\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^7} \, dx\) [83]

Optimal result

Integrand size = 42, antiderivative size = 463 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=-\frac {B^2 d^2 i^3 (c+d x)^4}{32 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B^2 d i^3 (c+d x)^5}{125 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B^2 i^3 (c+d x)^6}{108 (b c-a d)^3 g^7 (a+b x)^6}-\frac {B d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{18 (b c-a d)^3 g^7 (a+b x)^6}-\frac {d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 (b c-a d)^3 g^7 (a+b x)^6} \]

[Out]

-1/32*B^2*d^2*i^3*(d*x+c)^4/(-a*d+b*c)^3/g^7/(b*x+a)^4+4/125*b*B^2*d*i^3*(d*x+c)^5/(-a*d+b*c)^3/g^7/(b*x+a)^5-
1/108*b^2*B^2*i^3*(d*x+c)^6/(-a*d+b*c)^3/g^7/(b*x+a)^6-1/8*B*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a
*d+b*c)^3/g^7/(b*x+a)^4+4/25*b*B*d*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/18*b
^2*B*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^7/(b*x+a)^6-1/4*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x
+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^7/(b*x+a)^4+2/5*b*d*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^
7/(b*x+a)^5-1/6*b^2*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^7/(b*x+a)^6

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2562, 2395, 2342, 2341} \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=-\frac {b^2 i^3 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{6 g^7 (a+b x)^6 (b c-a d)^3}-\frac {b^2 B i^3 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{18 g^7 (a+b x)^6 (b c-a d)^3}-\frac {d^2 i^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^7 (a+b x)^4 (b c-a d)^3}-\frac {B d^2 i^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 g^7 (a+b x)^4 (b c-a d)^3}+\frac {2 b d i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 g^7 (a+b x)^5 (b c-a d)^3}+\frac {4 b B d i^3 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{25 g^7 (a+b x)^5 (b c-a d)^3}-\frac {b^2 B^2 i^3 (c+d x)^6}{108 g^7 (a+b x)^6 (b c-a d)^3}-\frac {B^2 d^2 i^3 (c+d x)^4}{32 g^7 (a+b x)^4 (b c-a d)^3}+\frac {4 b B^2 d i^3 (c+d x)^5}{125 g^7 (a+b x)^5 (b c-a d)^3} \]

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^7,x]

[Out]

-1/32*(B^2*d^2*i^3*(c + d*x)^4)/((b*c - a*d)^3*g^7*(a + b*x)^4) + (4*b*B^2*d*i^3*(c + d*x)^5)/(125*(b*c - a*d)
^3*g^7*(a + b*x)^5) - (b^2*B^2*i^3*(c + d*x)^6)/(108*(b*c - a*d)^3*g^7*(a + b*x)^6) - (B*d^2*i^3*(c + d*x)^4*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(b*c - a*d)^3*g^7*(a + b*x)^4) + (4*b*B*d*i^3*(c + d*x)^5*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(25*(b*c - a*d)^3*g^7*(a + b*x)^5) - (b^2*B*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))
/(c + d*x)]))/(18*(b*c - a*d)^3*g^7*(a + b*x)^6) - (d^2*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2
)/(4*(b*c - a*d)^3*g^7*(a + b*x)^4) + (2*b*d*i^3*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(b*c -
a*d)^3*g^7*(a + b*x)^5) - (b^2*i^3*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*(b*c - a*d)^3*g^7*(
a + b*x)^6)

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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && 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*} \text {integral}& = \frac {i^3 \text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))^2}{x^7} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^7} \\ & = \frac {i^3 \text {Subst}\left (\int \left (\frac {b^2 (A+B \log (e x))^2}{x^7}-\frac {2 b d (A+B \log (e x))^2}{x^6}+\frac {d^2 (A+B \log (e x))^2}{x^5}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^7} \\ & = \frac {\left (b^2 i^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^7} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^7}-\frac {\left (2 b d i^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^7}+\frac {\left (d^2 i^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^7} \\ & = -\frac {d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 (b c-a d)^3 g^7 (a+b x)^6}+\frac {\left (b^2 B i^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^7} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^3 g^7}-\frac {\left (4 b B d i^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{5 (b c-a d)^3 g^7}+\frac {\left (B d^2 i^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g^7} \\ & = -\frac {B^2 d^2 i^3 (c+d x)^4}{32 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B^2 d i^3 (c+d x)^5}{125 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B^2 i^3 (c+d x)^6}{108 (b c-a d)^3 g^7 (a+b x)^6}-\frac {B d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{18 (b c-a d)^3 g^7 (a+b x)^6}-\frac {d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 (b c-a d)^3 g^7 (a+b x)^6} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.69 (sec) , antiderivative size = 2606, normalized size of antiderivative = 5.63 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Result too large to show} \]

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^7,x]

[Out]

(i^3*(-6000*A*B*(b*c - a*d)^6 - 1000*B^2*(b*c - a*d)^6 + 25920*a*A*B*d*(-(b*c) + a*d)^5 + 5184*a*B^2*d*(-(b*c)
+ a*d)^5 - 25920*A*b*B*d*(b*c - a*d)^5*x - 5184*b*B^2*d*(b*c - a*d)^5*x + 32400*a*A*B*d^2*(b*c - a*d)^4*(a +
b*x) + 14580*a*B^2*d^2*(b*c - a*d)^4*(a + b*x) + 7200*A*B*d*(b*c - a*d)^5*(a + b*x) + 2640*B^2*d*(b*c - a*d)^5
*(a + b*x) + 32400*A*b*B*d^2*(b*c - a*d)^4*x*(a + b*x) + 14580*b*B^2*d^2*(b*c - a*d)^4*x*(a + b*x) - 49500*A*B
*d^2*(b*c - a*d)^4*(a + b*x)^2 - 15675*B^2*d^2*(b*c - a*d)^4*(a + b*x)^2 + 43200*a*A*B*d^3*(-(b*c) + a*d)^3*(a
+ b*x)^2 + 33840*a*B^2*d^3*(-(b*c) + a*d)^3*(a + b*x)^2 - 43200*A*b*B*d^3*(b*c - a*d)^3*x*(a + b*x)^2 - 33840
*b*B^2*d^3*(b*c - a*d)^3*x*(a + b*x)^2 + 64800*a*A*B*d^4*(b*c - a*d)^2*(a + b*x)^3 + 83160*a*B^2*d^4*(b*c - a*
d)^2*(a + b*x)^3 + 42000*A*B*d^3*(b*c - a*d)^3*(a + b*x)^3 + 34900*B^2*d^3*(b*c - a*d)^3*(a + b*x)^3 + 64800*A
*b*B*d^4*(b*c - a*d)^2*x*(a + b*x)^3 + 83160*b*B^2*d^4*(b*c - a*d)^2*x*(a + b*x)^3 - 63000*A*B*d^4*(b*c - a*d)
^2*(a + b*x)^4 - 83850*B^2*d^4*(b*c - a*d)^2*(a + b*x)^4 + 129600*a*A*B*d^5*(-(b*c) + a*d)*(a + b*x)^4 + 29592
0*a*B^2*d^5*(-(b*c) + a*d)*(a + b*x)^4 - 129600*A*b*B*d^5*(b*c - a*d)*x*(a + b*x)^4 - 295920*b*B^2*d^5*(b*c -
a*d)*x*(a + b*x)^4 + 126000*A*B*d^5*(b*c - a*d)*(a + b*x)^5 + 293700*B^2*d^5*(b*c - a*d)*(a + b*x)^5 - 129600*
a*A*B*d^6*(a + b*x)^5*Log[a + b*x] - 295920*a*B^2*d^6*(a + b*x)^5*Log[a + b*x] - 129600*A*b*B*d^6*x*(a + b*x)^
5*Log[a + b*x] - 295920*b*B^2*d^6*x*(a + b*x)^5*Log[a + b*x] + 126000*A*B*d^6*(a + b*x)^6*Log[a + b*x] + 29370
0*B^2*d^6*(a + b*x)^6*Log[a + b*x] + 64800*a*B^2*d^6*(a + b*x)^5*Log[a + b*x]^2 + 64800*b*B^2*d^6*x*(a + b*x)^
5*Log[a + b*x]^2 - 63000*B^2*d^6*(a + b*x)^6*Log[a + b*x]^2 - 6000*B^2*(b*c - a*d)^6*Log[(e*(a + b*x))/(c + d*
x)] + 25920*a*B^2*d*(-(b*c) + a*d)^5*Log[(e*(a + b*x))/(c + d*x)] - 25920*b*B^2*d*(b*c - a*d)^5*x*Log[(e*(a +
b*x))/(c + d*x)] + 32400*a*B^2*d^2*(b*c - a*d)^4*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 7200*B^2*d*(b*c - a*
d)^5*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 32400*b*B^2*d^2*(b*c - a*d)^4*x*(a + b*x)*Log[(e*(a + b*x))/(c +
d*x)] - 49500*B^2*d^2*(b*c - a*d)^4*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] + 43200*a*B^2*d^3*(-(b*c) + a*d)
^3*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] - 43200*b*B^2*d^3*(b*c - a*d)^3*x*(a + b*x)^2*Log[(e*(a + b*x))/(c
+ d*x)] + 64800*a*B^2*d^4*(b*c - a*d)^2*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)] + 42000*B^2*d^3*(b*c - a*d)^
3*(a + b*x)^3*Log[(e*(a + b*x))/(c + d*x)] + 64800*b*B^2*d^4*(b*c - a*d)^2*x*(a + b*x)^3*Log[(e*(a + b*x))/(c
+ d*x)] - 63000*B^2*d^4*(b*c - a*d)^2*(a + b*x)^4*Log[(e*(a + b*x))/(c + d*x)] + 129600*a*B^2*d^5*(-(b*c) + a*
d)*(a + b*x)^4*Log[(e*(a + b*x))/(c + d*x)] - 129600*b*B^2*d^5*(b*c - a*d)*x*(a + b*x)^4*Log[(e*(a + b*x))/(c
+ d*x)] + 126000*B^2*d^5*(b*c - a*d)*(a + b*x)^5*Log[(e*(a + b*x))/(c + d*x)] - 129600*a*B^2*d^6*(a + b*x)^5*L
og[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - 129600*b*B^2*d^6*x*(a + b*x)^5*Log[a + b*x]*Log[(e*(a + b*x))/(c +
d*x)] + 126000*B^2*d^6*(a + b*x)^6*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - 18000*(b*c - a*d)^6*(A + B*Log[
(e*(a + b*x))/(c + d*x)])^2 + 64800*d*(-(b*c) + a*d)^5*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 8100
0*d^2*(b*c - a*d)^4*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 36000*d^3*(-(b*c) + a*d)^3*(a + b*x)^
3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 129600*a*A*B*d^6*(a + b*x)^5*Log[c + d*x] + 295920*a*B^2*d^6*(a + b
*x)^5*Log[c + d*x] + 129600*A*b*B*d^6*x*(a + b*x)^5*Log[c + d*x] + 295920*b*B^2*d^6*x*(a + b*x)^5*Log[c + d*x]
- 126000*A*B*d^6*(a + b*x)^6*Log[c + d*x] - 293700*B^2*d^6*(a + b*x)^6*Log[c + d*x] - 129600*a*B^2*d^6*(a + b
*x)^5*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 129600*b*B^2*d^6*x*(a + b*x)^5*Log[(d*(a + b*x))/(-(b*c
) + a*d)]*Log[c + d*x] + 126000*B^2*d^6*(a + b*x)^6*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 129600*a*
B^2*d^6*(a + b*x)^5*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] + 129600*b*B^2*d^6*x*(a + b*x)^5*Log[(e*(a + b*x
))/(c + d*x)]*Log[c + d*x] - 126000*B^2*d^6*(a + b*x)^6*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] + 64800*a*B^
2*d^6*(a + b*x)^5*Log[c + d*x]^2 + 64800*b*B^2*d^6*x*(a + b*x)^5*Log[c + d*x]^2 - 63000*B^2*d^6*(a + b*x)^6*Lo
g[c + d*x]^2 - 129600*a*B^2*d^6*(a + b*x)^5*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 129600*b*B^2*d^6*x*(
a + b*x)^5*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 126000*B^2*d^6*(a + b*x)^6*Log[a + b*x]*Log[(b*(c + d
*x))/(b*c - a*d)] - 129600*a*B^2*d^6*(a + b*x)^5*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 129600*b*B^2*d^6*x
*(a + b*x)^5*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 126000*B^2*d^6*(a + b*x)^6*PolyLog[2, (d*(a + b*x))/(-
(b*c) + a*d)] - 129600*a*B^2*d^6*(a + b*x)^5*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 129600*b*B^2*d^6*x*(a + b
*x)^5*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 126000*B^2*d^6*(a + b*x)^6*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]
))/(108000*b^4*(b*c - a*d)^3*g^7*(a + b*x)^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1018\) vs. \(2(445)=890\).

Time = 2.52 (sec) , antiderivative size = 1019, normalized size of antiderivative = 2.20

method	result	size
parts	\(\text {Expression too large to display}\)	\(1019\)
derivativedivides	\(\text {Expression too large to display}\)	\(1082\)
default	\(\text {Expression too large to display}\)	\(1082\)
norman	\(\text {Expression too large to display}\)	\(2555\)
parallelrisch	\(\text {Expression too large to display}\)	\(2803\)
risch	\(\text {Expression too large to display}\)	\(6548\)

[In]

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x,method=_RETURNVERBOSE)

[Out]

i^3*A^2/g^7*(-3/5*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^4/(b*x+a)^5-1/3*d^3/b^4/(b*x+a)^3+3/4*d^2*(a*d-b*c)/b^4/(b*x
+a)^4-1/6*(-a^3*d^3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/b^4/(b*x+a)^6)-i^3*B^2/g^7/d^5*(a*d-b*c)^4*e^4*(d^7/(
a*d-b*c)^7*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/8/(b*e/d+(a*d-b*c)*e/d/(d
*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/32/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4)-2*d^6/(a*d-b*c)^7*b*e*(-1/5/(b*
e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))-2/125/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5)+d^5/(a*d-b*c)^7*e^2*b^2*(-1/6/(b*e/d+(a*d-b*c)*e/
d/(d*x+c))^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/18/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6*ln(b*e/d+(a*d-b*c)*e/d/(d*
x+c))-1/108/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6))-2*i^3*B*A/g^7/d^5*(a*d-b*c)^4*e^4*(d^7/(a*d-b*c)^7*(-1/4/(b*e/d+
(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4)-2*d^6/(a*d-b*c)
^7*b*e*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c)
)^5)+d^5/(a*d-b*c)^7*e^2*b^2*(-1/6/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/36/(b*e/d
+(a*d-b*c)*e/d/(d*x+c))^6))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1610 vs. \(2 (445) = 890\).

Time = 0.37 (sec) , antiderivative size = 1610, normalized size of antiderivative = 3.48 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, algorithm="fricas")

[Out]

-1/108000*(60*((60*A*B + 37*B^2)*b^6*c*d^5 - (60*A*B + 37*B^2)*a*b^5*d^6)*i^3*x^5 - 30*((60*A*B - 23*B^2)*b^6*
c^2*d^4 - 36*(20*A*B + 9*B^2)*a*b^5*c*d^5 + (660*A*B + 347*B^2)*a^2*b^4*d^6)*i^3*x^4 + 20*((1800*A^2 + 60*A*B
- 53*B^2)*b^6*c^3*d^3 - 27*(200*A^2 + 20*A*B - 11*B^2)*a*b^5*c^2*d^4 + 675*(8*A^2 + 4*A*B + B^2)*a^2*b^4*c*d^5
- (1800*A^2 + 2220*A*B + 919*B^2)*a^3*b^3*d^6)*i^3*x^3 + 15*((5400*A^2 + 1140*A*B + 73*B^2)*b^6*c^4*d^2 - 72*
(200*A^2 + 60*A*B + 7*B^2)*a*b^5*c^3*d^3 + 1350*(8*A^2 + 4*A*B + B^2)*a^2*b^4*c^2*d^4 - (1800*A^2 + 2220*A*B +
919*B^2)*a^4*b^2*d^6)*i^3*x^2 + 6*(8*(1350*A^2 + 390*A*B + 53*B^2)*b^6*c^5*d - 45*(600*A^2 + 220*A*B + 39*B^2
)*a*b^5*c^4*d^2 + 2250*(8*A^2 + 4*A*B + B^2)*a^2*b^4*c^3*d^3 - (1800*A^2 + 2220*A*B + 919*B^2)*a^5*b*d^6)*i^3*
x + (1000*(18*A^2 + 6*A*B + B^2)*b^6*c^6 - 1728*(25*A^2 + 10*A*B + 2*B^2)*a*b^5*c^5*d + 3375*(8*A^2 + 4*A*B +
B^2)*a^2*b^4*c^4*d^2 - (1800*A^2 + 2220*A*B + 919*B^2)*a^6*d^6)*i^3 + 1800*(B^2*b^6*d^6*i^3*x^6 + 6*B^2*a*b^5*
d^6*i^3*x^5 + 15*B^2*a^2*b^4*d^6*i^3*x^4 + 20*(B^2*b^6*c^3*d^3 - 3*B^2*a*b^5*c^2*d^4 + 3*B^2*a^2*b^4*c*d^5)*i^
3*x^3 + 15*(3*B^2*b^6*c^4*d^2 - 8*B^2*a*b^5*c^3*d^3 + 6*B^2*a^2*b^4*c^2*d^4)*i^3*x^2 + 6*(6*B^2*b^6*c^5*d - 15
*B^2*a*b^5*c^4*d^2 + 10*B^2*a^2*b^4*c^3*d^3)*i^3*x + (10*B^2*b^6*c^6 - 24*B^2*a*b^5*c^5*d + 15*B^2*a^2*b^4*c^4
*d^2)*i^3)*log((b*e*x + a*e)/(d*x + c))^2 + 60*((60*A*B + 37*B^2)*b^6*d^6*i^3*x^6 + 6*(10*B^2*b^6*c*d^5 + 3*(2
0*A*B + 9*B^2)*a*b^5*d^6)*i^3*x^5 - 15*(2*B^2*b^6*c^2*d^4 - 24*B^2*a*b^5*c*d^5 - 15*(4*A*B + B^2)*a^2*b^4*d^6)
*i^3*x^4 + 20*((60*A*B + B^2)*b^6*c^3*d^3 - 9*(20*A*B + B^2)*a*b^5*c^2*d^4 + 45*(4*A*B + B^2)*a^2*b^4*c*d^5)*i
^3*x^3 + 15*((180*A*B + 19*B^2)*b^6*c^4*d^2 - 24*(20*A*B + 3*B^2)*a*b^5*c^3*d^3 + 90*(4*A*B + B^2)*a^2*b^4*c^2
*d^4)*i^3*x^2 + 6*(4*(90*A*B + 13*B^2)*b^6*c^5*d - 15*(60*A*B + 11*B^2)*a*b^5*c^4*d^2 + 150*(4*A*B + B^2)*a^2*
b^4*c^3*d^3)*i^3*x + (100*(6*A*B + B^2)*b^6*c^6 - 288*(5*A*B + B^2)*a*b^5*c^5*d + 225*(4*A*B + B^2)*a^2*b^4*c^
4*d^2)*i^3)*log((b*e*x + a*e)/(d*x + c)))/((b^13*c^3 - 3*a*b^12*c^2*d + 3*a^2*b^11*c*d^2 - a^3*b^10*d^3)*g^7*x
^6 + 6*(a*b^12*c^3 - 3*a^2*b^11*c^2*d + 3*a^3*b^10*c*d^2 - a^4*b^9*d^3)*g^7*x^5 + 15*(a^2*b^11*c^3 - 3*a^3*b^1
0*c^2*d + 3*a^4*b^9*c*d^2 - a^5*b^8*d^3)*g^7*x^4 + 20*(a^3*b^10*c^3 - 3*a^4*b^9*c^2*d + 3*a^5*b^8*c*d^2 - a^6*
b^7*d^3)*g^7*x^3 + 15*(a^4*b^9*c^3 - 3*a^5*b^8*c^2*d + 3*a^6*b^7*c*d^2 - a^7*b^6*d^3)*g^7*x^2 + 6*(a^5*b^8*c^3
- 3*a^6*b^7*c^2*d + 3*a^7*b^6*c*d^2 - a^8*b^5*d^3)*g^7*x + (a^6*b^7*c^3 - 3*a^7*b^6*c^2*d + 3*a^8*b^5*c*d^2 -
a^9*b^4*d^3)*g^7)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Timed out} \]

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**7,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20330 vs. \(2 (445) = 890\).

Time = 2.08 (sec) , antiderivative size = 20330, normalized size of antiderivative = 43.91 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, algorithm="maxima")

[Out]

-1/10*(6*b*x + a)*B^2*c^2*d*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2
*b^6*g^7*x^4 + 20*a^3*b^5*g^7*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) - 1/20*(15*b^2*x^2 + 6
*a*b*x + a^2)*B^2*c*d^2*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7
*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x + a^6*b^3*g^7) - 1/60*(20*b^3*x^3 + 15*a*
b^2*x^2 + 6*a^2*b*x + a^3)*B^2*d^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5
+ 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7) + 1/10800*(60*
((60*b^5*d^5*x^5 - 10*b^5*c^5 + 62*a*b^4*c^4*d - 163*a^2*b^3*c^3*d^2 + 237*a^3*b^2*c^2*d^3 - 213*a^4*b*c*d^4 +
147*a^5*d^5 - 30*(b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 8*a*b^4*c*d^4 + 37*a^2*b^3*d^5)*x^3 - 15*
(b^5*c^3*d^2 - 7*a*b^4*c^2*d^3 + 23*a^2*b^3*c*d^4 - 57*a^3*b^2*d^5)*x^2 + 6*(2*b^5*c^4*d - 13*a*b^4*c^3*d^2 +
37*a^2*b^3*c^2*d^3 - 63*a^3*b^2*c*d^4 + 87*a^4*b*d^5)*x)/((b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 1
0*a^3*b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*g^7*x^6 + 6*(a*b^11*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3
*d^2 - 10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*d^5)*g^7*x^5 + 15*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a
^4*b^8*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*g^7*x^4 + 20*(a^3*b^9*c^5 - 5*a^4*b^8*c^4
*d + 10*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4 - a^8*b^4*d^5)*g^7*x^3 + 15*(a^4*b^8*c^5 - 5*a^
5*b^7*c^4*d + 10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c*d^4 - a^9*b^3*d^5)*g^7*x^2 + 6*(a^5*b^7*c^
5 - 5*a^6*b^6*c^4*d + 10*a^7*b^5*c^3*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9*b^3*c*d^4 - a^10*b^2*d^5)*g^7*x + (a^6*b
^6*c^5 - 5*a^7*b^5*c^4*d + 10*a^8*b^4*c^3*d^2 - 10*a^9*b^3*c^2*d^3 + 5*a^10*b^2*c*d^4 - a^11*b*d^5)*g^7) + 60*
d^6*log(b*x + a)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*
a^5*b^2*c*d^5 + a^6*b*d^6)*g^7) - 60*d^6*log(d*x + c)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*
b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*g^7))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (
100*b^6*c^6 - 864*a*b^5*c^5*d + 3375*a^2*b^4*c^4*d^2 - 8000*a^3*b^3*c^3*d^3 + 13500*a^4*b^2*c^2*d^4 - 21600*a^
5*b*c*d^5 + 13489*a^6*d^6 - 8820*(b^6*c*d^5 - a*b^5*d^6)*x^5 + 90*(29*b^6*c^2*d^4 - 548*a*b^5*c*d^5 + 519*a^2*
b^4*d^6)*x^4 - 60*(19*b^6*c^3*d^3 - 231*a*b^5*c^2*d^4 + 1875*a^2*b^4*c*d^5 - 1663*a^3*b^3*d^6)*x^3 + 15*(37*b^
6*c^4*d^2 - 376*a*b^5*c^3*d^3 + 1950*a^2*b^4*c^2*d^4 - 8800*a^3*b^3*c*d^5 + 7189*a^4*b^2*d^6)*x^2 + 1800*(b^6*
d^6*x^6 + 6*a*b^5*d^6*x^5 + 15*a^2*b^4*d^6*x^4 + 20*a^3*b^3*d^6*x^3 + 15*a^4*b^2*d^6*x^2 + 6*a^5*b*d^6*x + a^6
*d^6)*log(b*x + a)^2 + 1800*(b^6*d^6*x^6 + 6*a*b^5*d^6*x^5 + 15*a^2*b^4*d^6*x^4 + 20*a^3*b^3*d^6*x^3 + 15*a^4*
b^2*d^6*x^2 + 6*a^5*b*d^6*x + a^6*d^6)*log(d*x + c)^2 - 6*(44*b^6*c^5*d - 405*a*b^5*c^4*d^2 + 1750*a^2*b^4*c^3
*d^3 - 5000*a^3*b^3*c^2*d^4 + 13500*a^4*b^2*c*d^5 - 9889*a^5*b*d^6)*x - 8820*(b^6*d^6*x^6 + 6*a*b^5*d^6*x^5 +
15*a^2*b^4*d^6*x^4 + 20*a^3*b^3*d^6*x^3 + 15*a^4*b^2*d^6*x^2 + 6*a^5*b*d^6*x + a^6*d^6)*log(b*x + a) + 180*(49
*b^6*d^6*x^6 + 294*a*b^5*d^6*x^5 + 735*a^2*b^4*d^6*x^4 + 980*a^3*b^3*d^6*x^3 + 735*a^4*b^2*d^6*x^2 + 294*a^5*b
*d^6*x + 49*a^6*d^6 - 20*(b^6*d^6*x^6 + 6*a*b^5*d^6*x^5 + 15*a^2*b^4*d^6*x^4 + 20*a^3*b^3*d^6*x^3 + 15*a^4*b^2
*d^6*x^2 + 6*a^5*b*d^6*x + a^6*d^6)*log(b*x + a))*log(d*x + c))/(a^6*b^7*c^6*g^7 - 6*a^7*b^6*c^5*d*g^7 + 15*a^
8*b^5*c^4*d^2*g^7 - 20*a^9*b^4*c^3*d^3*g^7 + 15*a^10*b^3*c^2*d^4*g^7 - 6*a^11*b^2*c*d^5*g^7 + a^12*b*d^6*g^7 +
(b^13*c^6*g^7 - 6*a*b^12*c^5*d*g^7 + 15*a^2*b^11*c^4*d^2*g^7 - 20*a^3*b^10*c^3*d^3*g^7 + 15*a^4*b^9*c^2*d^4*g
^7 - 6*a^5*b^8*c*d^5*g^7 + a^6*b^7*d^6*g^7)*x^6 + 6*(a*b^12*c^6*g^7 - 6*a^2*b^11*c^5*d*g^7 + 15*a^3*b^10*c^4*d
^2*g^7 - 20*a^4*b^9*c^3*d^3*g^7 + 15*a^5*b^8*c^2*d^4*g^7 - 6*a^6*b^7*c*d^5*g^7 + a^7*b^6*d^6*g^7)*x^5 + 15*(a^
2*b^11*c^6*g^7 - 6*a^3*b^10*c^5*d*g^7 + 15*a^4*b^9*c^4*d^2*g^7 - 20*a^5*b^8*c^3*d^3*g^7 + 15*a^6*b^7*c^2*d^4*g
^7 - 6*a^7*b^6*c*d^5*g^7 + a^8*b^5*d^6*g^7)*x^4 + 20*(a^3*b^10*c^6*g^7 - 6*a^4*b^9*c^5*d*g^7 + 15*a^5*b^8*c^4*
d^2*g^7 - 20*a^6*b^7*c^3*d^3*g^7 + 15*a^7*b^6*c^2*d^4*g^7 - 6*a^8*b^5*c*d^5*g^7 + a^9*b^4*d^6*g^7)*x^3 + 15*(a
^4*b^9*c^6*g^7 - 6*a^5*b^8*c^5*d*g^7 + 15*a^6*b^7*c^4*d^2*g^7 - 20*a^7*b^6*c^3*d^3*g^7 + 15*a^8*b^5*c^2*d^4*g^
7 - 6*a^9*b^4*c*d^5*g^7 + a^10*b^3*d^6*g^7)*x^2 + 6*(a^5*b^8*c^6*g^7 - 6*a^6*b^7*c^5*d*g^7 + 15*a^7*b^6*c^4*d^
2*g^7 - 20*a^8*b^5*c^3*d^3*g^7 + 15*a^9*b^4*c^2*d^4*g^7 - 6*a^10*b^3*c*d^5*g^7 + a^11*b^2*d^6*g^7)*x))*B^2*c^3
*i^3 - 1/18000*(60*((22*a*b^5*c^5 - 140*a^2*b^4*c^4*d + 385*a^3*b^3*c^3*d^2 - 615*a^4*b^2*c^2*d^3 + 735*a^5*b*
c*d^4 - 87*a^6*d^5 + 60*(6*b^6*c*d^4 - a*b^5*d^5)*x^5 - 30*(6*b^6*c^2*d^3 - 67*a*b^5*c*d^4 + 11*a^2*b^4*d^5)*x
^4 + 20*(6*b^6*c^3*d^2 - 49*a*b^5*c^2*d^3 + 230*a^2*b^4*c*d^4 - 37*a^3*b^3*d^5)*x^3 - 15*(6*b^6*c^4*d - 43*a*b
^5*c^3*d^2 + 145*a^2*b^4*c^2*d^3 - 365*a^3*b^3*c*d^4 + 57*a^4*b^2*d^5)*x^2 + 6*(12*b^6*c^5 - 80*a*b^5*c^4*d +
235*a^2*b^4*c^3*d^2 - 415*a^3*b^3*c^2*d^3 + 585*a^4*b^2*c*d^4 - 87*a^5*b*d^5)*x)/((b^13*c^5 - 5*a*b^12*c^4*d +
10*a^2*b^11*c^3*d^2 - 10*a^3*b^10*c^2*d^3 + 5*a^4*b^9*c*d^4 - a^5*b^8*d^5)*g^7*x^6 + 6*(a*b^12*c^5 - 5*a^2*b^
11*c^4*d + 10*a^3*b^10*c^3*d^2 - 10*a^4*b^9*c^2*d^3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*g^7*x^5 + 15*(a^2*b^11*c^
5 - 5*a^3*b^10*c^4*d + 10*a^4*b^9*c^3*d^2 - 10*a^5*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*g^7*x^4 + 20*(
a^3*b^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b^8*c^3*d^2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d^5)*g^7*
x^3 + 15*(a^4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b^7*c^3*d^2 - 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*
d^5)*g^7*x^2 + 6*(a^5*b^8*c^5 - 5*a^6*b^7*c^4*d + 10*a^7*b^6*c^3*d^2 - 10*a^8*b^5*c^2*d^3 + 5*a^9*b^4*c*d^4 -
a^10*b^3*d^5)*g^7*x + (a^6*b^7*c^5 - 5*a^7*b^6*c^4*d + 10*a^8*b^5*c^3*d^2 - 10*a^9*b^4*c^2*d^3 + 5*a^10*b^3*c*
d^4 - a^11*b^2*d^5)*g^7) + 60*(6*b*c*d^5 - a*d^6)*log(b*x + a)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2
- 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*g^7) - 60*(6*b*c*d^5 - a*d^6)*log(d
*x + c)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c
*d^5 + a^6*b^2*d^6)*g^7))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (364*a*b^6*c^6 - 3294*a^2*b^5*c^5*d + 13875*a
^3*b^4*c^4*d^2 - 38000*a^4*b^3*c^3*d^3 + 94500*a^5*b^2*c^2*d^4 - 72114*a^6*b*c*d^5 + 4669*a^7*d^6 + 180*(274*b
^7*c^2*d^4 - 303*a*b^6*c*d^5 + 29*a^2*b^5*d^6)*x^5 - 90*(154*b^7*c^3*d^3 - 3057*a*b^6*c^2*d^4 + 3202*a^2*b^5*c
*d^5 - 299*a^3*b^4*d^6)*x^4 + 60*(94*b^7*c^4*d^2 - 1205*a*b^6*c^3*d^3 + 10401*a^2*b^5*c^2*d^4 - 10213*a^3*b^4*
c*d^5 + 923*a^4*b^3*d^6)*x^3 - 15*(162*b^7*c^5*d - 1753*a*b^6*c^4*d^2 + 9796*a^2*b^5*c^3*d^3 - 48150*a^3*b^4*c
^2*d^4 + 43714*a^4*b^3*c*d^5 - 3769*a^5*b^2*d^6)*x^2 - 1800*(6*a^6*b*c*d^5 - a^7*d^6 + (6*b^7*c*d^5 - a*b^6*d^
6)*x^6 + 6*(6*a*b^6*c*d^5 - a^2*b^5*d^6)*x^5 + 15*(6*a^2*b^5*c*d^5 - a^3*b^4*d^6)*x^4 + 20*(6*a^3*b^4*c*d^5 -
a^4*b^3*d^6)*x^3 + 15*(6*a^4*b^3*c*d^5 - a^5*b^2*d^6)*x^2 + 6*(6*a^5*b^2*c*d^5 - a^6*b*d^6)*x)*log(b*x + a)^2
- 1800*(6*a^6*b*c*d^5 - a^7*d^6 + (6*b^7*c*d^5 - a*b^6*d^6)*x^6 + 6*(6*a*b^6*c*d^5 - a^2*b^5*d^6)*x^5 + 15*(6*
a^2*b^5*c*d^5 - a^3*b^4*d^6)*x^4 + 20*(6*a^3*b^4*c*d^5 - a^4*b^3*d^6)*x^3 + 15*(6*a^4*b^3*c*d^5 - a^5*b^2*d^6)
*x^2 + 6*(6*a^5*b^2*c*d^5 - a^6*b*d^6)*x)*log(d*x + c)^2 + 6*(144*b^7*c^6 - 1454*a*b^6*c^5*d + 7005*a^2*b^5*c^
4*d^2 - 22750*a^3*b^4*c^3*d^3 + 71000*a^4*b^3*c^2*d^4 - 58614*a^5*b^2*c*d^5 + 4669*a^6*b*d^6)*x + 180*(274*a^6
*b*c*d^5 - 29*a^7*d^6 + (274*b^7*c*d^5 - 29*a*b^6*d^6)*x^6 + 6*(274*a*b^6*c*d^5 - 29*a^2*b^5*d^6)*x^5 + 15*(27
4*a^2*b^5*c*d^5 - 29*a^3*b^4*d^6)*x^4 + 20*(274*a^3*b^4*c*d^5 - 29*a^4*b^3*d^6)*x^3 + 15*(274*a^4*b^3*c*d^5 -
29*a^5*b^2*d^6)*x^2 + 6*(274*a^5*b^2*c*d^5 - 29*a^6*b*d^6)*x)*log(b*x + a) - 180*(274*a^6*b*c*d^5 - 29*a^7*d^6
+ (274*b^7*c*d^5 - 29*a*b^6*d^6)*x^6 + 6*(274*a*b^6*c*d^5 - 29*a^2*b^5*d^6)*x^5 + 15*(274*a^2*b^5*c*d^5 - 29*
a^3*b^4*d^6)*x^4 + 20*(274*a^3*b^4*c*d^5 - 29*a^4*b^3*d^6)*x^3 + 15*(274*a^4*b^3*c*d^5 - 29*a^5*b^2*d^6)*x^2 +
6*(274*a^5*b^2*c*d^5 - 29*a^6*b*d^6)*x - 20*(6*a^6*b*c*d^5 - a^7*d^6 + (6*b^7*c*d^5 - a*b^6*d^6)*x^6 + 6*(6*a
*b^6*c*d^5 - a^2*b^5*d^6)*x^5 + 15*(6*a^2*b^5*c*d^5 - a^3*b^4*d^6)*x^4 + 20*(6*a^3*b^4*c*d^5 - a^4*b^3*d^6)*x^
3 + 15*(6*a^4*b^3*c*d^5 - a^5*b^2*d^6)*x^2 + 6*(6*a^5*b^2*c*d^5 - a^6*b*d^6)*x)*log(b*x + a))*log(d*x + c))/(a
^6*b^8*c^6*g^7 - 6*a^7*b^7*c^5*d*g^7 + 15*a^8*b^6*c^4*d^2*g^7 - 20*a^9*b^5*c^3*d^3*g^7 + 15*a^10*b^4*c^2*d^4*g
^7 - 6*a^11*b^3*c*d^5*g^7 + a^12*b^2*d^6*g^7 + (b^14*c^6*g^7 - 6*a*b^13*c^5*d*g^7 + 15*a^2*b^12*c^4*d^2*g^7 -
20*a^3*b^11*c^3*d^3*g^7 + 15*a^4*b^10*c^2*d^4*g^7 - 6*a^5*b^9*c*d^5*g^7 + a^6*b^8*d^6*g^7)*x^6 + 6*(a*b^13*c^6
*g^7 - 6*a^2*b^12*c^5*d*g^7 + 15*a^3*b^11*c^4*d^2*g^7 - 20*a^4*b^10*c^3*d^3*g^7 + 15*a^5*b^9*c^2*d^4*g^7 - 6*a
^6*b^8*c*d^5*g^7 + a^7*b^7*d^6*g^7)*x^5 + 15*(a^2*b^12*c^6*g^7 - 6*a^3*b^11*c^5*d*g^7 + 15*a^4*b^10*c^4*d^2*g^
7 - 20*a^5*b^9*c^3*d^3*g^7 + 15*a^6*b^8*c^2*d^4*g^7 - 6*a^7*b^7*c*d^5*g^7 + a^8*b^6*d^6*g^7)*x^4 + 20*(a^3*b^1
1*c^6*g^7 - 6*a^4*b^10*c^5*d*g^7 + 15*a^5*b^9*c^4*d^2*g^7 - 20*a^6*b^8*c^3*d^3*g^7 + 15*a^7*b^7*c^2*d^4*g^7 -
6*a^8*b^6*c*d^5*g^7 + a^9*b^5*d^6*g^7)*x^3 + 15*(a^4*b^10*c^6*g^7 - 6*a^5*b^9*c^5*d*g^7 + 15*a^6*b^8*c^4*d^2*g
^7 - 20*a^7*b^7*c^3*d^3*g^7 + 15*a^8*b^6*c^2*d^4*g^7 - 6*a^9*b^5*c*d^5*g^7 + a^10*b^4*d^6*g^7)*x^2 + 6*(a^5*b^
9*c^6*g^7 - 6*a^6*b^8*c^5*d*g^7 + 15*a^7*b^7*c^4*d^2*g^7 - 20*a^8*b^6*c^3*d^3*g^7 + 15*a^9*b^5*c^2*d^4*g^7 - 6
*a^10*b^4*c*d^5*g^7 + a^11*b^3*d^6*g^7)*x))*B^2*c^2*d*i^3 - 1/36000*(60*((37*a^2*b^5*c^5 - 245*a^3*b^4*c^4*d +
730*a^4*b^3*c^3*d^2 - 1470*a^5*b^2*c^2*d^3 + 405*a^6*b*c*d^4 - 57*a^7*d^5 - 60*(15*b^7*c^2*d^3 - 6*a*b^6*c*d^
4 + a^2*b^5*d^5)*x^5 + 30*(15*b^7*c^3*d^2 - 171*a*b^6*c^2*d^3 + 67*a^2*b^5*c*d^4 - 11*a^3*b^4*d^5)*x^4 - 20*(1
5*b^7*c^4*d - 126*a*b^6*c^3*d^2 + 604*a^2*b^5*c^2*d^3 - 230*a^3*b^4*c*d^4 + 37*a^4*b^3*d^5)*x^3 + 15*(15*b^7*c
^5 - 111*a*b^6*c^4*d + 388*a^2*b^5*c^3*d^2 - 1000*a^3*b^4*c^2*d^3 + 365*a^4*b^3*c*d^4 - 57*a^5*b^2*d^5)*x^2 +
6*(27*a*b^6*c^5 - 185*a^2*b^5*c^4*d + 580*a^3*b^4*c^3*d^2 - 1270*a^4*b^3*c^2*d^3 + 405*a^5*b^2*c*d^4 - 57*a^6*
b*d^5)*x)/((b^14*c^5 - 5*a*b^13*c^4*d + 10*a^2*b^12*c^3*d^2 - 10*a^3*b^11*c^2*d^3 + 5*a^4*b^10*c*d^4 - a^5*b^9
*d^5)*g^7*x^6 + 6*(a*b^13*c^5 - 5*a^2*b^12*c^4*d + 10*a^3*b^11*c^3*d^2 - 10*a^4*b^10*c^2*d^3 + 5*a^5*b^9*c*d^4
- a^6*b^8*d^5)*g^7*x^5 + 15*(a^2*b^12*c^5 - 5*a^3*b^11*c^4*d + 10*a^4*b^10*c^3*d^2 - 10*a^5*b^9*c^2*d^3 + 5*a
^6*b^8*c*d^4 - a^7*b^7*d^5)*g^7*x^4 + 20*(a^3*b^11*c^5 - 5*a^4*b^10*c^4*d + 10*a^5*b^9*c^3*d^2 - 10*a^6*b^8*c^
2*d^3 + 5*a^7*b^7*c*d^4 - a^8*b^6*d^5)*g^7*x^3 + 15*(a^4*b^10*c^5 - 5*a^5*b^9*c^4*d + 10*a^6*b^8*c^3*d^2 - 10*
a^7*b^7*c^2*d^3 + 5*a^8*b^6*c*d^4 - a^9*b^5*d^5)*g^7*x^2 + 6*(a^5*b^9*c^5 - 5*a^6*b^8*c^4*d + 10*a^7*b^7*c^3*d
^2 - 10*a^8*b^6*c^2*d^3 + 5*a^9*b^5*c*d^4 - a^10*b^4*d^5)*g^7*x + (a^6*b^8*c^5 - 5*a^7*b^7*c^4*d + 10*a^8*b^6*
c^3*d^2 - 10*a^9*b^5*c^2*d^3 + 5*a^10*b^4*c*d^4 - a^11*b^3*d^5)*g^7) - 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*
d^6)*log(b*x + a)/((b^9*c^6 - 6*a*b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6
*a^5*b^4*c*d^5 + a^6*b^3*d^6)*g^7) + 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(d*x + c)/((b^9*c^6 - 6*a*
b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5 + a^6*b^3*d^6)*g^7)
)*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (919*a^2*b^6*c^6 - 8934*a^3*b^5*c^5*d + 43125*a^4*b^4*c^4*d^2 - 17000
0*a^5*b^3*c^3*d^3 + 155625*a^6*b^2*c^2*d^4 - 22794*a^7*b*c*d^5 + 2059*a^8*d^6 - 180*(625*b^8*c^3*d^3 - 779*a*b
^7*c^2*d^4 + 173*a^2*b^6*c*d^5 - 19*a^3*b^5*d^6)*x^5 + 90*(325*b^8*c^4*d^2 - 6934*a*b^7*c^3*d^3 + 8182*a^2*b^6
*c^2*d^4 - 1762*a^3*b^5*c*d^5 + 189*a^4*b^4*d^6)*x^4 - 60*(175*b^8*c^5*d - 2449*a*b^7*c^4*d^2 + 23290*a^2*b^6*
c^3*d^3 - 25786*a^3*b^5*c^2*d^4 + 5323*a^4*b^4*c*d^5 - 553*a^5*b^3*d^6)*x^3 + 15*(225*b^8*c^6 - 2802*a*b^7*c^5
*d + 18193*a^2*b^6*c^4*d^2 - 104356*a^3*b^5*c^3*d^3 + 107475*a^4*b^4*c^2*d^4 - 20794*a^5*b^3*c*d^5 + 2059*a^6*
b^2*d^6)*x^2 + 1800*(15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8*d^6 + (15*b^8*c^2*d^4 - 6*a*b^7*c*d^5 + a^2*b^6*
d^6)*x^6 + 6*(15*a*b^7*c^2*d^4 - 6*a^2*b^6*c*d^5 + a^3*b^5*d^6)*x^5 + 15*(15*a^2*b^6*c^2*d^4 - 6*a^3*b^5*c*d^5
+ a^4*b^4*d^6)*x^4 + 20*(15*a^3*b^5*c^2*d^4 - 6*a^4*b^4*c*d^5 + a^5*b^3*d^6)*x^3 + 15*(15*a^4*b^4*c^2*d^4 - 6
*a^5*b^3*c*d^5 + a^6*b^2*d^6)*x^2 + 6*(15*a^5*b^3*c^2*d^4 - 6*a^6*b^2*c*d^5 + a^7*b*d^6)*x)*log(b*x + a)^2 + 1
800*(15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8*d^6 + (15*b^8*c^2*d^4 - 6*a*b^7*c*d^5 + a^2*b^6*d^6)*x^6 + 6*(15
*a*b^7*c^2*d^4 - 6*a^2*b^6*c*d^5 + a^3*b^5*d^6)*x^5 + 15*(15*a^2*b^6*c^2*d^4 - 6*a^3*b^5*c*d^5 + a^4*b^4*d^6)*
x^4 + 20*(15*a^3*b^5*c^2*d^4 - 6*a^4*b^4*c*d^5 + a^5*b^3*d^6)*x^3 + 15*(15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 +
a^6*b^2*d^6)*x^2 + 6*(15*a^5*b^3*c^2*d^4 - 6*a^6*b^2*c*d^5 + a^7*b*d^6)*x)*log(d*x + c)^2 + 6*(549*a*b^7*c^6
- 5744*a^2*b^6*c^5*d + 30555*a^3*b^5*c^4*d^2 - 138250*a^4*b^4*c^3*d^3 + 133625*a^5*b^3*c^2*d^4 - 22794*a^6*b^2
*c*d^5 + 2059*a^7*b*d^6)*x - 180*(625*a^6*b^2*c^2*d^4 - 154*a^7*b*c*d^5 + 19*a^8*d^6 + (625*b^8*c^2*d^4 - 154*
a*b^7*c*d^5 + 19*a^2*b^6*d^6)*x^6 + 6*(625*a*b^7*c^2*d^4 - 154*a^2*b^6*c*d^5 + 19*a^3*b^5*d^6)*x^5 + 15*(625*a
^2*b^6*c^2*d^4 - 154*a^3*b^5*c*d^5 + 19*a^4*b^4*d^6)*x^4 + 20*(625*a^3*b^5*c^2*d^4 - 154*a^4*b^4*c*d^5 + 19*a^
5*b^3*d^6)*x^3 + 15*(625*a^4*b^4*c^2*d^4 - 154*a^5*b^3*c*d^5 + 19*a^6*b^2*d^6)*x^2 + 6*(625*a^5*b^3*c^2*d^4 -
154*a^6*b^2*c*d^5 + 19*a^7*b*d^6)*x)*log(b*x + a) + 180*(625*a^6*b^2*c^2*d^4 - 154*a^7*b*c*d^5 + 19*a^8*d^6 +
(625*b^8*c^2*d^4 - 154*a*b^7*c*d^5 + 19*a^2*b^6*d^6)*x^6 + 6*(625*a*b^7*c^2*d^4 - 154*a^2*b^6*c*d^5 + 19*a^3*b
^5*d^6)*x^5 + 15*(625*a^2*b^6*c^2*d^4 - 154*a^3*b^5*c*d^5 + 19*a^4*b^4*d^6)*x^4 + 20*(625*a^3*b^5*c^2*d^4 - 15
4*a^4*b^4*c*d^5 + 19*a^5*b^3*d^6)*x^3 + 15*(625*a^4*b^4*c^2*d^4 - 154*a^5*b^3*c*d^5 + 19*a^6*b^2*d^6)*x^2 + 6*
(625*a^5*b^3*c^2*d^4 - 154*a^6*b^2*c*d^5 + 19*a^7*b*d^6)*x - 20*(15*a^6*b^2*c^2*d^4 - 6*a^7*b*c*d^5 + a^8*d^6
+ (15*b^8*c^2*d^4 - 6*a*b^7*c*d^5 + a^2*b^6*d^6)*x^6 + 6*(15*a*b^7*c^2*d^4 - 6*a^2*b^6*c*d^5 + a^3*b^5*d^6)*x^
5 + 15*(15*a^2*b^6*c^2*d^4 - 6*a^3*b^5*c*d^5 + a^4*b^4*d^6)*x^4 + 20*(15*a^3*b^5*c^2*d^4 - 6*a^4*b^4*c*d^5 + a
^5*b^3*d^6)*x^3 + 15*(15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*x^2 + 6*(15*a^5*b^3*c^2*d^4 - 6*a^6*
b^2*c*d^5 + a^7*b*d^6)*x)*log(b*x + a))*log(d*x + c))/(a^6*b^9*c^6*g^7 - 6*a^7*b^8*c^5*d*g^7 + 15*a^8*b^7*c^4*
d^2*g^7 - 20*a^9*b^6*c^3*d^3*g^7 + 15*a^10*b^5*c^2*d^4*g^7 - 6*a^11*b^4*c*d^5*g^7 + a^12*b^3*d^6*g^7 + (b^15*c
^6*g^7 - 6*a*b^14*c^5*d*g^7 + 15*a^2*b^13*c^4*d^2*g^7 - 20*a^3*b^12*c^3*d^3*g^7 + 15*a^4*b^11*c^2*d^4*g^7 - 6*
a^5*b^10*c*d^5*g^7 + a^6*b^9*d^6*g^7)*x^6 + 6*(a*b^14*c^6*g^7 - 6*a^2*b^13*c^5*d*g^7 + 15*a^3*b^12*c^4*d^2*g^7
- 20*a^4*b^11*c^3*d^3*g^7 + 15*a^5*b^10*c^2*d^4*g^7 - 6*a^6*b^9*c*d^5*g^7 + a^7*b^8*d^6*g^7)*x^5 + 15*(a^2*b^
13*c^6*g^7 - 6*a^3*b^12*c^5*d*g^7 + 15*a^4*b^11*c^4*d^2*g^7 - 20*a^5*b^10*c^3*d^3*g^7 + 15*a^6*b^9*c^2*d^4*g^7
- 6*a^7*b^8*c*d^5*g^7 + a^8*b^7*d^6*g^7)*x^4 + 20*(a^3*b^12*c^6*g^7 - 6*a^4*b^11*c^5*d*g^7 + 15*a^5*b^10*c^4*
d^2*g^7 - 20*a^6*b^9*c^3*d^3*g^7 + 15*a^7*b^8*c^2*d^4*g^7 - 6*a^8*b^7*c*d^5*g^7 + a^9*b^6*d^6*g^7)*x^3 + 15*(a
^4*b^11*c^6*g^7 - 6*a^5*b^10*c^5*d*g^7 + 15*a^6*b^9*c^4*d^2*g^7 - 20*a^7*b^8*c^3*d^3*g^7 + 15*a^8*b^7*c^2*d^4*
g^7 - 6*a^9*b^6*c*d^5*g^7 + a^10*b^5*d^6*g^7)*x^2 + 6*(a^5*b^10*c^6*g^7 - 6*a^6*b^9*c^5*d*g^7 + 15*a^7*b^8*c^4
*d^2*g^7 - 20*a^8*b^7*c^3*d^3*g^7 + 15*a^9*b^6*c^2*d^4*g^7 - 6*a^10*b^5*c*d^5*g^7 + a^11*b^4*d^6*g^7)*x))*B^2*
c*d^2*i^3 - 1/108000*(60*((57*a^3*b^5*c^5 - 405*a^4*b^4*c^4*d + 1470*a^5*b^3*c^3*d^2 - 730*a^6*b^2*c^2*d^3 + 2
45*a^7*b*c*d^4 - 37*a^8*d^5 + 60*(20*b^8*c^3*d^2 - 15*a*b^7*c^2*d^3 + 6*a^2*b^6*c*d^4 - a^3*b^5*d^5)*x^5 - 30*
(20*b^8*c^4*d - 235*a*b^7*c^3*d^2 + 171*a^2*b^6*c^2*d^3 - 67*a^3*b^5*c*d^4 + 11*a^4*b^4*d^5)*x^4 + 20*(20*b^8*
c^5 - 175*a*b^7*c^4*d + 866*a^2*b^6*c^3*d^2 - 604*a^3*b^5*c^2*d^3 + 230*a^4*b^4*c*d^4 - 37*a^5*b^3*d^5)*x^3 +
15*(35*a*b^7*c^5 - 271*a^2*b^6*c^4*d + 1128*a^3*b^5*c^3*d^2 - 700*a^4*b^4*c^2*d^3 + 245*a^5*b^3*c*d^4 - 37*a^6
*b^2*d^5)*x^2 + 6*(47*a^2*b^6*c^5 - 345*a^3*b^5*c^4*d + 1320*a^4*b^4*c^3*d^2 - 730*a^5*b^3*c^2*d^3 + 245*a^6*b
^2*c*d^4 - 37*a^7*b*d^5)*x)/((b^15*c^5 - 5*a*b^14*c^4*d + 10*a^2*b^13*c^3*d^2 - 10*a^3*b^12*c^2*d^3 + 5*a^4*b^
11*c*d^4 - a^5*b^10*d^5)*g^7*x^6 + 6*(a*b^14*c^5 - 5*a^2*b^13*c^4*d + 10*a^3*b^12*c^3*d^2 - 10*a^4*b^11*c^2*d^
3 + 5*a^5*b^10*c*d^4 - a^6*b^9*d^5)*g^7*x^5 + 15*(a^2*b^13*c^5 - 5*a^3*b^12*c^4*d + 10*a^4*b^11*c^3*d^2 - 10*a
^5*b^10*c^2*d^3 + 5*a^6*b^9*c*d^4 - a^7*b^8*d^5)*g^7*x^4 + 20*(a^3*b^12*c^5 - 5*a^4*b^11*c^4*d + 10*a^5*b^10*c
^3*d^2 - 10*a^6*b^9*c^2*d^3 + 5*a^7*b^8*c*d^4 - a^8*b^7*d^5)*g^7*x^3 + 15*(a^4*b^11*c^5 - 5*a^5*b^10*c^4*d + 1
0*a^6*b^9*c^3*d^2 - 10*a^7*b^8*c^2*d^3 + 5*a^8*b^7*c*d^4 - a^9*b^6*d^5)*g^7*x^2 + 6*(a^5*b^10*c^5 - 5*a^6*b^9*
c^4*d + 10*a^7*b^8*c^3*d^2 - 10*a^8*b^7*c^2*d^3 + 5*a^9*b^6*c*d^4 - a^10*b^5*d^5)*g^7*x + (a^6*b^9*c^5 - 5*a^7
*b^8*c^4*d + 10*a^8*b^7*c^3*d^2 - 10*a^9*b^6*c^2*d^3 + 5*a^10*b^5*c*d^4 - a^11*b^4*d^5)*g^7) + 60*(20*b^3*c^3*
d^3 - 15*a*b^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(b*x + a)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2
- 20*a^3*b^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7) - 60*(20*b^3*c^3*d^3 - 15*a*b
^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(d*x + c)/((b^10*c^6 - 6*a*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b^
7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (
2059*a^3*b^6*c^6 - 22794*a^4*b^5*c^5*d + 155625*a^5*b^4*c^4*d^2 - 170000*a^6*b^3*c^3*d^3 + 43125*a^7*b^2*c^2*d
^4 - 8934*a^8*b*c*d^5 + 919*a^9*d^6 + 60*(2200*b^9*c^4*d^2 - 3175*a*b^8*c^3*d^3 + 1257*a^2*b^7*c^2*d^4 - 319*a
^3*b^6*c*d^5 + 37*a^4*b^5*d^6)*x^5 - 30*(1000*b^9*c^5*d - 24075*a*b^8*c^4*d^2 + 32822*a^2*b^7*c^3*d^3 - 12466*
a^3*b^6*c^2*d^4 + 3066*a^4*b^5*c*d^5 - 347*a^5*b^4*d^6)*x^4 + 20*(400*b^9*c^6 - 6825*a*b^8*c^5*d + 78267*a^2*b
^7*c^4*d^2 - 100130*a^3*b^6*c^3*d^3 + 35778*a^4*b^5*c^2*d^4 - 8409*a^5*b^4*c*d^5 + 919*a^6*b^3*d^6)*x^3 + 15*(
925*a*b^8*c^6 - 12222*a^2*b^7*c^5*d + 106533*a^3*b^6*c^4*d^2 - 128396*a^4*b^5*c^3*d^3 + 41175*a^5*b^4*c^2*d^4
- 8934*a^6*b^3*c*d^5 + 919*a^7*b^2*d^6)*x^2 - 1800*(20*a^6*b^3*c^3*d^3 - 15*a^7*b^2*c^2*d^4 + 6*a^8*b*c*d^5 -
a^9*d^6 + (20*b^9*c^3*d^3 - 15*a*b^8*c^2*d^4 + 6*a^2*b^7*c*d^5 - a^3*b^6*d^6)*x^6 + 6*(20*a*b^8*c^3*d^3 - 15*a
^2*b^7*c^2*d^4 + 6*a^3*b^6*c*d^5 - a^4*b^5*d^6)*x^5 + 15*(20*a^2*b^7*c^3*d^3 - 15*a^3*b^6*c^2*d^4 + 6*a^4*b^5*
c*d^5 - a^5*b^4*d^6)*x^4 + 20*(20*a^3*b^6*c^3*d^3 - 15*a^4*b^5*c^2*d^4 + 6*a^5*b^4*c*d^5 - a^6*b^3*d^6)*x^3 +
15*(20*a^4*b^5*c^3*d^3 - 15*a^5*b^4*c^2*d^4 + 6*a^6*b^3*c*d^5 - a^7*b^2*d^6)*x^2 + 6*(20*a^5*b^4*c^3*d^3 - 15*
a^6*b^3*c^2*d^4 + 6*a^7*b^2*c*d^5 - a^8*b*d^6)*x)*log(b*x + a)^2 - 1800*(20*a^6*b^3*c^3*d^3 - 15*a^7*b^2*c^2*d
^4 + 6*a^8*b*c*d^5 - a^9*d^6 + (20*b^9*c^3*d^3 - 15*a*b^8*c^2*d^4 + 6*a^2*b^7*c*d^5 - a^3*b^6*d^6)*x^6 + 6*(20
*a*b^8*c^3*d^3 - 15*a^2*b^7*c^2*d^4 + 6*a^3*b^6*c*d^5 - a^4*b^5*d^6)*x^5 + 15*(20*a^2*b^7*c^3*d^3 - 15*a^3*b^6
*c^2*d^4 + 6*a^4*b^5*c*d^5 - a^5*b^4*d^6)*x^4 + 20*(20*a^3*b^6*c^3*d^3 - 15*a^4*b^5*c^2*d^4 + 6*a^5*b^4*c*d^5
- a^6*b^3*d^6)*x^3 + 15*(20*a^4*b^5*c^3*d^3 - 15*a^5*b^4*c^2*d^4 + 6*a^6*b^3*c*d^5 - a^7*b^2*d^6)*x^2 + 6*(20*
a^5*b^4*c^3*d^3 - 15*a^6*b^3*c^2*d^4 + 6*a^7*b^2*c*d^5 - a^8*b*d^6)*x)*log(d*x + c)^2 + 6*(1489*a^2*b^7*c^6 -
17604*a^3*b^6*c^5*d + 132255*a^4*b^5*c^4*d^2 - 151250*a^5*b^4*c^3*d^3 + 43125*a^6*b^3*c^2*d^4 - 8934*a^7*b^2*c
*d^5 + 919*a^8*b*d^6)*x + 60*(2200*a^6*b^3*c^3*d^3 - 975*a^7*b^2*c^2*d^4 + 282*a^8*b*c*d^5 - 37*a^9*d^6 + (220
0*b^9*c^3*d^3 - 975*a*b^8*c^2*d^4 + 282*a^2*b^7*c*d^5 - 37*a^3*b^6*d^6)*x^6 + 6*(2200*a*b^8*c^3*d^3 - 975*a^2*
b^7*c^2*d^4 + 282*a^3*b^6*c*d^5 - 37*a^4*b^5*d^6)*x^5 + 15*(2200*a^2*b^7*c^3*d^3 - 975*a^3*b^6*c^2*d^4 + 282*a
^4*b^5*c*d^5 - 37*a^5*b^4*d^6)*x^4 + 20*(2200*a^3*b^6*c^3*d^3 - 975*a^4*b^5*c^2*d^4 + 282*a^5*b^4*c*d^5 - 37*a
^6*b^3*d^6)*x^3 + 15*(2200*a^4*b^5*c^3*d^3 - 975*a^5*b^4*c^2*d^4 + 282*a^6*b^3*c*d^5 - 37*a^7*b^2*d^6)*x^2 + 6
*(2200*a^5*b^4*c^3*d^3 - 975*a^6*b^3*c^2*d^4 + 282*a^7*b^2*c*d^5 - 37*a^8*b*d^6)*x)*log(b*x + a) - 60*(2200*a^
6*b^3*c^3*d^3 - 975*a^7*b^2*c^2*d^4 + 282*a^8*b*c*d^5 - 37*a^9*d^6 + (2200*b^9*c^3*d^3 - 975*a*b^8*c^2*d^4 + 2
82*a^2*b^7*c*d^5 - 37*a^3*b^6*d^6)*x^6 + 6*(2200*a*b^8*c^3*d^3 - 975*a^2*b^7*c^2*d^4 + 282*a^3*b^6*c*d^5 - 37*
a^4*b^5*d^6)*x^5 + 15*(2200*a^2*b^7*c^3*d^3 - 975*a^3*b^6*c^2*d^4 + 282*a^4*b^5*c*d^5 - 37*a^5*b^4*d^6)*x^4 +
20*(2200*a^3*b^6*c^3*d^3 - 975*a^4*b^5*c^2*d^4 + 282*a^5*b^4*c*d^5 - 37*a^6*b^3*d^6)*x^3 + 15*(2200*a^4*b^5*c^
3*d^3 - 975*a^5*b^4*c^2*d^4 + 282*a^6*b^3*c*d^5 - 37*a^7*b^2*d^6)*x^2 + 6*(2200*a^5*b^4*c^3*d^3 - 975*a^6*b^3*
c^2*d^4 + 282*a^7*b^2*c*d^5 - 37*a^8*b*d^6)*x - 60*(20*a^6*b^3*c^3*d^3 - 15*a^7*b^2*c^2*d^4 + 6*a^8*b*c*d^5 -
a^9*d^6 + (20*b^9*c^3*d^3 - 15*a*b^8*c^2*d^4 + 6*a^2*b^7*c*d^5 - a^3*b^6*d^6)*x^6 + 6*(20*a*b^8*c^3*d^3 - 15*a
^2*b^7*c^2*d^4 + 6*a^3*b^6*c*d^5 - a^4*b^5*d^6)*x^5 + 15*(20*a^2*b^7*c^3*d^3 - 15*a^3*b^6*c^2*d^4 + 6*a^4*b^5*
c*d^5 - a^5*b^4*d^6)*x^4 + 20*(20*a^3*b^6*c^3*d^3 - 15*a^4*b^5*c^2*d^4 + 6*a^5*b^4*c*d^5 - a^6*b^3*d^6)*x^3 +
15*(20*a^4*b^5*c^3*d^3 - 15*a^5*b^4*c^2*d^4 + 6*a^6*b^3*c*d^5 - a^7*b^2*d^6)*x^2 + 6*(20*a^5*b^4*c^3*d^3 - 15*
a^6*b^3*c^2*d^4 + 6*a^7*b^2*c*d^5 - a^8*b*d^6)*x)*log(b*x + a))*log(d*x + c))/(a^6*b^10*c^6*g^7 - 6*a^7*b^9*c^
5*d*g^7 + 15*a^8*b^8*c^4*d^2*g^7 - 20*a^9*b^7*c^3*d^3*g^7 + 15*a^10*b^6*c^2*d^4*g^7 - 6*a^11*b^5*c*d^5*g^7 + a
^12*b^4*d^6*g^7 + (b^16*c^6*g^7 - 6*a*b^15*c^5*d*g^7 + 15*a^2*b^14*c^4*d^2*g^7 - 20*a^3*b^13*c^3*d^3*g^7 + 15*
a^4*b^12*c^2*d^4*g^7 - 6*a^5*b^11*c*d^5*g^7 + a^6*b^10*d^6*g^7)*x^6 + 6*(a*b^15*c^6*g^7 - 6*a^2*b^14*c^5*d*g^7
+ 15*a^3*b^13*c^4*d^2*g^7 - 20*a^4*b^12*c^3*d^3*g^7 + 15*a^5*b^11*c^2*d^4*g^7 - 6*a^6*b^10*c*d^5*g^7 + a^7*b^
9*d^6*g^7)*x^5 + 15*(a^2*b^14*c^6*g^7 - 6*a^3*b^13*c^5*d*g^7 + 15*a^4*b^12*c^4*d^2*g^7 - 20*a^5*b^11*c^3*d^3*g
^7 + 15*a^6*b^10*c^2*d^4*g^7 - 6*a^7*b^9*c*d^5*g^7 + a^8*b^8*d^6*g^7)*x^4 + 20*(a^3*b^13*c^6*g^7 - 6*a^4*b^12*
c^5*d*g^7 + 15*a^5*b^11*c^4*d^2*g^7 - 20*a^6*b^10*c^3*d^3*g^7 + 15*a^7*b^9*c^2*d^4*g^7 - 6*a^8*b^8*c*d^5*g^7 +
a^9*b^7*d^6*g^7)*x^3 + 15*(a^4*b^12*c^6*g^7 - 6*a^5*b^11*c^5*d*g^7 + 15*a^6*b^10*c^4*d^2*g^7 - 20*a^7*b^9*c^3
*d^3*g^7 + 15*a^8*b^8*c^2*d^4*g^7 - 6*a^9*b^7*c*d^5*g^7 + a^10*b^6*d^6*g^7)*x^2 + 6*(a^5*b^11*c^6*g^7 - 6*a^6*
b^10*c^5*d*g^7 + 15*a^7*b^9*c^4*d^2*g^7 - 20*a^8*b^8*c^3*d^3*g^7 + 15*a^9*b^7*c^2*d^4*g^7 - 6*a^10*b^6*c*d^5*g
^7 + a^11*b^5*d^6*g^7)*x))*B^2*d^3*i^3 - 1/1800*A*B*d^3*i^3*(60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*
log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3
+ 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7) + (57*a^3*b^5*c^5 - 405*a^4*b^4*c^4*d + 1470*a^5*b^3*c^
3*d^2 - 730*a^6*b^2*c^2*d^3 + 245*a^7*b*c*d^4 - 37*a^8*d^5 + 60*(20*b^8*c^3*d^2 - 15*a*b^7*c^2*d^3 + 6*a^2*b^6
*c*d^4 - a^3*b^5*d^5)*x^5 - 30*(20*b^8*c^4*d - 235*a*b^7*c^3*d^2 + 171*a^2*b^6*c^2*d^3 - 67*a^3*b^5*c*d^4 + 11
*a^4*b^4*d^5)*x^4 + 20*(20*b^8*c^5 - 175*a*b^7*c^4*d + 866*a^2*b^6*c^3*d^2 - 604*a^3*b^5*c^2*d^3 + 230*a^4*b^4
*c*d^4 - 37*a^5*b^3*d^5)*x^3 + 15*(35*a*b^7*c^5 - 271*a^2*b^6*c^4*d + 1128*a^3*b^5*c^3*d^2 - 700*a^4*b^4*c^2*d
^3 + 245*a^5*b^3*c*d^4 - 37*a^6*b^2*d^5)*x^2 + 6*(47*a^2*b^6*c^5 - 345*a^3*b^5*c^4*d + 1320*a^4*b^4*c^3*d^2 -
730*a^5*b^3*c^2*d^3 + 245*a^6*b^2*c*d^4 - 37*a^7*b*d^5)*x)/((b^15*c^5 - 5*a*b^14*c^4*d + 10*a^2*b^13*c^3*d^2 -
10*a^3*b^12*c^2*d^3 + 5*a^4*b^11*c*d^4 - a^5*b^10*d^5)*g^7*x^6 + 6*(a*b^14*c^5 - 5*a^2*b^13*c^4*d + 10*a^3*b^
12*c^3*d^2 - 10*a^4*b^11*c^2*d^3 + 5*a^5*b^10*c*d^4 - a^6*b^9*d^5)*g^7*x^5 + 15*(a^2*b^13*c^5 - 5*a^3*b^12*c^4
*d + 10*a^4*b^11*c^3*d^2 - 10*a^5*b^10*c^2*d^3 + 5*a^6*b^9*c*d^4 - a^7*b^8*d^5)*g^7*x^4 + 20*(a^3*b^12*c^5 - 5
*a^4*b^11*c^4*d + 10*a^5*b^10*c^3*d^2 - 10*a^6*b^9*c^2*d^3 + 5*a^7*b^8*c*d^4 - a^8*b^7*d^5)*g^7*x^3 + 15*(a^4*
b^11*c^5 - 5*a^5*b^10*c^4*d + 10*a^6*b^9*c^3*d^2 - 10*a^7*b^8*c^2*d^3 + 5*a^8*b^7*c*d^4 - a^9*b^6*d^5)*g^7*x^2
+ 6*(a^5*b^10*c^5 - 5*a^6*b^9*c^4*d + 10*a^7*b^8*c^3*d^2 - 10*a^8*b^7*c^2*d^3 + 5*a^9*b^6*c*d^4 - a^10*b^5*d^
5)*g^7*x + (a^6*b^9*c^5 - 5*a^7*b^8*c^4*d + 10*a^8*b^7*c^3*d^2 - 10*a^9*b^6*c^2*d^3 + 5*a^10*b^5*c*d^4 - a^11*
b^4*d^5)*g^7) + 60*(20*b^3*c^3*d^3 - 15*a*b^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(b*x + a)/((b^10*c^6 - 6*a
*b^9*c^5*d + 15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7
) - 60*(20*b^3*c^3*d^3 - 15*a*b^2*c^2*d^4 + 6*a^2*b*c*d^5 - a^3*d^6)*log(d*x + c)/((b^10*c^6 - 6*a*b^9*c^5*d +
15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*g^7)) - 1/600*A
*B*c*d^2*i^3*(60*(15*b^2*x^2 + 6*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^9*g^7*x^6 + 6*a*b^8*g^7*
x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b^5*g^7*x^2 + 6*a^5*b^4*g^7*x + a^6*b^3*g^7) + (37*a^2*
b^5*c^5 - 245*a^3*b^4*c^4*d + 730*a^4*b^3*c^3*d^2 - 1470*a^5*b^2*c^2*d^3 + 405*a^6*b*c*d^4 - 57*a^7*d^5 - 60*(
15*b^7*c^2*d^3 - 6*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 + 30*(15*b^7*c^3*d^2 - 171*a*b^6*c^2*d^3 + 67*a^2*b^5*c*d^4
- 11*a^3*b^4*d^5)*x^4 - 20*(15*b^7*c^4*d - 126*a*b^6*c^3*d^2 + 604*a^2*b^5*c^2*d^3 - 230*a^3*b^4*c*d^4 + 37*a^
4*b^3*d^5)*x^3 + 15*(15*b^7*c^5 - 111*a*b^6*c^4*d + 388*a^2*b^5*c^3*d^2 - 1000*a^3*b^4*c^2*d^3 + 365*a^4*b^3*c
*d^4 - 57*a^5*b^2*d^5)*x^2 + 6*(27*a*b^6*c^5 - 185*a^2*b^5*c^4*d + 580*a^3*b^4*c^3*d^2 - 1270*a^4*b^3*c^2*d^3
+ 405*a^5*b^2*c*d^4 - 57*a^6*b*d^5)*x)/((b^14*c^5 - 5*a*b^13*c^4*d + 10*a^2*b^12*c^3*d^2 - 10*a^3*b^11*c^2*d^3
+ 5*a^4*b^10*c*d^4 - a^5*b^9*d^5)*g^7*x^6 + 6*(a*b^13*c^5 - 5*a^2*b^12*c^4*d + 10*a^3*b^11*c^3*d^2 - 10*a^4*b
^10*c^2*d^3 + 5*a^5*b^9*c*d^4 - a^6*b^8*d^5)*g^7*x^5 + 15*(a^2*b^12*c^5 - 5*a^3*b^11*c^4*d + 10*a^4*b^10*c^3*d
^2 - 10*a^5*b^9*c^2*d^3 + 5*a^6*b^8*c*d^4 - a^7*b^7*d^5)*g^7*x^4 + 20*(a^3*b^11*c^5 - 5*a^4*b^10*c^4*d + 10*a^
5*b^9*c^3*d^2 - 10*a^6*b^8*c^2*d^3 + 5*a^7*b^7*c*d^4 - a^8*b^6*d^5)*g^7*x^3 + 15*(a^4*b^10*c^5 - 5*a^5*b^9*c^4
*d + 10*a^6*b^8*c^3*d^2 - 10*a^7*b^7*c^2*d^3 + 5*a^8*b^6*c*d^4 - a^9*b^5*d^5)*g^7*x^2 + 6*(a^5*b^9*c^5 - 5*a^6
*b^8*c^4*d + 10*a^7*b^7*c^3*d^2 - 10*a^8*b^6*c^2*d^3 + 5*a^9*b^5*c*d^4 - a^10*b^4*d^5)*g^7*x + (a^6*b^8*c^5 -
5*a^7*b^7*c^4*d + 10*a^8*b^6*c^3*d^2 - 10*a^9*b^5*c^2*d^3 + 5*a^10*b^4*c*d^4 - a^11*b^3*d^5)*g^7) - 60*(15*b^2
*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*log(b*x + a)/((b^9*c^6 - 6*a*b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3
*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*b^4*c*d^5 + a^6*b^3*d^6)*g^7) + 60*(15*b^2*c^2*d^4 - 6*a*b*c*d^5 + a^2*d^6)*
log(d*x + c)/((b^9*c^6 - 6*a*b^8*c^5*d + 15*a^2*b^7*c^4*d^2 - 20*a^3*b^6*c^3*d^3 + 15*a^4*b^5*c^2*d^4 - 6*a^5*
b^4*c*d^5 + a^6*b^3*d^6)*g^7)) - 1/300*A*B*c^2*d*i^3*(60*(6*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8
*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x^4 + 20*a^3*b^5*g^7*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x +
a^6*b^2*g^7) + (22*a*b^5*c^5 - 140*a^2*b^4*c^4*d + 385*a^3*b^3*c^3*d^2 - 615*a^4*b^2*c^2*d^3 + 735*a^5*b*c*d^4
- 87*a^6*d^5 + 60*(6*b^6*c*d^4 - a*b^5*d^5)*x^5 - 30*(6*b^6*c^2*d^3 - 67*a*b^5*c*d^4 + 11*a^2*b^4*d^5)*x^4 +
20*(6*b^6*c^3*d^2 - 49*a*b^5*c^2*d^3 + 230*a^2*b^4*c*d^4 - 37*a^3*b^3*d^5)*x^3 - 15*(6*b^6*c^4*d - 43*a*b^5*c^
3*d^2 + 145*a^2*b^4*c^2*d^3 - 365*a^3*b^3*c*d^4 + 57*a^4*b^2*d^5)*x^2 + 6*(12*b^6*c^5 - 80*a*b^5*c^4*d + 235*a
^2*b^4*c^3*d^2 - 415*a^3*b^3*c^2*d^3 + 585*a^4*b^2*c*d^4 - 87*a^5*b*d^5)*x)/((b^13*c^5 - 5*a*b^12*c^4*d + 10*a
^2*b^11*c^3*d^2 - 10*a^3*b^10*c^2*d^3 + 5*a^4*b^9*c*d^4 - a^5*b^8*d^5)*g^7*x^6 + 6*(a*b^12*c^5 - 5*a^2*b^11*c^
4*d + 10*a^3*b^10*c^3*d^2 - 10*a^4*b^9*c^2*d^3 + 5*a^5*b^8*c*d^4 - a^6*b^7*d^5)*g^7*x^5 + 15*(a^2*b^11*c^5 - 5
*a^3*b^10*c^4*d + 10*a^4*b^9*c^3*d^2 - 10*a^5*b^8*c^2*d^3 + 5*a^6*b^7*c*d^4 - a^7*b^6*d^5)*g^7*x^4 + 20*(a^3*b
^10*c^5 - 5*a^4*b^9*c^4*d + 10*a^5*b^8*c^3*d^2 - 10*a^6*b^7*c^2*d^3 + 5*a^7*b^6*c*d^4 - a^8*b^5*d^5)*g^7*x^3 +
15*(a^4*b^9*c^5 - 5*a^5*b^8*c^4*d + 10*a^6*b^7*c^3*d^2 - 10*a^7*b^6*c^2*d^3 + 5*a^8*b^5*c*d^4 - a^9*b^4*d^5)*
g^7*x^2 + 6*(a^5*b^8*c^5 - 5*a^6*b^7*c^4*d + 10*a^7*b^6*c^3*d^2 - 10*a^8*b^5*c^2*d^3 + 5*a^9*b^4*c*d^4 - a^10*
b^3*d^5)*g^7*x + (a^6*b^7*c^5 - 5*a^7*b^6*c^4*d + 10*a^8*b^5*c^3*d^2 - 10*a^9*b^4*c^2*d^3 + 5*a^10*b^3*c*d^4 -
a^11*b^2*d^5)*g^7) + 60*(6*b*c*d^5 - a*d^6)*log(b*x + a)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*
a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5 + a^6*b^2*d^6)*g^7) - 60*(6*b*c*d^5 - a*d^6)*log(d*x +
c)/((b^8*c^6 - 6*a*b^7*c^5*d + 15*a^2*b^6*c^4*d^2 - 20*a^3*b^5*c^3*d^3 + 15*a^4*b^4*c^2*d^4 - 6*a^5*b^3*c*d^5
+ a^6*b^2*d^6)*g^7)) + 1/180*A*B*c^3*i^3*((60*b^5*d^5*x^5 - 10*b^5*c^5 + 62*a*b^4*c^4*d - 163*a^2*b^3*c^3*d^2
+ 237*a^3*b^2*c^2*d^3 - 213*a^4*b*c*d^4 + 147*a^5*d^5 - 30*(b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 -
8*a*b^4*c*d^4 + 37*a^2*b^3*d^5)*x^3 - 15*(b^5*c^3*d^2 - 7*a*b^4*c^2*d^3 + 23*a^2*b^3*c*d^4 - 57*a^3*b^2*d^5)*x
^2 + 6*(2*b^5*c^4*d - 13*a*b^4*c^3*d^2 + 37*a^2*b^3*c^2*d^3 - 63*a^3*b^2*c*d^4 + 87*a^4*b*d^5)*x)/((b^12*c^5 -
5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 10*a^3*b^9*c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*g^7*x^6 + 6*(a*b^1
1*c^5 - 5*a^2*b^10*c^4*d + 10*a^3*b^9*c^3*d^2 - 10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^6*b^6*d^5)*g^7*x^5 +
15*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^8*c^3*d^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*
g^7*x^4 + 20*(a^3*b^9*c^5 - 5*a^4*b^8*c^4*d + 10*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + 5*a^7*b^5*c*d^4 - a^8*
b^4*d^5)*g^7*x^3 + 15*(a^4*b^8*c^5 - 5*a^5*b^7*c^4*d + 10*a^6*b^6*c^3*d^2 - 10*a^7*b^5*c^2*d^3 + 5*a^8*b^4*c*d
^4 - a^9*b^3*d^5)*g^7*x^2 + 6*(a^5*b^7*c^5 - 5*a^6*b^6*c^4*d + 10*a^7*b^5*c^3*d^2 - 10*a^8*b^4*c^2*d^3 + 5*a^9
*b^3*c*d^4 - a^10*b^2*d^5)*g^7*x + (a^6*b^6*c^5 - 5*a^7*b^5*c^4*d + 10*a^8*b^4*c^3*d^2 - 10*a^9*b^3*c^2*d^3 +
5*a^10*b^2*c*d^4 - a^11*b*d^5)*g^7) - 60*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 +
15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a^4*b^3*g^7*x^2 + 6*a^5*b^2*g^7*x + a^6*b*g^7) + 60*d^6*log(b*x
+ a)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3 + 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^
5 + a^6*b*d^6)*g^7) - 60*d^6*log(d*x + c)/((b^7*c^6 - 6*a*b^6*c^5*d + 15*a^2*b^5*c^4*d^2 - 20*a^3*b^4*c^3*d^3
+ 15*a^4*b^3*c^2*d^4 - 6*a^5*b^2*c*d^5 + a^6*b*d^6)*g^7)) - 1/6*B^2*c^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c
))^2/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a^4*b^3*g^7*x^2 + 6*a^5*b^2
*g^7*x + a^6*b*g^7) - 1/10*(6*b*x + a)*A^2*c^2*d*i^3/(b^8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x^4 + 20*
a^3*b^5*g^7*x^3 + 15*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) - 1/20*(15*b^2*x^2 + 6*a*b*x + a^2)*A^2*
c*d^2*i^3/(b^9*g^7*x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b^5*g^7*x^2 + 6*a^
5*b^4*g^7*x + a^6*b^3*g^7) - 1/60*(20*b^3*x^3 + 15*a*b^2*x^2 + 6*a^2*b*x + a^3)*A^2*d^3*i^3/(b^10*g^7*x^6 + 6*
a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 + 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7)
- 1/6*A^2*c^3*i^3/(b^7*g^7*x^6 + 6*a*b^6*g^7*x^5 + 15*a^2*b^5*g^7*x^4 + 20*a^3*b^4*g^7*x^3 + 15*a^4*b^3*g^7*x^
2 + 6*a^5*b^2*g^7*x + a^6*b*g^7)

Giac [A] (verification not implemented)

none

Time = 0.66 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.68 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=-\frac {1}{108000} \, {\left (\frac {1800 \, {\left (10 \, B^{2} b^{2} e^{7} i^{3} - \frac {24 \, {\left (b e x + a e\right )} B^{2} b d e^{6} i^{3}}{d x + c} + \frac {15 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b e x + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b e x + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}} + \frac {60 \, {\left (600 \, A B b^{2} e^{7} i^{3} + 100 \, B^{2} b^{2} e^{7} i^{3} - \frac {1440 \, {\left (b e x + a e\right )} A B b d e^{6} i^{3}}{d x + c} - \frac {288 \, {\left (b e x + a e\right )} B^{2} b d e^{6} i^{3}}{d x + c} + \frac {900 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}} + \frac {225 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b e x + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b e x + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}} + \frac {18000 \, A^{2} b^{2} e^{7} i^{3} + 6000 \, A B b^{2} e^{7} i^{3} + 1000 \, B^{2} b^{2} e^{7} i^{3} - \frac {43200 \, {\left (b e x + a e\right )} A^{2} b d e^{6} i^{3}}{d x + c} - \frac {17280 \, {\left (b e x + a e\right )} A B b d e^{6} i^{3}}{d x + c} - \frac {3456 \, {\left (b e x + a e\right )} B^{2} b d e^{6} i^{3}}{d x + c} + \frac {27000 \, {\left (b e x + a e\right )}^{2} A^{2} d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}} + \frac {13500 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}} + \frac {3375 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{5} i^{3}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{6} b^{2} c^{2} g^{7}}{{\left (d x + c\right )}^{6}} - \frac {2 \, {\left (b e x + a e\right )}^{6} a b c d g^{7}}{{\left (d x + c\right )}^{6}} + \frac {{\left (b e x + a e\right )}^{6} a^{2} d^{2} g^{7}}{{\left (d x + c\right )}^{6}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, algorithm="giac")

[Out]

-1/108000*(1800*(10*B^2*b^2*e^7*i^3 - 24*(b*e*x + a*e)*B^2*b*d*e^6*i^3/(d*x + c) + 15*(b*e*x + a*e)^2*B^2*d^2*
e^5*i^3/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x + a*e)
^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^6) + 60*(600*A*B*b^2*e^7*i^3 + 100*B^2*b^2*
e^7*i^3 - 1440*(b*e*x + a*e)*A*B*b*d*e^6*i^3/(d*x + c) - 288*(b*e*x + a*e)*B^2*b*d*e^6*i^3/(d*x + c) + 900*(b*
e*x + a*e)^2*A*B*d^2*e^5*i^3/(d*x + c)^2 + 225*(b*e*x + a*e)^2*B^2*d^2*e^5*i^3/(d*x + c)^2)*log((b*e*x + a*e)/
(d*x + c))/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x + a*e)^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e
)^6*a^2*d^2*g^7/(d*x + c)^6) + (18000*A^2*b^2*e^7*i^3 + 6000*A*B*b^2*e^7*i^3 + 1000*B^2*b^2*e^7*i^3 - 43200*(b
*e*x + a*e)*A^2*b*d*e^6*i^3/(d*x + c) - 17280*(b*e*x + a*e)*A*B*b*d*e^6*i^3/(d*x + c) - 3456*(b*e*x + a*e)*B^2
*b*d*e^6*i^3/(d*x + c) + 27000*(b*e*x + a*e)^2*A^2*d^2*e^5*i^3/(d*x + c)^2 + 13500*(b*e*x + a*e)^2*A*B*d^2*e^5
*i^3/(d*x + c)^2 + 3375*(b*e*x + a*e)^2*B^2*d^2*e^5*i^3/(d*x + c)^2)/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6
- 2*(b*e*x + a*e)^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^6))*(b*c/((b*c*e - a*d*e)*
(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 10.60 (sec) , antiderivative size = 6275, normalized size of antiderivative = 13.55 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \]

[In]

int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x)^7,x)

[Out]

((1800*A^2*a^5*d^5*i^3 + 18000*A^2*b^5*c^5*i^3 + 919*B^2*a^5*d^5*i^3 + 1000*B^2*b^5*c^5*i^3 + 2220*A*B*a^5*d^5
*i^3 + 6000*A*B*b^5*c^5*i^3 - 25200*A^2*a*b^4*c^4*d*i^3 + 1800*A^2*a^4*b*c*d^4*i^3 - 2456*B^2*a*b^4*c^4*d*i^3
+ 919*B^2*a^4*b*c*d^4*i^3 + 1800*A^2*a^2*b^3*c^3*d^2*i^3 + 1800*A^2*a^3*b^2*c^2*d^3*i^3 + 919*B^2*a^2*b^3*c^3*
d^2*i^3 + 919*B^2*a^3*b^2*c^2*d^3*i^3 + 2220*A*B*a^2*b^3*c^3*d^2*i^3 + 2220*A*B*a^3*b^2*c^2*d^3*i^3 - 11280*A*
B*a*b^4*c^4*d*i^3 + 2220*A*B*a^4*b*c*d^4*i^3)/(60*(a*d - b*c)) + (x^4*(347*B^2*a*b^4*d^5*i^3 + 23*B^2*b^5*c*d^
4*i^3 + 660*A*B*a*b^4*d^5*i^3 - 60*A*B*b^5*c*d^4*i^3))/(2*(a*d - b*c)) + (x^2*(1800*A^2*a^3*b^2*d^5*i^3 + 919*
B^2*a^3*b^2*d^5*i^3 + 5400*A^2*b^5*c^3*d^2*i^3 + 73*B^2*b^5*c^3*d^2*i^3 - 9000*A^2*a*b^4*c^2*d^3*i^3 + 1800*A^
2*a^2*b^3*c*d^4*i^3 - 431*B^2*a*b^4*c^2*d^3*i^3 + 919*B^2*a^2*b^3*c*d^4*i^3 + 2220*A*B*a^3*b^2*d^5*i^3 + 1140*
A*B*b^5*c^3*d^2*i^3 - 3180*A*B*a*b^4*c^2*d^3*i^3 + 2220*A*B*a^2*b^3*c*d^4*i^3))/(4*(a*d - b*c)) + (x^3*(1800*A
^2*a^2*b^3*d^5*i^3 + 919*B^2*a^2*b^3*d^5*i^3 + 1800*A^2*b^5*c^2*d^3*i^3 - 53*B^2*b^5*c^2*d^3*i^3 - 3600*A^2*a*
b^4*c*d^4*i^3 + 244*B^2*a*b^4*c*d^4*i^3 + 2220*A*B*a^2*b^3*d^5*i^3 + 60*A*B*b^5*c^2*d^3*i^3 - 480*A*B*a*b^4*c*
d^4*i^3))/(3*(a*d - b*c)) + (x*(1800*A^2*a^4*b*d^5*i^3 + 919*B^2*a^4*b*d^5*i^3 + 10800*A^2*b^5*c^4*d*i^3 + 424
*B^2*b^5*c^4*d*i^3 - 16200*A^2*a*b^4*c^3*d^2*i^3 + 1800*A^2*a^3*b^2*c*d^4*i^3 - 1331*B^2*a*b^4*c^3*d^2*i^3 + 9
19*B^2*a^3*b^2*c*d^4*i^3 + 2220*A*B*a^4*b*d^5*i^3 + 3120*A*B*b^5*c^4*d*i^3 + 1800*A^2*a^2*b^3*c^2*d^3*i^3 + 91
9*B^2*a^2*b^3*c^2*d^3*i^3 - 6780*A*B*a*b^4*c^3*d^2*i^3 + 2220*A*B*a^3*b^2*c*d^4*i^3 + 2220*A*B*a^2*b^3*c^2*d^3
*i^3))/(10*(a*d - b*c)) + (d*x^5*(37*B^2*b^5*d^4*i^3 + 60*A*B*b^5*d^4*i^3))/(a*d - b*c))/(x*(10800*a^5*b^6*c*g
^7 - 10800*a^6*b^5*d*g^7) - x^5*(10800*a^2*b^9*d*g^7 - 10800*a*b^10*c*g^7) + x^6*(1800*b^11*c*g^7 - 1800*a*b^1
0*d*g^7) + x^2*(27000*a^4*b^7*c*g^7 - 27000*a^5*b^6*d*g^7) + x^4*(27000*a^2*b^9*c*g^7 - 27000*a^3*b^8*d*g^7) +
x^3*(36000*a^3*b^8*c*g^7 - 36000*a^4*b^7*d*g^7) + 1800*a^6*b^5*c*g^7 - 1800*a^7*b^4*d*g^7) - log((e*(a + b*x)
)/(c + d*x))^2*((x*(a*(b*((B^2*a*d^3*i^3)/(60*b^5*g^7) + (B^2*c*d^2*i^3)/(20*b^4*g^7)) + (B^2*a*d^3*i^3)/(15*b
^4*g^7) + (B^2*c*d^2*i^3)/(5*b^3*g^7)) + b*(a*((B^2*a*d^3*i^3)/(60*b^5*g^7) + (B^2*c*d^2*i^3)/(20*b^4*g^7)) +
(B^2*c^2*d*i^3)/(10*b^3*g^7)) + (B^2*c^2*d*i^3)/(2*b^2*g^7)) + x^2*(b*(b*((B^2*a*d^3*i^3)/(60*b^5*g^7) + (B^2*
c*d^2*i^3)/(20*b^4*g^7)) + (B^2*a*d^3*i^3)/(15*b^4*g^7) + (B^2*c*d^2*i^3)/(5*b^3*g^7)) + (B^2*a*d^3*i^3)/(6*b^
3*g^7) + (B^2*c*d^2*i^3)/(2*b^2*g^7)) + a*(a*((B^2*a*d^3*i^3)/(60*b^5*g^7) + (B^2*c*d^2*i^3)/(20*b^4*g^7)) + (
B^2*c^2*d*i^3)/(10*b^3*g^7)) + (B^2*c^3*i^3)/(6*b^2*g^7) + (B^2*d^3*i^3*x^3)/(3*b^2*g^7))/(6*a^5*x + a^6/b + b
^5*x^6 + 15*a^4*b*x^2 + 6*a*b^4*x^5 + 20*a^3*b^2*x^3 + 15*a^2*b^3*x^4) - (B^2*d^6*i^3)/(60*b^4*g^7*(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))*(a*(a*((B*d*i^3*(9*A*b*c - B*a*d +
B*b*c))/(90*b^5*g^7) + (A*B*a*d^2*i^3)/(30*b^5*g^7)) + (B*i^3*(36*A*b^2*c^2 - 3*B*a^2*d^2 + 5*B*b^2*c^2 - 2*B*
a*b*c*d))/(180*b^5*g^7)) + x^2*(b*(b*((B*d*i^3*(9*A*b*c - B*a*d + B*b*c))/(90*b^5*g^7) + (A*B*a*d^2*i^3)/(30*b
^5*g^7)) + (2*B*d*i^3*(9*A*b*c - B*a*d + B*b*c))/(45*b^4*g^7) + (2*A*B*a*d^2*i^3)/(15*b^4*g^7)) + (B*d*i^3*(9*
A*b*c - B*a*d + B*b*c))/(9*b^3*g^7) + (A*B*a*d^2*i^3)/(3*b^3*g^7) + (B^2*d^6*i^3*(b*((20*a^4*d^4 + b^4*c^4 + 2
1*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 35*a^3*b*c*d^3)/(15*d^5) + b*(a*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*
b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b*d^4)) + (2
0*a^4*d^4 + b^4*c^4 + 21*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 35*a^3*b*c*d^3)/(60*b*d^5)) + a*(b*(a*((6*a^2*d^2 +
b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2
*b*c*d^2)/(60*b*d^4)) + a*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a
^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c))/(3*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21
*a^2*b*c*d^2)/(20*d^4))) + a*(a*(b*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2
)) + (6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c))/(3*d^2)) - a*((b^2*c - a*b*d)/(6*d^2) - (b*(
a*d - b*c))/(3*d^2)) + (b^3*c^2 + 6*a^2*b*d^2 - 7*a*b^2*c*d)/(10*d^3)) - (b^4*c^3 - 15*a^3*b*d^3 + 21*a^2*b^2*
c*d^2 - 7*a*b^3*c^2*d)/(10*d^4) + b*(b*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b
*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b*d^4)) + a*(b*((6*a^2*d^2 + b^2*c^2 - 7*
a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c)
)/(3*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(20*d^4))) + (b^5*c^4 + 20*a^4*b*d^4 - 35
*a^3*b^2*c*d^3 + 21*a^2*b^3*c^2*d^2 - 7*a*b^4*c^3*d)/(6*d^5)))/(30*b^4*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2))) + x*(b*(a*((B*d*i^3*(9*A*b*c - B*a*d + B*b*c))/(90*b^5*g^7) + (A*B*a*d^2*i^3)/(30*b^5*g^7))
+ (B*i^3*(36*A*b^2*c^2 - 3*B*a^2*d^2 + 5*B*b^2*c^2 - 2*B*a*b*c*d))/(180*b^5*g^7)) + a*(b*((B*d*i^3*(9*A*b*c -
B*a*d + B*b*c))/(90*b^5*g^7) + (A*B*a*d^2*i^3)/(30*b^5*g^7)) + (2*B*d*i^3*(9*A*b*c - B*a*d + B*b*c))/(45*b^4*
g^7) + (2*A*B*a*d^2*i^3)/(15*b^4*g^7)) + (B*i^3*(36*A*b^2*c^2 - 3*B*a^2*d^2 + 5*B*b^2*c^2 - 2*B*a*b*c*d))/(36*
b^4*g^7) + (B^2*d^6*i^3*(a*((20*a^4*d^4 + b^4*c^4 + 21*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 35*a^3*b*c*d^3)/(15*d
^5) + b*(a*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3 - b^3*c
^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b*d^4)) + (20*a^4*d^4 + b^4*c^4 + 21*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d
- 35*a^3*b*c*d^3)/(60*b*d^5)) + a*(b*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d
^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b*d^4)) + a*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*
b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c))/
(3*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(20*d^4))) + (15*a^5*d^5 - b^5*c^5 - 21*a^2
*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 7*a*b^4*c^4*d - 35*a^4*b*c*d^4)/(6*d^6) + b*(a*(a*(a*((6*a^2*d^2 + b^2*c^2
- 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2
)/(60*b*d^4)) + (20*a^4*d^4 + b^4*c^4 + 21*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 35*a^3*b*c*d^3)/(60*b*d^5)) + (15
*a^5*d^5 - b^5*c^5 - 21*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 7*a*b^4*c^4*d - 35*a^4*b*c*d^4)/(30*b*d^6))))/(
30*b^4*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + x^3*((2*A*B*d^2*i^3)/(3*b^2*g^7) + (B^2*d^6
*i^3*(b*(a*(b*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2
*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c))/(3*d^2)) - a*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2))
+ (b^3*c^2 + 6*a^2*b*d^2 - 7*a*b^2*c*d)/(10*d^3)) - (b^4*c^3 - 15*a^3*b*d^3 + 21*a^2*b^2*c*d^2 - 7*a*b^3*c^2*d
)/(10*d^4) + b*(b*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3
- b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b*d^4)) + a*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) +
(a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d - b*c))/(3*d^2)) + (15*a^3*
d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(20*d^4))) - (b^5*c^3 - 15*a^3*b^2*d^3 + 21*a^2*b^3*c*d^2 - 7*
a*b^4*c^2*d)/(6*d^4) + a*((2*(b^4*c^2 + 6*a^2*b^2*d^2 - 7*a*b^3*c*d))/(15*d^3) + b*(b*(b*((6*a^2*d^2 + b^2*c^2
- 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(15*d^3) + (a*(a*d -
b*c))/(3*d^2)) - a*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c^2 + 6*a^2*b*d^2 - 7*a*b^2*c*d
)/(10*d^3)) - a*(b*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c - a*b^2*d)/(6*d^2)))))/(30*b^4
*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (B*i^3*(60*A*b^3*c^3 - 6*B*a^3*d^3 + 11*B*b^3*c^3
- 8*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(180*b^5*d*g^7) + (B^2*d^6*i^3*(a*(a*(a*(a*((6*a^2*d^2 + b^2*c^2 - 7*a*
b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (15*a^3*d^3 - b^3*c^3 + 7*a*b^2*c^2*d - 21*a^2*b*c*d^2)/(60*b
*d^4)) + (20*a^4*d^4 + b^4*c^4 + 21*a^2*b^2*c^2*d^2 - 7*a*b^3*c^3*d - 35*a^3*b*c*d^3)/(60*b*d^5)) + (15*a^5*d^
5 - b^5*c^5 - 21*a^2*b^3*c^3*d^2 + 35*a^3*b^2*c^2*d^3 + 7*a*b^4*c^4*d - 35*a^4*b*c*d^4)/(30*b*d^6)) + (6*a^6*d
^6 + b^6*c^6 + 21*a^2*b^4*c^4*d^2 - 35*a^3*b^3*c^3*d^3 + 35*a^4*b^2*c^2*d^4 - 7*a*b^5*c^5*d - 21*a^5*b*c*d^5)/
(6*b*d^7)))/(30*b^4*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*d^6*i^3*x^5*((b^5*c - a*b^
4*d)/(6*d^2) + b*(b*(b*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c - a*b^2*d)/(6*d^2)) + (b^4
*c - a*b^3*d)/(6*d^2))))/(30*b^4*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*d^6*i^3*x^4*(
(b^5*c^2 + 6*a^2*b^3*d^2 - 7*a*b^4*c*d)/(6*d^3) + b*((2*(b^4*c^2 + 6*a^2*b^2*d^2 - 7*a*b^3*c*d))/(15*d^3) + b*
(b*(b*((6*a^2*d^2 + b^2*c^2 - 7*a*b*c*d)/(30*b*d^3) + (a*(a*d - b*c))/(6*b*d^2)) + (6*a^2*d^2 + b^2*c^2 - 7*a*
b*c*d)/(15*d^3) + (a*(a*d - b*c))/(3*d^2)) - a*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c^2
+ 6*a^2*b*d^2 - 7*a*b^2*c*d)/(10*d^3)) - a*(b*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c - a
*b^2*d)/(6*d^2))) - a*(b*(b*((b^2*c - a*b*d)/(6*d^2) - (b*(a*d - b*c))/(3*d^2)) + (b^3*c - a*b^2*d)/(6*d^2)) +
(b^4*c - a*b^3*d)/(6*d^2))))/(30*b^4*g^7*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((6*a^5*x)/d
+ a^6/(b*d) + (b^5*x^6)/d + (15*a^4*b*x^2)/d + (6*a*b^4*x^5)/d + (20*a^3*b^2*x^3)/d + (15*a^2*b^3*x^4)/d) - (B
*d^6*i^3*atan((B*d^6*i^3*(60*A + 37*B)*(1800*b^7*c^3*g^7 + 1800*a^3*b^4*d^3*g^7 - 1800*a*b^6*c^2*d*g^7 - 1800*
a^2*b^5*c*d^2*g^7)*1i)/(1800*b^4*g^7*(37*B^2*d^6*i^3 + 60*A*B*d^6*i^3)*(a*d - b*c)^3) + (B*d^7*i^3*x*(60*A + 3
7*B)*(b^6*c^2*g^7 + a^2*b^4*d^2*g^7 - 2*a*b^5*c*d*g^7)*2i)/(b^3*g^7*(37*B^2*d^6*i^3 + 60*A*B*d^6*i^3)*(a*d - b
*c)^3))*(60*A + 37*B)*1i)/(900*b^4*g^7*(a*d - b*c)^3)