3.3.2 \(\int \frac {(a g+b g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^3} \, dx\) [202]

Optimal. Leaf size=676 \[ \frac {B^2 (b c-a d) g^3 n^2 (a+b x)^2}{4 d^2 i^3 (c+d x)^2}-\frac {4 A b B (b c-a d) g^3 n (a+b x)}{d^3 i^3 (c+d x)}+\frac {4 b B^2 (b c-a d) g^3 n^2 (a+b x)}{d^3 i^3 (c+d x)}-\frac {4 b B^2 (b c-a d) g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}-\frac {B (b c-a d) g^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^2 i^3 (c+d x)^2}+\frac {b^2 g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^3}+\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d^2 i^3 (c+d x)^2}+\frac {2 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 i^3 (c+d x)}+\frac {2 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^3}+\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^4 i^3}+\frac {2 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac {6 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}-\frac {6 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.41, antiderivative size = 676, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2561, 2395, 2333, 2332, 2342, 2341, 2355, 2354, 2438, 2421, 6724} \begin {gather*} \frac {6 b^2 B g^3 n (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^4 i^3}+\frac {2 b^2 B^2 g^3 n^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}-\frac {6 b^2 B^2 g^3 n^2 (b c-a d) \text {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^3}+\frac {3 b^2 g^3 (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^4 i^3}+\frac {2 b^2 B g^3 n (b c-a d) \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 )}{d^4 i^3}+\frac {b^2 g^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^3 i^3}+\frac {2 b g^3 (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}{d^3 i^3 (c+d x)}-\frac {4 A b B g^3 n (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {g^3 (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^2 i^3 (c+d x)^2}-\frac {B g^3 n (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^2 i^3 (c+d x)^2}-\frac {4 b B^2 g^3 n (a+b x) (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^3 i^3 (c+d x)}+\frac {4 b B^2 g^3 n^2 (a+b x) (b c-a d)}{d^3 i^3 (c+d x)}+\frac {B^2 g^3 n^2 (a+b x)^2 (b c-a d)}{4 d^2 i^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 2395

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

Rule 2421

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

Rule 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 {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(202 c+202 d x)^3} \, dx &=\int \left (\frac {b^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3 (c+d x)^3}+\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3 (c+d x)^2}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^3 g^3\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{8242408 d^3}-\frac {\left (3 b^2 (b c-a d) g^3\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}{8242408 d^3}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2} \, dx}{8242408 d^3}-\frac {\left ((b c-a d)^3 g^3\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)^3} \, dx}{8242408 d^3}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {\left (b^3 B g^3 n\right ) \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}{4121204 d^3}+\frac {\left (3 b^2 B (b c-a d) g^3 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}{4121204 d^4}+\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^2} \, dx}{4121204 d^4}-\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) (c+d x)^3} \, dx}{8242408 d^4}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}+\frac {\left (3 b^2 B (b c-a d) g^3 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}{4121204 d^4}-\frac {\left (b^3 B (b c-a d) g^3 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}{4121204 d^3}+\frac {\left (3 b B (b c-a d)^3 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{4121204 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{8242408 d^4}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {\left (b^3 B (b c-a d) g^3 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}{4121204 d^3}+\frac {\left (3 b^2 B (b c-a d)^2 g^3 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}{4121204 d^4}+\frac {\left (3 b B (b c-a d)^3 g^3 n\right ) \int \left (\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{4121204 d^4}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \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}{8242408 d^4}\\ &=\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}+\frac {\left (a b^3 B g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4121204 d^3}-\frac {\left (b^3 B c g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4121204 d^3}-\frac {\left (b^3 B (b c-a d) g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{8242408 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{4121204 d^4}+\frac {\left (b^2 B (b c-a d) g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{8242408 d^3}-\frac {\left (3 b^2 B (b c-a d) g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{4121204 d^3}+\frac {\left (3 b^2 B (b c-a d)^2 g^3 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}{4121204 d^4}+\frac {\left (b B (b c-a d)^2 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{8242408 d^3}-\frac {\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{4121204 d^3}+\frac {\left (B (b c-a d)^3 g^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{8242408 d^3}\\ &=-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3 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}{4121204 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 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}{4121204 d^3}+\frac {\left (b^3 B^2 c g^3 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}{4121204 d^4}-\frac {\left (a b^2 B^2 g^3 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}{4121204 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3 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}{8242408 d^4}-\frac {\left (b^2 B^2 (b c-a d) g^3 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}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^4}+\frac {\left (b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{8242408 d^4}-\frac {\left (3 b B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{4121204 d^4}+\frac {\left (B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{16484816 d^4}\\ &=-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}+\frac {\left (3 b^3 B (b c-a d) g^3 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}{4121204 d^4}-\frac {\left (3 b^2 B (b c-a d) g^3 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}{4121204 d^3}+\frac {\left (b^3 B^2 c g^3 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}{4121204 d^4}-\frac {\left (a b^2 B^2 g^3 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}{4121204 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3 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}{8242408 d^4}-\frac {\left (b^2 B^2 (b c-a d) g^3 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}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^4}+\frac {\left (b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{8242408 d^4}-\frac {\left (3 b B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{4121204 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{16484816 d^4}\\ &=-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}+\frac {\left (3 A b^3 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4121204 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 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}{4121204 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4121204 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^3}+\frac {\left (b^4 B^2 c g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4121204 d^4}-\frac {\left (a b^3 B^2 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4121204 d^3}-\frac {\left (b^3 B^2 c g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4121204 d^3}+\frac {\left (a b^2 B^2 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4121204 d^2}+\frac {\left (b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{8242408 d^4}-\frac {\left (b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{8242408 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{4121204 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{4121204 d^4}-\frac {\left (b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{8242408 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{8242408 d^3}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{4121204 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{4121204 d^3}+\frac {\left (b B^2 (b c-a d)^3 g^3 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}{8242408 d^4}-\frac {\left (3 b B^2 (b c-a d)^3 g^3 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}{4121204 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3 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}{16484816 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4121204 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 n\right ) \int \frac {\log \left ((a+b x)^n\right ) \log (c+d x)}{a+b x} \, dx}{4121204 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 n\right ) \int \frac {\log (c+d x) \log \left ((c+d x)^{-n}\right )}{a+b x} \, dx}{4121204 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4121204 d^3}-\frac {\left (b^3 B^2 c g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4121204 d^4}-\frac {\left (a b^2 B^2 g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4121204 d^3}-\frac {\left (a b^3 B^2 g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4121204 d^3}-\frac {\left (b^3 B^2 c g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4121204 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{8242408 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{4121204 d^4}+\frac {\left (b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{8242408 d^4}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log ^2(c+d x)}{a+b x} \, dx}{8242408 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{4121204 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{8242408 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log ^2(c+d x)}{c+d x} \, dx}{8242408 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{4121204 d^3}+\frac {\left (3 b^3 B^2 (b c-a d) g^3 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}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (3 A b^2 B (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (b^3 B^2 c g^3 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 )}{4121204 d^4}-\frac {\left (a b^2 B^2 g^3 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 )}{4121204 d^3}+\frac {\left (b^2 B^2 (b c-a d) g^3 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 )}{8242408 d^4}+\frac {\left (b^2 B^2 (b c-a d) g^3 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 )}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,c+d x\right )}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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}{4121204 d^3}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {\left (3 b B^2 (b c-a d) g^3\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 )}{8242408 d^3}-\frac {\left (3 b B^2 (b c-a d) g^3 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 )}{4121204 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 n^2\right ) \text {Subst}\left (\int x^2 \, dx,x,\log (c+d x)\right )}{8242408 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b B^2 (b c-a d) g^3 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 )}{4121204 d^3}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {b^2 B^2 (b c-a d) g^3 n^2 \log ^3(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log ^2(c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {b^2 B^2 (b c-a d) g^3 n^2 \log ^3(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {\left (3 b^3 B^2 (b c-a d) g^3\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 )}{8242408 d^5}-\frac {\left (3 b^3 B^2 (b c-a d) g^3 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 )}{8242408 d^5}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {b^2 B^2 (b c-a d) g^3 n^2 \log ^3(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {b^2 B^2 (b c-a d) g^3 n^2 \log ^3(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+2 \frac {\left (3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}\\ &=\frac {B^2 (b c-a d)^3 g^3 n^2}{32969632 d^4 (c+d x)^2}-\frac {9 b B^2 (b c-a d)^2 g^3 n^2}{16484816 d^4 (c+d x)}-\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x)}{16484816 d^4}-\frac {a b^2 B^2 g^3 n^2 \log ^2(a+b x)}{8242408 d^3}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(a+b x)}{16484816 d^4}-\frac {B (b c-a d)^3 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16484816 d^4 (c+d x)^2}+\frac {5 b B (b c-a d)^2 g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4 (c+d x)}+\frac {a b^2 B g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4121204 d^3}+\frac {5 b^2 B (b c-a d) g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8242408 d^4}+\frac {b^3 g^3 x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^3}+\frac {(b c-a d)^3 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{16484816 d^4 (c+d x)^2}-\frac {3 b (b c-a d)^2 g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{8242408 d^4 (c+d x)}+\frac {9 b^2 B^2 (b c-a d) g^3 n^2 \log (c+d x)}{16484816 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{8242408 d^4}-\frac {b^3 B c g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{4121204 d^4}-\frac {5 b^2 B (b c-a d) g^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{8242408 d^4}-\frac {3 b^2 (b c-a d) g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{8242408 d^4}-\frac {3 A b^2 B (b c-a d) g^3 n \log ^2(c+d x)}{8242408 d^4}-\frac {b^3 B^2 c g^3 n^2 \log ^2(c+d x)}{8242408 d^4}-\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log ^2(c+d x)}{16484816 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log ^2(c+d x)}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{8242408 d^4}-\frac {b^2 B^2 (b c-a d) g^3 n^2 \log ^3(c+d x)}{8242408 d^4}+\frac {a b^2 B^2 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left ((c+d x)^{-n}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {a b^2 B^2 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^3}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{8242408 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}+\frac {3 A b^2 B (b c-a d) g^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {b^3 B^2 c g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}+\frac {5 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{8242408 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}-\frac {3 b^2 B^2 (b c-a d) g^3 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 )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{4121204 d^4}+\frac {3 b^2 B^2 (b c-a d) g^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{4121204 d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(4669\) vs. \(2(676)=1352\).
time = 7.62, size = 4669, normalized size = 6.91 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*(4*b^3*d*x*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + (2*(b*c - a*d)^3*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2)/(c + d*x)^2 - (12*b*(b*c - a*d)^2*(A + B
*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2)/(c + d*x) - 12*b^2*(b*c - a*d)*(A + B*Log[e
*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2*Log[c + d*x] + (6*a^2*b*B*d^2*n*(-A - B*Log[e*((a
+ b*x)/(c + d*x))^n] + B*n*Log[(a + b*x)/(c + d*x)])*(-(b^2*c^3) + 4*a*b*c^2*d - 3*a^2*c*d^2 - 2*b^2*c^2*d*x +
6*a*b*c*d^2*x - 4*a^2*d^3*x - 2*b*(b*c - 2*a*d)*(c + d*x)^2*Log[a + b*x] + 2*(b*c - a*d)^2*(c + 2*d*x)*Log[(a
+ b*x)/(c + d*x)] + 2*b^2*c^3*Log[c + d*x] - 4*a*b*c^2*d*Log[c + d*x] + 4*b^2*c^2*d*x*Log[c + d*x] - 8*a*b*c*
d^2*x*Log[c + d*x] + 2*b^2*c*d^2*x^2*Log[c + d*x] - 4*a*b*d^3*x^2*Log[c + d*x]))/((b*c - a*d)^2*(c + d*x)^2) +
(2*a^3*B*d^3*n*(-A - B*Log[e*((a + b*x)/(c + d*x))^n] + B*n*Log[(a + b*x)/(c + d*x)])*(-(b^2*c^2) + 4*a*b*c*d
- a^2*d^2 + 2*b^2*c*d*x + 2*a*b*d^2*x + 2*b^2*d^2*x^2 - 2*b^2*(c + d*x)^2*Log[a/b + x] + 2*(b*c - a*d)^2*Log[
(a + b*x)/(c + d*x)] + 2*b^2*c^2*Log[(b*(c + d*x))/(b*c - a*d)] + 4*b^2*c*d*x*Log[(b*(c + d*x))/(b*c - a*d)] +
2*b^2*d^2*x^2*Log[(b*(c + d*x))/(b*c - a*d)]))/((b*c - a*d)^2*(c + d*x)^2) + 6*a*b^2*B*d*n*(A + B*Log[e*((a +
b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(-2*Log[c/d + x]^2 - (8*c*(1 + Log[c/d + x]))/(c + d*x) +
(c^2*(1 + 2*Log[c/d + x]))/(c + d*x)^2 + 8*c*(Log[a/b + x]/(c + d*x) + (b*(Log[a + b*x] - Log[c + d*x]))/(-(b*
c) + a*d)) + 2*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)])*((c*(3*c + 4*d*x))/(c + d*x)^2 + 2*Lo
g[c + d*x]) + (2*c^2*(-Log[a/b + x] + (b*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c +
d*x]))/(b*c - a*d)^2))/(c + d*x)^2 + 4*(Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + PolyLog[2, (d*(a + b*x)
)/(-(b*c) + a*d)])) - 2*b^3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(-4*d*(a
/b + x)*(-1 + Log[a/b + x]) + 4*(c + d*x)*(-1 + Log[c/d + x]) - 6*c*Log[c/d + x]^2 - (12*c^2*(1 + Log[c/d + x]
))/(c + d*x) + (c^3*(1 + 2*Log[c/d + x]))/(c + d*x)^2 - 12*c^2*(-(Log[a/b + x]/(c + d*x)) + (b*(Log[a + b*x] -
Log[c + d*x]))/(b*c - a*d)) + 2*(-Log[a/b + x] + Log[c/d + x] + Log[(a + b*x)/(c + d*x)])*(-2*d*x + (c^2*(5*c
+ 6*d*x))/(c + d*x)^2 + 6*c*Log[c + d*x]) + (2*c^3*(-Log[a/b + x] + (b*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log
[a + b*x] - b*(c + d*x)*Log[c + d*x]))/(b*c - a*d)^2))/(c + d*x)^2 + 12*c*(Log[a/b + x]*Log[(b*(c + d*x))/(b*c
- a*d)] + PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])) + (3*a^2*b*B^2*d^2*n^2*(2*c*Log[(a + b*x)/(c + d*x)]^2 -
4*(c + d*x)*Log[(a + b*x)/(c + d*x)]^2 - (4*(c + d*x)*(2*b*c - 2*a*d + 2*b*(c + d*x)*Log[a + b*x] - 2*(b*c -
a*d)*Log[(a + b*x)/(c + d*x)] - 2*b*(c + d*x)*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] - 2*b*(c + d*x)*Log[c + d*
x] - 2*b*(c + d*x)*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + b*(c + d*x)*(Log[a + b*x]*(Log[a
+ b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + b*(c + d*x)*(Log[(b
*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)])))/(b*c - a*d) + (c*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x) + 2*b^2*(c + d*x)^2
*Log[a + b*x] - 2*(b*c - a*d)^2*Log[(a + b*x)/(c + d*x)] - 4*b*(b*c - a*d)*(c + d*x)*Log[(a + b*x)/(c + d*x)]
- 4*b^2*(c + d*x)^2*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] - 2*b^2*(c + d*x)^2*Log[c + d*x] + 4*b*(c + d*x)*(b*
c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x]) - 4*b^2*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]*Lo
g[(b*c - a*d)/(b*c + b*d*x)] + 2*b^2*(c + d*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)
]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*b^2*(c + d*x)^2*(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d
*(a + b*x))/(-(b*c) + a*d)] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(b*
c - a*d)^2))/(c + d*x)^2 - (a^3*B^2*d^3*n^2*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x) + 2*b^2*(c + d*x)^2*Log
[a + b*x] - 2*(b*c - a*d)^2*Log[(a + b*x)/(c + d*x)] - 4*b*(b*c - a*d)*(c + d*x)*Log[(a + b*x)/(c + d*x)] - 4*
b^2*(c + d*x)^2*Log[a + b*x]*Log[(a + b*x)/(c + d*x)] + 2*(b*c - a*d)^2*Log[(a + b*x)/(c + d*x)]^2 - 2*b^2*(c
+ d*x)^2*Log[c + d*x] + 4*b*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x]) - 4*b^
2*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]*Log[(b*c - a*d)/(b*c + b*d*x)] + 2*b^2*(c + d*x)^2*(Log[a + b*x]*(Log[a
+ b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*b^2*(c + d*x)^2*
(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c) + a*d)] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*Pol
yLog[2, (b*(c + d*x))/(b*c - a*d)])))/((b*c - a*d)^2*(c + d*x)^2) + 2*b^3*B^2*n^2*((2*d*(a + b*x)*Log[(a + b*x
)/(c + d*x)]^2)/b + (c^3*Log[(a + b*x)/(c + d*x)]^2)/(c + d*x)^2 - (6*c^2*Log[(a + b*x)/(c + d*x)]^2)/(c + d*x
) + 6*c*Log[(a + b*x)/(c + d*x)]^2*Log[(b*c - a...

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Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (d i x +c i \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b^2*log(b*x + a)/(2*I*b^2*c^2*d - 4*I*a*b*c*d^2 + 2*I*a^2*d^3) - b^2*log(d*x + c)/(2*I*b^2*c^2*d - 4*I*a*b
*c*d^2 + 2*I*a^2*d^3) - (2*b*d*x + 3*b*c - a*d)/(-4*I*b*c^3*d + 4*I*a*c^2*d^2 - 4*(I*b*c*d^3 - I*a*d^4)*x^2 -
8*(I*b*c^2*d^2 - I*a*c*d^3)*x))*A*B*a^3*g^3*n - 6*A*B*a^2*b*g^3*n*((b^2*c - 2*a*b*d)*log(b*x + a)/(2*I*b^2*c^2
*d^2 - 4*I*a*b*c*d^3 + 2*I*a^2*d^4) - (b^2*c - 2*a*b*d)*log(d*x + c)/(2*I*b^2*c^2*d^2 - 4*I*a*b*c*d^3 + 2*I*a^
2*d^4) - (b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/(-4*I*b*c^3*d^2 + 4*I*a*c^2*d^3 - 4*(I*b*c*d^4 - I*a*d^5)*x
^2 - 8*(I*b*c^2*d^3 - I*a*c*d^4)*x)) + A^2*b^3*g^3*((6*c^2*d*x + 5*c^3)/(2*I*d^6*x^2 + 4*I*c*d^5*x + 2*I*c^2*d
^4) + I*x/d^3 - 3*I*c*log(d*x + c)/d^4) - 3*A^2*a*b^2*g^3*((4*c*d*x + 3*c^2)/(2*I*d^5*x^2 + 4*I*c*d^4*x + 2*I*
c^2*d^3) - I*log(d*x + c)/d^3) + 6*(2*d*x + c)*A*B*a^2*b*g^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(2*I*d^4*x
^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + 3*(2*d*x + c)*A^2*a^2*b*g^3/(2*I*d^4*x^2 + 4*I*c*d^3*x + 2*I*c^2*d^2) + 2*A*
B*a^3*g^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e)/(2*I*d^3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d) + A^2*a^3*g^3/(2*I*d^
3*x^2 + 4*I*c*d^2*x + 2*I*c^2*d) + 1/2*(2*I*B^2*b^3*d^3*g^3*x^3 + 4*I*B^2*b^3*c*d^2*g^3*x^2 - 2*(2*I*b^3*c^2*d
*g^3 - 6*I*a*b^2*c*d^2*g^3 + 3*I*a^2*b*d^3*g^3)*B^2*x + (-5*I*b^3*c^3*g^3 + 9*I*a*b^2*c^2*d*g^3 - 3*I*a^2*b*c*
d^2*g^3 - I*a^3*d^3*g^3)*B^2 - 6*((I*b^3*c*d^2*g^3 - I*a*b^2*d^3*g^3)*B^2*x^2 + 2*(I*b^3*c^2*d*g^3 - I*a*b^2*c
*d^2*g^3)*B^2*x + (I*b^3*c^3*g^3 - I*a*b^2*c^2*d*g^3)*B^2)*log(d*x + c))*log((d*x + c)^n)^2/(d^6*x^2 + 2*c*d^5
*x + c^2*d^4) - integrate((-3*I*B^2*a^2*b*d^3*g^3*x - I*B^2*a^3*d^3*g^3 + (-2*I*A*B*b^3*d^3*g^3 - I*B^2*b^3*d^
3*g^3)*x^3 - 3*(2*I*A*B*a*b^2*d^3*g^3 + I*B^2*a*b^2*d^3*g^3)*x^2 + (-I*B^2*b^3*d^3*g^3*x^3 - 3*I*B^2*a*b^2*d^3
*g^3*x^2 - 3*I*B^2*a^2*b*d^3*g^3*x - I*B^2*a^3*d^3*g^3)*log((b*x + a)^n)^2 - 2*(3*I*B^2*a^2*b*d^3*g^3*x + I*B^
2*a^3*d^3*g^3 + (I*A*B*b^3*d^3*g^3 + I*B^2*b^3*d^3*g^3)*x^3 + 3*(I*A*B*a*b^2*d^3*g^3 + I*B^2*a*b^2*d^3*g^3)*x^
2)*log((b*x + a)^n) - (2*(2*I*b^3*c^2*d*g^3*n - 6*I*a*b^2*c*d^2*g^3*n + 3*a^2*b*d^3*g^3*(I*n - I))*B^2*x + 2*(
B^2*b^3*d^3*g^3*(-I*n - I) - I*A*B*b^3*d^3*g^3)*x^3 - (-5*I*b^3*c^3*g^3*n + 9*I*a*b^2*c^2*d*g^3*n - 3*I*a^2*b*
c*d^2*g^3*n + a^3*d^3*g^3*(-I*n + 2*I))*B^2 + 2*(-3*I*A*B*a*b^2*d^3*g^3 + (-2*I*b^3*c*d^2*g^3*n - 3*I*a*b^2*d^
3*g^3)*B^2)*x^2 + 6*((I*b^3*c*d^2*g^3*n - I*a*b^2*d^3*g^3*n)*B^2*x^2 + 2*(I*b^3*c^2*d*g^3*n - I*a*b^2*c*d^2*g^
3*n)*B^2*x + (I*b^3*c^3*g^3*n - I*a*b^2*c^2*d*g^3*n)*B^2)*log(d*x + c) + 2*(-I*B^2*b^3*d^3*g^3*x^3 - 3*I*B^2*a
*b^2*d^3*g^3*x^2 - 3*I*B^2*a^2*b*d^3*g^3*x - I*B^2*a^3*d^3*g^3)*log((b*x + a)^n))*log((d*x + c)^n))/(d^6*x^3 +
3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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