
*d*g + 3*a^2*b*c^3*d^2*g - a^3*c^2*d^3*g + (b^3*c^3*d^2*g - 3*a*b^2*c^2*d^3*g + 3*a^2*b*c*d^4*g - a^3*d^5*g)*x
^2 + 2*(b^3*c^4*d*g - 3*a*b^2*c^3*d^2*g + 3*a^2*b*c^2*d^3*g - a^3*c*d^4*g)*x) + 1/2*A^2*(2*b^2*log(b*x + a)/((
-I*b^3*c^3 + 3*I*a*b^2*c^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*g) - 2*b^2*log(d*x + c)/((-I*b^3*c^3 + 3*I*a*b^2*c
^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*g) + (2*b*d*x + 3*b*c - a*d)/((-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)
*g*x^2 + 2*(-I*b^2*c^3*d + 2*I*a*b*c^2*d^2 - I*a^2*c*d^3)*g*x + (-I*b^2*c^4 + 2*I*a*b*c^3*d - I*a^2*c^2*d^2)*g
))

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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. 873 vs. \(2 (375) = 750\).
time = 0.45, size = 873, normalized size = 2.17 \begin {gather*} \frac {18 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} c^{2} + 24 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b c d + 6 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} a^{2} d^{2} + 4 \, {\left (i \, B^{2} b^{2} d^{2} n^{2} x^{2} + 2 i \, B^{2} b^{2} c d n^{2} x + i \, B^{2} b^{2} c^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{3} + 3 \, {\left (15 i \, B^{2} b^{2} c^{2} - 16 i \, B^{2} a b c d + i \, B^{2} a^{2} d^{2}\right )} n^{2} + 6 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{2} c^{2} n + {\left (-4 i \, B^{2} a b c d + i \, B^{2} a^{2} d^{2}\right )} n^{2} + {\left (-3 i \, B^{2} b^{2} d^{2} n^{2} + 2 \, {\left (i \, A B + i \, B^{2}\right )} b^{2} d^{2} n\right )} x^{2} + 2 \, {\left (2 \, {\left (i \, A B + i \, B^{2}\right )} b^{2} c d n + {\left (-2 i \, B^{2} b^{2} c d - i \, B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 6 \, {\left (7 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} c^{2} + 8 \, {\left (i \, A B + i \, B^{2}\right )} a b c d + {\left (-i \, A B - i \, B^{2}\right )} a^{2} d^{2}\right )} n + 6 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} c d + 2 \, {\left (-i \, A^{2} - 2 i \, A B - i \, B^{2}\right )} a b d^{2} + 7 \, {\left (i \, B^{2} b^{2} c d - i \, B^{2} a b d^{2}\right )} n^{2} + 6 \, {\left ({\left (-i \, A B - i \, B^{2}\right )} b^{2} c d + {\left (i \, A B + i \, B^{2}\right )} a b d^{2}\right )} n\right )} x + 6 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} c^{2} + {\left (8 i \, B^{2} a b c d - i \, B^{2} a^{2} d^{2}\right )} n^{2} + {\left (7 i \, B^{2} b^{2} d^{2} n^{2} + 6 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} d^{2} n + 2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (4 \, {\left (-i \, A B - i \, B^{2}\right )} a b c d + {\left (i \, A B + i \, B^{2}\right )} a^{2} d^{2}\right )} n + 2 \, {\left (2 \, {\left (i \, A^{2} + 2 i \, A B + i \, B^{2}\right )} b^{2} c d + {\left (4 i \, B^{2} b^{2} c d + 3 i \, B^{2} a b d^{2}\right )} n^{2} + 2 \, {\left (2 \, {\left (-i \, A B - i \, B^{2}\right )} b^{2} c d + {\left (-i \, A B - i \, B^{2}\right )} a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{12 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g\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)/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

1/12*(18*(I*A^2 + 2*I*A*B + I*B^2)*b^2*c^2 + 24*(-I*A^2 - 2*I*A*B - I*B^2)*a*b*c*d + 6*(I*A^2 + 2*I*A*B + I*B^
2)*a^2*d^2 + 4*(I*B^2*b^2*d^2*n^2*x^2 + 2*I*B^2*b^2*c*d*n^2*x + I*B^2*b^2*c^2*n^2)*log((b*x + a)/(d*x + c))^3
+ 3*(15*I*B^2*b^2*c^2 - 16*I*B^2*a*b*c*d + I*B^2*a^2*d^2)*n^2 + 6*(2*(I*A*B + I*B^2)*b^2*c^2*n + (-4*I*B^2*a*b
*c*d + I*B^2*a^2*d^2)*n^2 + (-3*I*B^2*b^2*d^2*n^2 + 2*(I*A*B + I*B^2)*b^2*d^2*n)*x^2 + 2*(2*(I*A*B + I*B^2)*b^
2*c*d*n + (-2*I*B^2*b^2*c*d - I*B^2*a*b*d^2)*n^2)*x)*log((b*x + a)/(d*x + c))^2 + 6*(7*(-I*A*B - I*B^2)*b^2*c^
2 + 8*(I*A*B + I*B^2)*a*b*c*d + (-I*A*B - I*B^2)*a^2*d^2)*n + 6*(2*(I*A^2 + 2*I*A*B + I*B^2)*b^2*c*d + 2*(-I*A
^2 - 2*I*A*B - I*B^2)*a*b*d^2 + 7*(I*B^2*b^2*c*d - I*B^2*a*b*d^2)*n^2 + 6*((-I*A*B - I*B^2)*b^2*c*d + (I*A*B +
I*B^2)*a*b*d^2)*n)*x + 6*(2*(I*A^2 + 2*I*A*B + I*B^2)*b^2*c^2 + (8*I*B^2*a*b*c*d - I*B^2*a^2*d^2)*n^2 + (7*I*
B^2*b^2*d^2*n^2 + 6*(-I*A*B - I*B^2)*b^2*d^2*n + 2*(I*A^2 + 2*I*A*B + I*B^2)*b^2*d^2)*x^2 + 2*(4*(-I*A*B - I*B
^2)*a*b*c*d + (I*A*B + I*B^2)*a^2*d^2)*n + 2*(2*(I*A^2 + 2*I*A*B + I*B^2)*b^2*c*d + (4*I*B^2*b^2*c*d + 3*I*B^2
*a*b*d^2)*n^2 + 2*(2*(-I*A*B - I*B^2)*b^2*c*d + (-I*A*B - I*B^2)*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^
3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 - a^3*d^5)*g*x^2 + 2*(b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^
3 - a^3*c*d^4)*g*x + (b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3)*g)

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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)/(d*i*x+c*i)**3,x)

[Out]

Timed out

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Giac [A]
time = 3.74, size = 647, normalized size = 1.61 \begin {gather*} -\frac {1}{12} \, {\left (-\frac {4 i \, B^{2} b^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{3}}{b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g} + 6 \, {\left (\frac {4 i \, {\left (b x + a\right )} B^{2} b d n^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}} - \frac {i \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} + \frac {2 \, {\left (-i \, A B b^{2} n - i \, B^{2} b^{2} n\right )}}{b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (\frac {{\left (-i \, B^{2} d^{2} n^{2} + 2 i \, A B d^{2} n + 2 i \, B^{2} d^{2} n\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (-i \, B^{2} b d n^{2} + i \, A B b d n + i \, B^{2} b d n\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {12 \, {\left (A^{2} b^{2} + 2 \, A B b^{2} + B^{2} b^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{i \, b^{2} c^{2} g - 2 i \, a b c d g + i \, a^{2} d^{2} g} - \frac {3 \, {\left (i \, B^{2} d^{2} n^{2} - 2 i \, A B d^{2} n - 2 i \, B^{2} d^{2} n + 2 i \, A^{2} d^{2} + 4 i \, A B d^{2} + 2 i \, B^{2} d^{2}\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}^{2}} + \frac {24 \, {\left (2 i \, B^{2} b d n^{2} - 2 i \, A B b d n - 2 i \, B^{2} b d n + i \, A^{2} b d + 2 i \, A B b d + i \, B^{2} b d\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{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)/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

-1/12*(-4*I*B^2*b^2*n^2*log((b*x + a)/(d*x + c))^3/(b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g) + 6*(4*I*(b*x + a)*B^
2*b*d*n^2/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(d*x + c)) - I*(b*x + a)^2*B^2*d^2*n^2/((b^2*c^2*g - 2*a*b*c*
d*g + a^2*d^2*g)*(d*x + c)^2) + 2*(-I*A*B*b^2*n - I*B^2*b^2*n)/(b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g))*log((b*x
+ a)/(d*x + c))^2 - 6*((-I*B^2*d^2*n^2 + 2*I*A*B*d^2*n + 2*I*B^2*d^2*n)*(b*x + a)^2/((b^2*c^2*g - 2*a*b*c*d*g
+ a^2*d^2*g)*(d*x + c)^2) - 8*(-I*B^2*b*d*n^2 + I*A*B*b*d*n + I*B^2*b*d*n)*(b*x + a)/((b^2*c^2*g - 2*a*b*c*d*
g + a^2*d^2*g)*(d*x + c)))*log((b*x + a)/(d*x + c)) + 12*(A^2*b^2 + 2*A*B*b^2 + B^2*b^2)*log((b*x + a)/(d*x +
c))/(I*b^2*c^2*g - 2*I*a*b*c*d*g + I*a^2*d^2*g) - 3*(I*B^2*d^2*n^2 - 2*I*A*B*d^2*n - 2*I*B^2*d^2*n + 2*I*A^2*d
^2 + 4*I*A*B*d^2 + 2*I*B^2*d^2)*(b*x + a)^2/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(d*x + c)^2) + 24*(2*I*B^2*
b*d*n^2 - 2*I*A*B*b*d*n - 2*I*B^2*b*d*n + I*A^2*b*d + 2*I*A*B*b*d + I*B^2*b*d)*(b*x + a)/((b^2*c^2*g - 2*a*b*c
*d*g + a^2*d^2*g)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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Mupad [B]
time = 8.73, size = 1007, normalized size = 2.50 \begin {gather*} {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )}{2\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B^2\,b^2\,\left (\frac {c\,g\,i^3\,n\,\left (a\,d-b\,c\right )}{2\,b}-\frac {g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,\left (a\,d-b\,c\right )}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (g\,c^2\,i^3+2\,g\,c\,d\,i^3\,x+g\,d^2\,i^3\,x^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-6\,A^2\,b\,c+B^2\,a\,d\,n^2-15\,B^2\,b\,c\,n^2-2\,A\,B\,a\,d\,n+14\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}-\frac {x\,\left (2\,b\,d\,A^2-6\,b\,d\,A\,B\,n+7\,b\,d\,B^2\,n^2\right )}{a\,d-b\,c}}{x^2\,\left (2\,a\,d^3\,g\,i^3-2\,b\,c\,d^2\,g\,i^3\right )+x\,\left (4\,a\,c\,d^2\,g\,i^3-4\,b\,c^2\,d\,g\,i^3\right )-2\,b\,c^3\,g\,i^3+2\,a\,c^2\,d\,g\,i^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B^2\,n}{x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3}+\frac {b^2\,\left (3\,B^2\,n-2\,A\,B\right )\,\left (\frac {c\,g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2}{2\,b}-\frac {g\,i^3\,n\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {d\,g\,i^3\,n\,x\,{\left (a\,d-b\,c\right )}^2}{b}\right )}{g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (x^2\,\left (a\,d^3\,g\,i^3-b\,c\,d^2\,g\,i^3\right )+x\,\left (2\,a\,c\,d^2\,g\,i^3-2\,b\,c^2\,d\,g\,i^3\right )-b\,c^3\,g\,i^3+a\,c^2\,d\,g\,i^3\right )}\right )-\frac {B^2\,b^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{3\,g\,i^3\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {b^2\,\mathrm {atan}\left (\frac {b^2\,\left (\frac {g\,a^3\,d^3\,i^3-g\,a^2\,b\,c\,d^2\,i^3-g\,a\,b^2\,c^2\,d\,i^3+g\,b^3\,c^3\,i^3}{g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3}+2\,b\,d\,x\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,\left (g\,a^2\,d^2\,i^3-2\,g\,a\,b\,c\,d\,i^3+g\,b^2\,c^2\,i^3\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,\left (2\,A^2\,b^2-6\,A\,B\,b^2\,n+7\,B^2\,b^2\,n^2\right )}\right )\,\left (A^2-3\,A\,B\,n+\frac {7\,B^2\,n^2}{2}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \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)*(c*i + d*i*x)^3),x)

[Out]

log(e*((a + b*x)/(c + d*x))^n)^2*((b^2*(3*B^2*n - 2*A*B))/(2*g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*
d)) + (B^2*b^2*((c*g*i^3*n*(a*d - b*c))/(2*b) - (g*i^3*n*(a*d - b*c)*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*(a*
d - b*c))/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(c^2*g*i^3 + d^2*g*i^3*x^2 + 2*c*d*g*i^3*x)
)) - ((2*A^2*a*d - 6*A^2*b*c + B^2*a*d*n^2 - 15*B^2*b*c*n^2 - 2*A*B*a*d*n + 14*A*B*b*c*n)/(2*(a*d - b*c)) - (x
*(2*A^2*b*d + 7*B^2*b*d*n^2 - 6*A*B*b*d*n))/(a*d - b*c))/(x^2*(2*a*d^3*g*i^3 - 2*b*c*d^2*g*i^3) + x*(4*a*c*d^2
*g*i^3 - 4*b*c^2*d*g*i^3) - 2*b*c^3*g*i^3 + 2*a*c^2*d*g*i^3) - log(e*((a + b*x)/(c + d*x))^n)*((B^2*n)/(x^2*(a
*d^3*g*i^3 - b*c*d^2*g*i^3) + x*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g*i^3 + a*c^2*d*g*i^3) + (b^2*(3*B
^2*n - 2*A*B)*((c*g*i^3*n*(a*d - b*c)^2)/(2*b) - (g*i^3*n*(a*d - b*c)^2*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*
(a*d - b*c)^2)/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^2*(a*d^3*g*i^3 - b*c*d^2*g*i^3) + x
*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g*i^3 + a*c^2*d*g*i^3))) + (b^2*atan((b^2*((a^3*d^3*g*i^3 + b^3*c
^3*g*i^3 - a*b^2*c^2*d*g*i^3 - a^2*b*c*d^2*g*i^3)/(a^2*d^2*g*i^3 + b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3) + 2*b*d*x)
*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*(a^2*d^2*g*i^3 + b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3)*2i)/(g*i^3*(a*d - b*c)^3*(2
*A^2*b^2 + 7*B^2*b^2*n^2 - 6*A*B*b^2*n)))*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*2i)/(g*i^3*(a*d - b*c)^3) - (B^2*b^2
*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))

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