3.2.77 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^6} \, dx\) [177]

Optimal. Leaf size=493 \[ -\frac {2 B^2 d^2 i^2 n^2 (c+d x)^3}{27 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B^2 d i^2 n^2 (c+d x)^4}{16 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B^2 i^2 n^2 (c+d x)^5}{125 (b c-a d)^3 g^6 (a+b x)^5}-\frac {2 B d^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B i^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 (b c-a d)^3 g^6 (a+b x)^5} \]

[Out]

-2/27*B^2*d^2*i^2*n^2*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/16*b*B^2*d*i^2*n^2*(d*x+c)^4/(-a*d+b*c)^3/g^6/(b*
x+a)^4-2/125*b^2*B^2*i^2*n^2*(d*x+c)^5/(-a*d+b*c)^3/g^6/(b*x+a)^5-2/9*B*d^2*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)
/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/4*b*B*d*i^2*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^
3/g^6/(b*x+a)^4-2/25*b^2*B*i^2*n*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^5-1/3*d^2*
i^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/2*b*d*i^2*(d*x+c)^4*(A+B*ln(e*((b
*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/5*b^2*i^2*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+
b*c)^3/g^6/(b*x+a)^5

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 493, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {2561, 2395, 2342, 2341} \begin {gather*} -\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {2 b^2 B i^2 n (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^6 (a+b x)^3 (b c-a d)^3}-\frac {2 B d^2 i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^6 (a+b x)^4 (b c-a d)^3}+\frac {b B d i^2 n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^3}-\frac {2 b^2 B^2 i^2 n^2 (c+d x)^5}{125 g^6 (a+b x)^5 (b c-a d)^3}-\frac {2 B^2 d^2 i^2 n^2 (c+d x)^3}{27 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B^2 d i^2 n^2 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^3} \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)^6,x]

[Out]

(-2*B^2*d^2*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B^2*d*i^2*n^2*(c + d*x)^4)/(16*(b*c -
a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B^2*i^2*n^2*(c + d*x)^5)/(125*(b*c - a*d)^3*g^6*(a + b*x)^5) - (2*B*d^2*i^2*
n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*
x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*c - a*d)^3*g^6*(a + b*x)^4) - (2*b^2*B*i^2*n*(c + d*x)^5*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])^2)/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2)/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)

Rule 2341

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

Rule 2342

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

Rule 2395

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

Rule 2561

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

Rubi steps

\begin {align*} \int \frac {(177 c+177 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)^6} \, dx &=\int \left (\frac {31329 (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 g^6 (a+b x)^6}+\frac {62658 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^6 (a+b x)^5}+\frac {31329 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac {\left (31329 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+b x)^4} \, dx}{b^2 g^6}+\frac {(62658 d (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^6}+\frac {\left (31329 (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+b x)^6} \, dx}{b^2 g^6}\\ &=-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {\left (20886 B 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 )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac {(31329 B d (b c-a d) n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (62658 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 )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {\left (20886 B d^2 (b c-a d) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (31329 B d (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)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (62658 B (b c-a d)^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {\left (20886 B d^2 (b c-a d) n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \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)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac {\left (31329 B d (b c-a d)^2 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \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)^3}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac {\left (62658 B (b c-a d)^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 )}{(b c-a d) (a+b x)^6}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2 \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)^4}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 (a+b x)}+\frac {d^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^3 g^6}\\ &=-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {\left (62658 B 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)^4} \, dx}{5 b^2 g^6}+\frac {\left (20886 B 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)^4} \, dx}{b^2 g^6}-\frac {\left (31329 B 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)^4} \, dx}{b^2 g^6}-\frac {\left (62658 B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac {\left (20886 B d^5 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 (b c-a d)^3 g^6}+\frac {\left (31329 B d^5 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 (b c-a d)^3 g^6}+\frac {\left (62658 B d^6 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B d^6 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B d^6 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{5 b^2 (b c-a d)^2 g^6}+\frac {\left (20886 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d)^2 g^6}-\frac {\left (31329 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d)^2 g^6}-\frac {\left (62658 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{5 b^2 (b c-a d) g^6}-\frac {\left (20886 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 (b c-a d) g^6}+\frac {\left (31329 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 (b c-a d) g^6}-\frac {(62658 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)^5} \, dx}{5 b^2 g^6}+\frac {(31329 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)^5} \, dx}{b^2 g^6}+\frac {\left (62658 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)^6} \, dx}{5 b^2 g^6}\\ &=-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{5 b^3 g^6}+\frac {\left (6962 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac {\left (10443 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (62658 B^2 d^5 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}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (62658 B^2 d^5 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}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 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 (b c-a d)^3 g^6}-\frac {\left (20886 B^2 d^5 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^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 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 (b c-a d)^3 g^6}+\frac {\left (31329 B^2 d^5 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^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^4 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d)^2 g^6}+\frac {\left (20886 B^2 d^4 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d)^2 g^6}-\frac {\left (31329 B^2 d^4 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d)^2 g^6}-\frac {\left (31329 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}-\frac {\left (10443 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 (b c-a d) g^6}+\frac {\left (31329 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 (b c-a d) g^6}-\frac {\left (31329 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac {\left (31329 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac {\left (62658 B^2 (b c-a d)^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}\\ &=-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 g^6}-\frac {\left (10443 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (31329 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^6}+\frac {\left (62658 B^2 d^5 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}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (62658 B^2 d^5 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}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 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 (b c-a d)^3 g^6}-\frac {\left (20886 B^2 d^5 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^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 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 (b c-a d)^3 g^6}+\frac {\left (31329 B^2 d^5 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^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^4 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}+\frac {\left (20886 B^2 d^4 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^6}-\frac {\left (31329 B^2 d^4 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^6}+\frac {\left (20886 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{5 b^3 g^6}+\frac {\left (6962 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac {\left (10443 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{b^3 g^6}-\frac {\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac {\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac {\left (62658 B^2 (b c-a d)^3 n^2\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}\\ &=-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{5 b^3 g^6}-\frac {\left (10443 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac {\left (31329 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^3 g^6}+\frac {\left (62658 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac {\left (62658 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac {\left (20886 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac {\left (31329 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac {\left (62658 B^2 d^6 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^6 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (20886 B^2 d^6 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^6 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (31329 B^2 d^6 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^6 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^4 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{5 b^3 (b c-a d) g^6}+\frac {\left (20886 B^2 d^4 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 (b c-a d) g^6}-\frac {\left (31329 B^2 d^4 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 (b c-a d) g^6}+\frac {\left (20886 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{5 b^3 g^6}+\frac {\left (6962 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}-\frac {\left (10443 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{b^3 g^6}-\frac {\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{10 b^3 g^6}+\frac {\left (31329 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b^3 g^6}+\frac {\left (62658 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{25 b^3 g^6}\\ &=-\frac {62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac {219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac {149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac {45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^6 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^6 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^6 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}\\ &=-\frac {62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac {219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac {149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac {45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}+\frac {10443 B^2 d^5 n^2 \log ^2(a+b x)}{10 b^3 (b c-a d)^3 g^6}-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {10443 B^2 d^5 n^2 \log ^2(c+d x)}{10 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^5 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 )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (62658 B^2 d^5 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 )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 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 (b c-a d)^3 g^6}+\frac {\left (20886 B^2 d^5 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^3 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 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 (b c-a d)^3 g^6}-\frac {\left (31329 B^2 d^5 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^3 (b c-a d)^3 g^6}\\ &=-\frac {62658 B^2 (b c-a d)^2 n^2}{125 b^3 g^6 (a+b x)^5}-\frac {219303 B^2 d (b c-a d) n^2}{400 b^3 g^6 (a+b x)^4}+\frac {149683 B^2 d^2 n^2}{300 b^3 g^6 (a+b x)^3}-\frac {45253 B^2 d^3 n^2}{200 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {163607 B^2 d^4 n^2}{100 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {163607 B^2 d^5 n^2 \log (a+b x)}{100 b^3 (b c-a d)^3 g^6}+\frac {10443 B^2 d^5 n^2 \log ^2(a+b x)}{10 b^3 (b c-a d)^3 g^6}-\frac {62658 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {93987 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {3481 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 g^6 (a+b x)^3}+\frac {10443 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {10443 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {10443 B d^5 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {31329 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {31329 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {10443 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^6 (a+b x)^3}+\frac {163607 B^2 d^5 n^2 \log (c+d x)}{100 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {10443 B d^5 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{5 b^3 (b c-a d)^3 g^6}+\frac {10443 B^2 d^5 n^2 \log ^2(c+d x)}{10 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}-\frac {10443 B^2 d^5 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b^3 (b c-a d)^3 g^6}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.39, size = 2320, normalized size = 4.71 \begin {gather*} \text {Result too large to show} \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)^6,x]

[Out]

-1/54000*(i^2*(10800*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 27000*d*(b*c - a*d)^4*(a + b*x)*
(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 18000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c +
d*x))^n])^2 + 1000*B*d^2*n*(a + b*x)^2*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 18*d*(b*c -
a*d)^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[e*((a + b
*x)/(c + d*x))^n]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*d^3*(a + b*x)
^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 36*B*d^2*n*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a
+ b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*n*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*
(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*n*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b
*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) - 18*B*
d^3*n*(a + b*x)^3*(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)]) + 18*B*d^3*n*(a + b*x)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x]
+ 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 375*B*d*n*(a + b*x)*(36*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^2*(b*c - a*d)^2*(
a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)
/(c + d*x))^n]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^4*(a + b*x)^
4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 144*B*d^3*n*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a
+ b*x] - d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*n*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2
*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*n*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*
c - a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[
c + d*x]) + 3*B*n*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3
*(-(b*c) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*n*
(a + b*x)^4*(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)]) - 72*B*d^4*n*(a + b*x)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)])) + 6*B*n*(-225*a*B*d*(b*c - a*d)^4*n + 144*B*(b*c - a*d)^5*n - 225*b*B*d*
(b*c - a*d)^4*n*x + 300*a*B*d^2*(b*c - a*d)^3*n*(a + b*x) - 180*B*d*(b*c - a*d)^4*n*(a + b*x) + 300*b*B*d^2*(b
*c - a*d)^3*n*x*(a + b*x) - 450*a*B*d^3*(b*c - a*d)^2*n*(a + b*x)^2 + 640*B*d^2*(b*c - a*d)^3*n*(a + b*x)^2 -
450*b*B*d^3*(b*c - a*d)^2*n*x*(a + b*x)^2 + 900*a*B*d^4*(b*c - a*d)*n*(a + b*x)^3 - 1860*B*d^3*(b*c - a*d)^2*n
*(a + b*x)^3 + 900*b*B*d^4*(b*c - a*d)*n*x*(a + b*x)^3 + 3600*b*B*c*d^4*n*(a + b*x)^4 - 3600*a*B*d^5*n*(a + b*
x)^4 + 3720*B*d^4*(b*c - a*d)*n*(a + b*x)^4 + 900*a*B*d^5*n*(a + b*x)^4*Log[a + b*x] + 900*b*B*d^5*n*x*(a + b*
x)^4*Log[a + b*x] + 7320*B*d^5*n*(a + b*x)^5*Log[a + b*x] + 720*(b*c - a*d)^5*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]) - 900*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 1200*d^2*(b*c - a*d)^3*(a + b*
x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 1800*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]) + 3600*d^4*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 3600*d^5*(a + b*x)^5*Log
[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 900*a*B*d^5*n*(a + b*x)^4*Log[c + d*x] - 900*b*B*d^5*n*x*(a
+ b*x)^4*Log[c + d*x] - 7320*B*d^5*n*(a + b*x)^5*Log[c + d*x] - 3600*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/
(c + d*x))^n])*Log[c + d*x] - 1800*B*d^5*n*(a + b*x)^5*(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)]) + 1800*B*d^5*n*(a + b*x)^5*((2*Log[(d*(a + b*x))/(-(b*c
) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))))/(b^3*(b*c - a*d)^3*g^6*(a
+ b*x)^5)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{6}}\, dx\]

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)^6,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)^6,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 10900 vs. \(2 (454) = 908\).
time = 1.43, size = 10900, normalized size = 22.11 \begin {gather*} \text {Too large to display} \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)^6,x, algorithm="maxima")

[Out]

1/150*A*B*c^2*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 137*a
^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^
4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^
2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*
d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4
*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2
+ 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c^4
- 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*
a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x + c
)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6)) + 1
/900*A*B*d^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*
(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 -
9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d
^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*x)/((b
^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 - 4*a^2
*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*c^3*d
+ 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5*b^7*c
^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^7
*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 + a^9*b
^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6
*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d
^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b
^3*d^5)*g^6)) + 1/300*A*B*c*d*n*((27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c*d^3 + 7
7*a^5*d^4 - 60*(5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3 - 10*(1
0*b^5*c^3*d - 67*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^3*d + 23
2*a^2*b^3*c^2*d^2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a
^3*b^8*c*d^3 + a^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 +
a^5*b^6*d^4)*g^6*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*
g^6*x^3 + 10*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x^2 + 5*(
a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^4 - 4*a^
6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x + a)/((
b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6) + 60*(
5*b*c*d^4 - a*d^5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^
3*c*d^4 - a^5*b^2*d^5)*g^6)) + 1/10*(5*b*x + a)*B^2*c*d*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2/(b^7*g^6*x^5
+ 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) + 1/30*(10*b^2*x^
2 + 5*a*b*x + a^2)*B^2*d^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^
6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) + 1/9000*(60*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 -
63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*
(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 -
77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*
(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 -
4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3
*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4
*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b
^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c
^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1902 vs. \(2 (454) = 908\).
time = 0.45, size = 1902, normalized size = 3.86 \begin {gather*} \text {Too large to display} \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)^6,x, algorithm="fricas")

[Out]

1/54000*(10800*(A^2 + 2*A*B + B^2)*b^5*c^5 - 27000*(A^2 + 2*A*B + B^2)*a*b^4*c^4*d + 18000*(A^2 + 2*A*B + B^2)
*a^2*b^3*c^3*d^2 - 1800*(A^2 + 2*A*B + B^2)*a^5*d^5 + 60*(47*(B^2*b^5*c*d^4 - B^2*a*b^4*d^5)*n^2 + 60*((A*B +
B^2)*b^5*c*d^4 - (A*B + B^2)*a*b^4*d^5)*n)*x^4 + 30*((13*B^2*b^5*c^2*d^3 + 350*B^2*a*b^4*c*d^4 - 363*B^2*a^2*b
^3*d^5)*n^2 - 60*((A*B + B^2)*b^5*c^2*d^3 - 10*(A*B + B^2)*a*b^4*c*d^4 + 9*(A*B + B^2)*a^2*b^3*d^5)*n)*x^3 + (
864*B^2*b^5*c^5 - 3375*B^2*a*b^4*c^4*d + 4000*B^2*a^2*b^3*c^3*d^2 - 1489*B^2*a^5*d^5)*n^2 + 10*(1800*(A^2 + 2*
A*B + B^2)*b^5*c^3*d^2 - 5400*(A^2 + 2*A*B + B^2)*a*b^4*c^2*d^3 + 5400*(A^2 + 2*A*B + B^2)*a^2*b^3*c*d^4 - 180
0*(A^2 + 2*A*B + B^2)*a^3*b^2*d^5 - (86*B^2*b^5*c^3*d^2 - 375*B^2*a*b^4*c^2*d^3 - 1200*B^2*a^2*b^3*c*d^4 + 148
9*B^2*a^3*b^2*d^5)*n^2 + 60*(2*(A*B + B^2)*b^5*c^3*d^2 - 15*(A*B + B^2)*a*b^4*c^2*d^3 + 60*(A*B + B^2)*a^2*b^3
*c*d^4 - 47*(A*B + B^2)*a^3*b^2*d^5)*n)*x^2 + 1800*(B^2*b^5*d^5*n^2*x^5 + 5*B^2*a*b^4*d^5*n^2*x^4 + 10*B^2*a^2
*b^3*d^5*n^2*x^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 3*B^2*a^2*b^3*c*d^4)*n^2*x^2 + 5*(3*B^2*b^5*c^4
*d - 8*B^2*a*b^4*c^3*d^2 + 6*B^2*a^2*b^3*c^2*d^3)*n^2*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3
*c^3*d^2)*n^2)*log((b*x + a)/(d*x + c))^2 + 60*(72*(A*B + B^2)*b^5*c^5 - 225*(A*B + B^2)*a*b^4*c^4*d + 200*(A*
B + B^2)*a^2*b^3*c^3*d^2 - 47*(A*B + B^2)*a^5*d^5)*n + 5*(5400*(A^2 + 2*A*B + B^2)*b^5*c^4*d - 14400*(A^2 + 2*
A*B + B^2)*a*b^4*c^3*d^2 + 10800*(A^2 + 2*A*B + B^2)*a^2*b^3*c^2*d^3 - 1800*(A^2 + 2*A*B + B^2)*a^4*b*d^5 + (1
89*B^2*b^5*c^4*d - 1100*B^2*a*b^4*c^3*d^2 + 2400*B^2*a^2*b^3*c^2*d^3 - 1489*B^2*a^4*b*d^5)*n^2 + 60*(27*(A*B +
B^2)*b^5*c^4*d - 100*(A*B + B^2)*a*b^4*c^3*d^2 + 120*(A*B + B^2)*a^2*b^3*c^2*d^3 - 47*(A*B + B^2)*a^4*b*d^5)*
n)*x + 60*((47*B^2*b^5*d^5*n^2 + 60*(A*B + B^2)*b^5*d^5*n)*x^5 + 5*(60*(A*B + B^2)*a*b^4*d^5*n + (12*B^2*b^5*c
*d^4 + 35*B^2*a*b^4*d^5)*n^2)*x^4 + 10*(60*(A*B + B^2)*a^2*b^3*d^5*n - (3*B^2*b^5*c^2*d^3 - 30*B^2*a*b^4*c*d^4
- 20*B^2*a^2*b^3*d^5)*n^2)*x^3 + (72*B^2*b^5*c^5 - 225*B^2*a*b^4*c^4*d + 200*B^2*a^2*b^3*c^3*d^2)*n^2 + 10*((
2*B^2*b^5*c^3*d^2 - 15*B^2*a*b^4*c^2*d^3 + 60*B^2*a^2*b^3*c*d^4)*n^2 + 60*((A*B + B^2)*b^5*c^3*d^2 - 3*(A*B +
B^2)*a*b^4*c^2*d^3 + 3*(A*B + B^2)*a^2*b^3*c*d^4)*n)*x^2 + 60*(6*(A*B + B^2)*b^5*c^5 - 15*(A*B + B^2)*a*b^4*c^
4*d + 10*(A*B + B^2)*a^2*b^3*c^3*d^2)*n + 5*((27*B^2*b^5*c^4*d - 100*B^2*a*b^4*c^3*d^2 + 120*B^2*a^2*b^3*c^2*d
^3)*n^2 + 60*(3*(A*B + B^2)*b^5*c^4*d - 8*(A*B + B^2)*a*b^4*c^3*d^2 + 6*(A*B + B^2)*a^2*b^3*c^2*d^3)*n)*x)*log
((b*x + a)/(d*x + c)))/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^6*x^5 + 5*(a*b^10*c^3 -
3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g^6*x^4 + 10*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d^2
- a^5*b^6*d^3)*g^6*x^3 + 10*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^6*x^2 + 5*(a^4*
b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*c*d^2 - a^7*b^4*d^3)*g^6*x + (a^5*b^6*c^3 - 3*a^6*b^5*c^2*d + 3*a^7*b^4*
c*d^2 - a^8*b^3*d^3)*g^6)

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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((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**6,x)

[Out]

Timed out

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Giac [A]
time = 13.69, size = 811, normalized size = 1.65 \begin {gather*} \frac {1}{54000} \, {\left (\frac {1800 \, {\left (6 \, B^{2} b^{2} n^{2} - \frac {15 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {10 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {60 \, {\left (72 \, B^{2} b^{2} n^{2} - \frac {225 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {200 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}} + 360 \, A B b^{2} n + 360 \, B^{2} b^{2} n - \frac {900 \, {\left (b x + a\right )} A B b d n}{d x + c} - \frac {900 \, {\left (b x + a\right )} B^{2} b d n}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} A B d^{2} n}{{\left (d x + c\right )}^{2}} + \frac {600 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {864 \, B^{2} b^{2} n^{2} - \frac {3375 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {4000 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}} + 4320 \, A B b^{2} n + 4320 \, B^{2} b^{2} n - \frac {13500 \, {\left (b x + a\right )} A B b d n}{d x + c} - \frac {13500 \, {\left (b x + a\right )} B^{2} b d n}{d x + c} + \frac {12000 \, {\left (b x + a\right )}^{2} A B d^{2} n}{{\left (d x + c\right )}^{2}} + \frac {12000 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n}{{\left (d x + c\right )}^{2}} + 10800 \, A^{2} b^{2} + 21600 \, A B b^{2} + 10800 \, B^{2} b^{2} - \frac {27000 \, {\left (b x + a\right )} A^{2} b d}{d x + c} - \frac {54000 \, {\left (b x + a\right )} A B b d}{d x + c} - \frac {27000 \, {\left (b x + a\right )} B^{2} b d}{d x + c} + \frac {18000 \, {\left (b x + a\right )}^{2} A^{2} d^{2}}{{\left (d x + c\right )}^{2}} + \frac {36000 \, {\left (b x + a\right )}^{2} A B d^{2}}{{\left (d x + c\right )}^{2}} + \frac {18000 \, {\left (b x + a\right )}^{2} B^{2} d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

1/54000*(1800*(6*B^2*b^2*n^2 - 15*(b*x + a)*B^2*b*d*n^2/(d*x + c) + 10*(b*x + a)^2*B^2*d^2*n^2/(d*x + c)^2)*lo
g((b*x + a)/(d*x + c))^2/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x +
a)^5*a^2*d^2*g^6/(d*x + c)^5) + 60*(72*B^2*b^2*n^2 - 225*(b*x + a)*B^2*b*d*n^2/(d*x + c) + 200*(b*x + a)^2*B^
2*d^2*n^2/(d*x + c)^2 + 360*A*B*b^2*n + 360*B^2*b^2*n - 900*(b*x + a)*A*B*b*d*n/(d*x + c) - 900*(b*x + a)*B^2*
b*d*n/(d*x + c) + 600*(b*x + a)^2*A*B*d^2*n/(d*x + c)^2 + 600*(b*x + a)^2*B^2*d^2*n/(d*x + c)^2)*log((b*x + a)
/(d*x + c))/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^5*a^2*d^2
*g^6/(d*x + c)^5) + (864*B^2*b^2*n^2 - 3375*(b*x + a)*B^2*b*d*n^2/(d*x + c) + 4000*(b*x + a)^2*B^2*d^2*n^2/(d*
x + c)^2 + 4320*A*B*b^2*n + 4320*B^2*b^2*n - 13500*(b*x + a)*A*B*b*d*n/(d*x + c) - 13500*(b*x + a)*B^2*b*d*n/(
d*x + c) + 12000*(b*x + a)^2*A*B*d^2*n/(d*x + c)^2 + 12000*(b*x + a)^2*B^2*d^2*n/(d*x + c)^2 + 10800*A^2*b^2 +
21600*A*B*b^2 + 10800*B^2*b^2 - 27000*(b*x + a)*A^2*b*d/(d*x + c) - 54000*(b*x + a)*A*B*b*d/(d*x + c) - 27000
*(b*x + a)*B^2*b*d/(d*x + c) + 18000*(b*x + a)^2*A^2*d^2/(d*x + c)^2 + 36000*(b*x + a)^2*A*B*d^2/(d*x + c)^2 +
18000*(b*x + a)^2*B^2*d^2/(d*x + c)^2)/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x
+ c)^5 + (b*x + a)^5*a^2*d^2*g^6/(d*x + c)^5))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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Mupad [B]
time = 11.15, size = 2500, normalized size = 5.07 \begin {gather*} \text {Too large to display} \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)^6,x)

[Out]

((1800*A^2*a^4*d^4*i^2 + 10800*A^2*b^4*c^4*i^2 + 1489*B^2*a^4*d^4*i^2*n^2 + 864*B^2*b^4*c^4*i^2*n^2 - 16200*A^
2*a*b^3*c^3*d*i^2 + 1800*A^2*a^3*b*c*d^3*i^2 + 2820*A*B*a^4*d^4*i^2*n + 4320*A*B*b^4*c^4*i^2*n + 1800*A^2*a^2*
b^2*c^2*d^2*i^2 + 1489*B^2*a^2*b^2*c^2*d^2*i^2*n^2 - 2511*B^2*a*b^3*c^3*d*i^2*n^2 + 1489*B^2*a^3*b*c*d^3*i^2*n
^2 + 2820*A*B*a^2*b^2*c^2*d^2*i^2*n - 9180*A*B*a*b^3*c^3*d*i^2*n + 2820*A*B*a^3*b*c*d^3*i^2*n)/(60*(a*d - b*c)
) + (x*(1800*A^2*a^3*b*d^4*i^2 + 5400*A^2*b^4*c^3*d*i^2 - 9000*A^2*a*b^3*c^2*d^2*i^2 + 1800*A^2*a^2*b^2*c*d^3*
i^2 + 1489*B^2*a^3*b*d^4*i^2*n^2 + 189*B^2*b^4*c^3*d*i^2*n^2 + 1620*A*B*b^4*c^3*d*i^2*n - 911*B^2*a*b^3*c^2*d^
2*i^2*n^2 + 1489*B^2*a^2*b^2*c*d^3*i^2*n^2 + 2820*A*B*a^3*b*d^4*i^2*n - 4380*A*B*a*b^3*c^2*d^2*i^2*n + 2820*A*
B*a^2*b^2*c*d^3*i^2*n))/(12*(a*d - b*c)) + (x^2*(1800*A^2*a^2*b^2*d^4*i^2 + 1800*A^2*b^4*c^2*d^2*i^2 - 3600*A^
2*a*b^3*c*d^3*i^2 + 1489*B^2*a^2*b^2*d^4*i^2*n^2 - 86*B^2*b^4*c^2*d^2*i^2*n^2 + 2820*A*B*a^2*b^2*d^4*i^2*n + 1
20*A*B*b^4*c^2*d^2*i^2*n + 289*B^2*a*b^3*c*d^3*i^2*n^2 - 780*A*B*a*b^3*c*d^3*i^2*n))/(6*(a*d - b*c)) + (x^3*(3
63*B^2*a*b^3*d^4*i^2*n^2 + 13*B^2*b^4*c*d^3*i^2*n^2 - 60*A*B*b^4*c*d^3*i^2*n + 540*A*B*a*b^3*d^4*i^2*n))/(2*(a
*d - b*c)) + (d*x^4*(47*B^2*b^4*d^3*i^2*n^2 + 60*A*B*b^4*d^3*i^2*n))/(a*d - b*c))/(x*(4500*a^4*b^5*c*g^6 - 450
0*a^5*b^4*d*g^6) - x^4*(4500*a^2*b^7*d*g^6 - 4500*a*b^8*c*g^6) + x^5*(900*b^9*c*g^6 - 900*a*b^8*d*g^6) + x^2*(
9000*a^3*b^6*c*g^6 - 9000*a^4*b^5*d*g^6) + x^3*(9000*a^2*b^7*c*g^6 - 9000*a^3*b^6*d*g^6) + 900*a^5*b^4*c*g^6 -
900*a^6*b^3*d*g^6) - log(e*((a + b*x)/(c + d*x))^n)^2*((a*((B^2*c*d*i^2)/(10*b^2) + (B^2*a*d^2*i^2)/(30*b^3))
+ x*(b*((B^2*c*d*i^2)/(10*b^2) + (B^2*a*d^2*i^2)/(30*b^3)) + (2*B^2*c*d*i^2)/(5*b) + (2*B^2*a*d^2*i^2)/(15*b^
2)) + (B^2*c^2*i^2)/(5*b) + (B^2*d^2*i^2*x^2)/(3*b))/(a^5*g^6 + b^5*g^6*x^5 + 5*a*b^4*g^6*x^4 + 10*a^3*b^2*g^6
*x^2 + 10*a^2*b^3*g^6*x^3 + 5*a^4*b*g^6*x) - (B^2*d^5*i^2)/(30*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*
a^2*b*c*d^2))) - log(e*((a + b*x)/(c + d*x))^n)*((a*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2
+ 3*A*B*b*c*d*i^2) + x*(b*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 3*A*B*b*c*d*i^2) + 4*A
*B*a*b*d^2*i^2 + 12*A*B*b^2*c*d*i^2 - 2*B^2*a*b*d^2*i^2*n + 2*B^2*b^2*c*d*i^2*n) + 6*A*B*b^2*c^2*i^2 - B^2*a^2
*d^2*i^2*n + B^2*b^2*c^2*i^2*n + 10*A*B*b^2*d^2*i^2*x^2)/(15*a^5*b^3*g^6 + 15*b^8*g^6*x^5 + 75*a^4*b^4*g^6*x +
75*a*b^7*g^6*x^4 + 150*a^3*b^5*g^6*x^2 + 150*a^2*b^6*g^6*x^3) + (B^2*d^5*i^2*(x^3*(b*(b*(b*((3*a*b^3*g^6*n*(a
*d - b*c))/d + (3*b^3*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (6*a*b^4*g^6*n*(a*d - b*c))/d + (3*b^4*g^6*n
*(a*d - b*c)*(5*a*d - b*c))/(2*d^2)) + (9*a*b^5*g^6*n*(a*d - b*c))/d + (9*b^5*g^6*n*(a*d - b*c)*(5*a*d - b*c))
/(4*d^2)) + (12*a*b^6*g^6*n*(a*d - b*c))/d + (3*b^6*g^6*n*(a*d - b*c)*(5*a*d - b*c))/d^2) + x*(a*(a*(b*((3*a*b
^3*g^6*n*(a*d - b*c))/d + (3*b^3*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (6*a*b^4*g^6*n*(a*d - b*c))/d + (
3*b^4*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(2*d^2)) + b*(a*((3*a*b^3*g^6*n*(a*d - b*c))/d + (3*b^3*g^6*n*(a*d - b*
c)*(5*a*d - b*c))/(4*d^2)) + (b^3*g^6*n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(2*d^3)) + (3*b^4*g^6*
n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(2*d^3)) + b*(a*(a*((3*a*b^3*g^6*n*(a*d - b*c))/d + (3*b^3*g
^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (b^3*g^6*n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(2*d^3))
+ (3*b^3*g^6*n*(a*d - b*c)*(10*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2))/(4*d^4)) + (3*b^4*g^6*n*(
a*d - b*c)*(10*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2))/d^4) + x^2*(a*(b*(b*((3*a*b^3*g^6*n*(a*d -
b*c))/d + (3*b^3*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (6*a*b^4*g^6*n*(a*d - b*c))/d + (3*b^4*g^6*n*(a*
d - b*c)*(5*a*d - b*c))/(2*d^2)) + (9*a*b^5*g^6*n*(a*d - b*c))/d + (9*b^5*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*
d^2)) + b*(a*(b*((3*a*b^3*g^6*n*(a*d - b*c))/d + (3*b^3*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (6*a*b^4*g
^6*n*(a*d - b*c))/d + (3*b^4*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(2*d^2)) + b*(a*((3*a*b^3*g^6*n*(a*d - b*c))/d +
(3*b^3*g^6*n*(a*d - b*c)*(5*a*d - b*c))/(4*d^2)) + (b^3*g^6*n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))
/(2*d^3)) + (3*b^4*g^6*n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(2*d^3)) + (3*b^5*g^6*n*(a*d - b*c)*(
10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/d^3) + a*(a*(a*((3*a*b^3*g^6*n*(a*d - b*c))/d + (3*b^3*g^6*n*(a*d - b*c)*(5
*a*d - b*c))/(4*d^2)) + (b^3*g^6*n*(a*d - b*c)*(10*a^2*d^2 + b^2*c^2 - 5*a*b*c*d))/(2*d^3)) + (3*b^3*g^6*n*(a*
d - b*c)*(10*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2))/(4*d^4)) + (15*b^7*g^6*n*x^4*(a*d - b*c))/d
+ (3*b^3*g^6*n*(a*d - b*c)*(5*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3))/d^5))/
(15*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(15*a^5*b^3*g^6 + 15*b^8*g^6*x^5 + 75*a^4*b^4*
g^6*x + 75*a*b^7*g^6*x^4 + 150*a^3*b^5*g^6*x^2 + 150*a^2*b^6*g^6*x^3))) - (B*d^5*i^2*n*atan((B*d^5*i^2*n*(60*A
+ 47*B*n)*((b^6*c^3*g^6 + a^3*b^3*d^3*g^6 - a*...

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