3.2.72 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{a g+b g x} \, dx\) [172]

Optimal. Leaf size=572 \[ -\frac {B d (b c-a d) i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {d (b c-a d) i^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g}+\frac {i^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b g}+\frac {2 B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b^3 g}+\frac {B^2 (b c-a d)^2 i^2 n^2 \log (c+d x)}{b^3 g}+\frac {B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B^2 (b c-a d)^2 i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g}-\frac {B^2 (b c-a d)^2 i^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B^2 (b c-a d)^2 i^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2561, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31} \begin {gather*} \frac {2 B i^2 n (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {2 B^2 i^2 n^2 (b c-a d)^2 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 g}-\frac {B^2 i^2 n^2 (b c-a d)^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {2 B^2 i^2 n^2 (b c-a d)^2 \text {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g}+\frac {d i^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g}-\frac {B d i^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {2 B i^2 n (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}-\frac {i^2 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^3 g}+\frac {B i^2 n (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g}+\frac {i^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 b g}+\frac {B^2 i^2 n^2 (b c-a d)^2 \log (c+d x)}{b^3 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2351

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

Rule 2354

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

Rule 2355

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

Rule 2356

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

Rule 2379

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

Rule 2389

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2561

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(172 c+172 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx &=\int \left (\frac {29584 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {172 d (172 c+172 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 (a g+b g x)}\right ) \, dx\\ &=\frac {\left (29584 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx}{b^2}+\frac {(172 d) \int (172 c+172 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}+\frac {(29584 d (b c-a d)) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b^2 g}\\ &=\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {(B n) \int \frac {29584 (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b g}-\frac {(59168 B d (b c-a d) n) \int \frac {(b c-a d) 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}{b^2 g}-\frac {\left (59168 B (b c-a d)^2 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 (a g+b g x)}{a+b x} \, dx}{b^3 g}\\ &=\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {(29584 B (b c-a d) n) \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{b g}-\frac {\left (59168 B (b c-a d)^2 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 (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b^3 g}-\frac {\left (59168 B d (b c-a d)^2 n\right ) \int \frac {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}{b^2 g}\\ &=\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {(29584 B (b c-a d) n) \int \left (\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (a+b x)}\right ) \, dx}{b g}-\frac {\left (59168 B d (b c-a d)^2 n\right ) \int \left (-\frac {a \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)}+\frac {c \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)}\right ) \, dx}{b^2 g}-\frac {\left (59168 B (b c-a d)^3 n\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(a+b x) (c+d x)} \, dx}{b^3 g}\\ &=\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {(29584 B d (b c-a d) n) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 g}+\frac {(59168 a B d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g}-\frac {(59168 B c d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 g}-\frac {\left (29584 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 B (b c-a d)^3 n\right ) \int \left (\frac {d \left (-A-B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (c+d x)}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (a g+b g x)}{(b c-a d) (a+b x)}\right ) \, dx}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {\left (29584 B^2 d (b c-a d) n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{b^2 g}-\frac {\left (59168 B (b c-a 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 (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 B d (b c-a 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 (a g+b g x)}{c+d x} \, dx}{b^3 g}+\frac {\left (59168 B^2 c (b c-a 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}{b^2 g}-\frac {\left (59168 a B^2 d (b c-a 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}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^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}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {\left (59168 B (b c-a d)^2 n\right ) \int \left (\frac {A \log (a g+b g x)}{a+b x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x}\right ) \, dx}{b^2 g}-\frac {\left (59168 B d (b c-a d)^2 n\right ) \int \left (\frac {A \log (a g+b g x)}{-c-d x}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x}\right ) \, dx}{b^3 g}+\frac {\left (59168 B^2 c (b c-a 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}{b^2 g}-\frac {\left (59168 a B^2 d (b c-a 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}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^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}{b^3 g}+\frac {\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}-\frac {\left (59168 A B (b c-a d)^2 n\right ) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 B^2 (b c-a d)^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 A B d (b c-a d)^2 n\right ) \int \frac {\log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac {\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}+\frac {\left (59168 B^2 c (b c-a d) n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b g}-\frac {\left (59168 a B^2 d (b c-a d) n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 B^2 c d (b c-a d) n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^2 g}+\frac {\left (59168 a B^2 d^2 (b c-a d) n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g}-\frac {\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {\left (59168 A B (b c-a d)^2 n\right ) \int \frac {\log \left (\frac {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx}{b^2}-\frac {\left (59168 A B (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}-\frac {\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac {\left (59168 B^2 d (b c-a d)^2 n\right ) \int \frac {\log \left ((c+d x)^{-n}\right ) \log (a g+b g x)}{-c-d x} \, dx}{b^3 g}-\frac {\left (59168 B^2 c (b c-a d) n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^2 g}-\frac {\left (59168 a B^2 d (b c-a d) n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g}-\frac {\left (59168 a B^2 d (b c-a d) n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g}-\frac {\left (59168 B^2 c d (b c-a d) n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 g}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log ^2(a g+b g x)}{a+b x} \, dx}{b^2 g}-\frac {\left (29584 B^2 d (b c-a d)^2 n^2\right ) \int \frac {\log ^2(a g+b g x)}{c+d x} \, dx}{b^3 g}-\frac {\left (59168 B^2 d (b c-a 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 (a g+b g x)}{-c-d x} \, dx}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {\left (59168 A B (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac {\left (59168 A B (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^n\right ) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{x} \, dx,x,-c-d x\right )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (\frac {-b c g+a d g}{d}+\frac {b g x}{d}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right ) \log (a g+b g x)}{a g+b g x} \, dx}{b^2}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {g \log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b^3 g^2}-\frac {\left (59168 B^2 c (b c-a 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 )}{b^2 g}-\frac {\left (59168 a B^2 d (b c-a 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 )}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^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 )}{b^3 g}-\frac {\left (59168 B^2 (b c-a 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 {b g (-c-d x)}{-b c g+a d g}\right )}{a g+b g x} \, dx}{b^2}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac {\left (29584 B^2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {\log ^2\left (x^{-n}\right )}{\frac {-b c g+a d g}{d}+\frac {b g x}{d}} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (59168 B^2 (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\left (\frac {-b c+a d}{d}-\frac {b x}{d}\right )^n\right )}{\frac {-b c+a d}{d}-\frac {b x}{d}} \, dx,x,-c-d x\right )}{b^2 d g}+\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{d}-\frac {b g x}{d}\right )}{\frac {-b c g+a d g}{d}-\frac {b g x}{d}} \, dx,x,-c-d x\right )}{b^2 d}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {b g \left (\frac {b c g-a d g}{b g}+\frac {d x}{b g}\right )}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}-\frac {\left (59168 B^2 (b c-a 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 ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a 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 ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {\left (59168 B^2 (b c-a d)^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 )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n\right ) \text {Subst}\left (\int \frac {\log \left (x^{-n}\right ) \log \left (1+\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}+\frac {\left (29584 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (g (a+b x))\right )}{b^3 g}-\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {-b c g+a d g}{b g}-\frac {d x}{b g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}+\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a 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 ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {\left (29584 B^2 d (b c-a 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 )}{b^4 g}-\frac {\left (29584 B^2 d (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{\frac {-b c g+a d g}{b g}-\frac {d x}{b g}} \, dx,x,a g+b g x\right )}{b^4 g^2}-\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b g x}{-b c g+a d g}\right )}{x} \, dx,x,c+d x\right )}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \log (g (a+b x)) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a 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 ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {\left (59168 B^2 (b c-a d)^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 )}{b^3 g}-\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x) \log \left (1-\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a 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 ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {\left (59168 B^2 (b c-a d)^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 )}{b^3 g}-\frac {\left (59168 B^2 (b c-a d)^2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{-b c g+a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b^3 g}\\ &=-\frac {29584 A B d (b c-a d) n x}{b^2 g}-\frac {29584 a B^2 d (b c-a d) n^2 \log ^2(a+b x)}{b^3 g}+\frac {14792 B^2 (b c-a d)^2 n^2 \log ^2(a+b x)}{b^3 g}-\frac {29584 A B (b c-a d)^2 n \log ^2(g (a+b x))}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^3(g (a+b x))}{3 b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log (-c-d x)}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log (g (a+b x)) \log \left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log (-c-d x)}{b^3 g}-\frac {29584 B^2 d (b c-a d) n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b^3 g}+\frac {59168 a B d (b c-a d) n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}-\frac {29584 B (b c-a d)^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g}+\frac {29584 d (b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g}+\frac {14792 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log (c+d x)}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2 g}-\frac {59168 B c (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{b^2 g}-\frac {29584 B^2 c (b c-a d) n^2 \log ^2(c+d x)}{b^2 g}+\frac {59168 a B^2 d (b c-a d) n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 n^2 \log ^2(g (a+b x)) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}+\frac {29584 B^2 (b c-a d)^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 \log (g (a+b x)) \log ^2\left ((c+d x)^{-n}\right )}{b^3 g}+\frac {29584 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left (\frac {b (c+d x)}{b c-a d}\right ) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \log (a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(a g+b g x)}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log ^2(a g+b g x)}{b^3 g}+\frac {59168 A B (b c-a d)^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 a B^2 d (b c-a d) n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {29584 B^2 (b c-a d)^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 (b c-a d)^2 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a 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 ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}+\frac {59168 B^2 c (b c-a d) n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac {59168 B^2 (b c-a d)^2 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g}-\frac {59168 B^2 (b c-a d)^2 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1659\) vs. \(2(572)=1144\).
time = 1.97, size = 1659, normalized size = 2.90 \begin {gather*} \frac {i^2 \left (6 b d (2 b c-a d) x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2+3 b^2 d^2 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2+6 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right )^2-12 b B c n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (a d \log ^2\left (\frac {a}{b}+x\right )-2 a d \log \left (\frac {a}{b}+x\right ) (1+\log (a+b x))+2 \left (-b c+a d+\log \left (\frac {c}{d}+x\right ) \left (b c+a d \log (a+b x)-a d \log \left (\frac {d (a+b x)}{-b c+a d}\right )\right )+(-b d x+a d \log (a+b x)) \log \left (\frac {a+b x}{c+d x}\right )\right )-2 a d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )+6 b^2 B c^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (\log ^2\left (\frac {a}{b}+x\right )-2 \log (a+b x) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right )-2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {a+b x}{c+d x}\right )\right ) \left (-4 a d^2 (a+b x) \left (-1+\log \left (\frac {a}{b}+x\right )\right )+2 a^2 d^2 \log ^2\left (\frac {a}{b}+x\right )+4 a b d (c+d x) \left (-1+\log \left (\frac {c}{d}+x\right )\right )+d^2 \left (b x (2 a-b x)+2 b^2 x^2 \log \left (\frac {a}{b}+x\right )-2 a^2 \log (a+b x)\right )-2 d^2 \left (b x (-2 a+b x)+2 a^2 \log (a+b x)\right ) \left (\log \left (\frac {a}{b}+x\right )-\log \left (\frac {c}{d}+x\right )-\log \left (\frac {a+b x}{c+d x}\right )\right )+b^2 \left (d x (-2 c+d x)-2 d^2 x^2 \log \left (\frac {c}{d}+x\right )+2 c^2 \log (c+d x)\right )-4 a^2 d^2 \left (\log \left (\frac {c}{d}+x\right ) \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+4 b B^2 c n^2 \left (\log \left (\frac {a+b x}{c+d x}\right ) \left (-a d \log ^2\left (\frac {a+b x}{c+d x}\right )+6 (b c-a d) \log \left (\frac {b c-a d}{b c+b d x}\right )+3 d \log \left (\frac {a+b x}{c+d x}\right ) \left (a+b x+a \log \left (\frac {b c-a d}{b c+b d x}\right )\right )\right )+6 \left (b c-a d+a d \log \left (\frac {a+b x}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-6 a d \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )-B^2 n^2 \left (6 b^2 c^2 \log \left (\frac {b (b c-a d)}{c+d x}\right )+6 a^2 d^2 \log \left (\frac {b (b c-a d)}{c+d x}\right )-12 a b c d \log \left (\frac {b^2 (b c-a d)}{c+d x}\right )+6 a b c d \log \left (\frac {a+b x}{c+d x}\right )-6 a^2 d^2 \log \left (\frac {a+b x}{c+d x}\right )+6 b^2 c d x \log \left (\frac {a+b x}{c+d x}\right )-6 a b d^2 x \log \left (\frac {a+b x}{c+d x}\right )+9 a^2 d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+6 a b d^2 x \log ^2\left (\frac {a+b x}{c+d x}\right )-3 b^2 d^2 x^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 a^2 d^2 \log ^3\left (\frac {a+b x}{c+d x}\right )+6 b^2 c^2 \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+12 a b c d \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-18 a^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+6 a^2 d^2 \log ^2\left (\frac {a+b x}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+6 \left (b^2 c^2+2 a b c d-3 a^2 d^2+2 a^2 d^2 \log \left (\frac {a+b x}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )-12 a^2 d^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )\right )-6 b^2 B^2 c^2 n^2 \left (\log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {a+b x}{c+d x}\right )-2 \log \left (\frac {a+b x}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )\right )\right )}{6 b^3 g} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(i^2*(6*b*d*(2*b*c - a*d)*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 3*b^2*d^
2*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 6*(b*c - a*d)^2*Log[a + b*x]*(
A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 - 12*b*B*c*n*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + Log[a + b*x]) + 2*(-
(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + (-(b*d*x) + a*d*
Log[a + b*x])*Log[(a + b*x)/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 6*b^2*B*c^2*n*(A + B*
Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(Log[a/b + x]^2 - 2*Log[a + b*x]*(Log[a/b + x]
- Log[c/d + x] - Log[(a + b*x)/(c + d*x)]) - 2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b
*(c + d*x))/(b*c - a*d)])) + 3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(-4*a
*d^2*(a + b*x)*(-1 + Log[a/b + x]) + 2*a^2*d^2*Log[a/b + x]^2 + 4*a*b*d*(c + d*x)*(-1 + Log[c/d + x]) + d^2*(b
*x*(2*a - b*x) + 2*b^2*x^2*Log[a/b + x] - 2*a^2*Log[a + b*x]) - 2*d^2*(b*x*(-2*a + b*x) + 2*a^2*Log[a + b*x])*
(Log[a/b + x] - Log[c/d + x] - Log[(a + b*x)/(c + d*x)]) + b^2*(d*x*(-2*c + d*x) - 2*d^2*x^2*Log[c/d + x] + 2*
c^2*Log[c + d*x]) - 4*a^2*d^2*(Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + PolyLog[2, (b*(c + d*x))/(b*c
- a*d)])) + 4*b*B^2*c*n^2*(Log[(a + b*x)/(c + d*x)]*(-(a*d*Log[(a + b*x)/(c + d*x)]^2) + 6*(b*c - a*d)*Log[(b*
c - a*d)/(b*c + b*d*x)] + 3*d*Log[(a + b*x)/(c + d*x)]*(a + b*x + a*Log[(b*c - a*d)/(b*c + b*d*x)])) + 6*(b*c
- a*d + a*d*Log[(a + b*x)/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 6*a*d*PolyLog[3, (d*(a + b*x))
/(b*(c + d*x))]) - B^2*n^2*(6*b^2*c^2*Log[(b*(b*c - a*d))/(c + d*x)] + 6*a^2*d^2*Log[(b*(b*c - a*d))/(c + d*x)
] - 12*a*b*c*d*Log[(b^2*(b*c - a*d))/(c + d*x)] + 6*a*b*c*d*Log[(a + b*x)/(c + d*x)] - 6*a^2*d^2*Log[(a + b*x)
/(c + d*x)] + 6*b^2*c*d*x*Log[(a + b*x)/(c + d*x)] - 6*a*b*d^2*x*Log[(a + b*x)/(c + d*x)] + 9*a^2*d^2*Log[(a +
b*x)/(c + d*x)]^2 + 6*a*b*d^2*x*Log[(a + b*x)/(c + d*x)]^2 - 3*b^2*d^2*x^2*Log[(a + b*x)/(c + d*x)]^2 - 2*a^2
*d^2*Log[(a + b*x)/(c + d*x)]^3 + 6*b^2*c^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + 12*a*b*c
*d*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] - 18*a^2*d^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*
d)/(b*c + b*d*x)] + 6*a^2*d^2*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a*d)/(b*c + b*d*x)] + 6*(b^2*c^2 + 2*a*b*c
*d - 3*a^2*d^2 + 2*a^2*d^2*Log[(a + b*x)/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))] - 12*a^2*d^2*Poly
Log[3, (d*(a + b*x))/(b*(c + d*x))]) - 6*b^2*B^2*c^2*n^2*(Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(a + b*x)/(c +
d*x)]^2 - 2*Log[(a + b*x)/(c + d*x)]*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 2*PolyLog[3, (b*(c + d*x))/(d*
(a + b*x))])))/(6*b^3*g)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{b g x +a g}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i^{2} \left (\int \frac {A^{2} c^{2}}{a + b x}\, dx + \int \frac {A^{2} d^{2} x^{2}}{a + b x}\, dx + \int \frac {B^{2} c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {2 A^{2} c d x}{a + b x}\, dx + \int \frac {B^{2} d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a + b x}\, dx + \int \frac {2 A B d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx + \int \frac {2 B^{2} c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a + b x}\, dx + \int \frac {4 A B c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a + b x}\, dx\right )}{g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

i**2*(Integral(A**2*c**2/(a + b*x), x) + Integral(A**2*d**2*x**2/(a + b*x), x) + Integral(B**2*c**2*log(e*(a/(
c + d*x) + b*x/(c + d*x))**n)**2/(a + b*x), x) + Integral(2*A*B*c**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(
a + b*x), x) + Integral(2*A**2*c*d*x/(a + b*x), x) + Integral(B**2*d**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x
))**n)**2/(a + b*x), x) + Integral(2*A*B*d**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a + b*x), x) + Int
egral(2*B**2*c*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(a + b*x), x) + Integral(4*A*B*c*d*x*log(e*(a/(c
+ d*x) + b*x/(c + d*x))**n)/(a + b*x), x))/g

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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