\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) [150]

Optimal result

Integrand size = 43, antiderivative size = 477 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {B d^4 n (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 n (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d n (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B n (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {2 b B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2561, 45, 2372, 2338} \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {b^4 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^5}+\frac {d^4 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac {4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac {b^4 B n (c+d x)^3}{9 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {b^3 B d n (c+d x)^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 B d^2 n (c+d x)}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac {B d^4 n (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5}+\frac {2 b B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^4 i^2 (b c-a d)^5} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2561

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x)^4 \left (A+B \log \left (e x^n\right )\right )}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \\ & = \frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}-\frac {(B n) \text {Subst}\left (\int \left (d^4-\frac {b^4}{3 x^4}+\frac {2 b^3 d}{x^3}-\frac {6 b^2 d^2}{x^2}-\frac {4 b d^3 \log (x)}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \\ & = -\frac {B d^4 n (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 n (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d n (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B n (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {\left (4 b B d^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \\ & = -\frac {B d^4 n (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 n (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d n (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B n (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {2 b B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.72 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.15 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {\frac {b B (b c-a d)^3 n}{(a+b x)^3}-\frac {6 b B d (b c-a d)^2 n}{(a+b x)^2}+\frac {27 b^2 B c d^2 n}{a+b x}-\frac {27 a b B d^3 n}{a+b x}+\frac {12 b B d^2 (b c-a d) n}{a+b x}-\frac {9 b B c d^3 n}{c+d x}+\frac {9 a B d^4 n}{c+d x}+30 b B d^3 n \log (a+b x)+\frac {3 b (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3}-\frac {9 b d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {27 b d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}-\frac {9 d^3 (-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+36 b d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-30 b B d^3 n \log (c+d x)-36 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-18 b B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+18 b B d^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{9 (b c-a d)^5 g^4 i^2} \]

[In]

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

[Out]

-1/9*((b*B*(b*c - a*d)^3*n)/(a + b*x)^3 - (6*b*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (27*b^2*B*c*d^2*n)/(a + b*x)
- (27*a*b*B*d^3*n)/(a + b*x) + (12*b*B*d^2*(b*c - a*d)*n)/(a + b*x) - (9*b*B*c*d^3*n)/(c + d*x) + (9*a*B*d^4*
n)/(c + d*x) + 30*b*B*d^3*n*Log[a + b*x] + (3*b*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x
)^3 - (9*b*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (27*b*d^2*(b*c - a*d)*(A + B*
Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (9*d^3*(-(b*c) + a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c
+ d*x) + 36*b*d^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 30*b*B*d^3*n*Log[c + d*x] - 36*b*d^3*(
A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 18*b*B*d^3*n*(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*B*d^3*n*((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*c - a*d)^5*g^4*i^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1909\) vs. \(2(473)=946\).

Time = 44.86 (sec) , antiderivative size = 1910, normalized size of antiderivative = 4.00

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1910\)

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x,method=_RETURNVERBOSE)

[Out]

1/9*(-90*A*x*a^6*b^3*c^5*d^2*n+45*A*x*a^5*b^4*c^6*d*n+54*B*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c^5*d^2*n-18*B*ln
(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^6*d*n+18*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b^4*c^4*d^3-27*B*x^3*a^7*b^2*
c^2*d^5*n^2+126*B*x^3*a^6*b^3*c^3*d^4*n^2-77*B*x^3*a^5*b^4*c^4*d^3*n^2-27*B*x^3*a^4*b^5*c^5*d^2*n^2+6*B*x^3*a^
3*b^6*c^6*d*n^2+108*A*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^3*d^4+36*A*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4
*c^4*d^3+27*A*x^3*a^7*b^2*c^2*d^5*n+63*A*x^3*a^6*b^3*c^3*d^4*n-96*A*x^3*a^5*b^4*c^4*d^3*n+9*A*x^3*a^3*b^6*c^6*
d*n+9*A*x*a^9*c^2*d^5*n-9*A*x*a^4*b^5*c^7*n+18*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*b*c^4*d^3-9*B*ln(e*((b*x+a)/(
d*x+c))^n)*a^9*c^3*d^4*n+3*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^7*n+36*A*ln(e*((b*x+a)/(d*x+c))^n)*a^8*b*c^4*
d^3-B*x^3*a^2*b^7*c^7*n^2-3*A*x^3*a^2*b^7*c^7*n-3*B*x^2*a^3*b^6*c^7*n^2-9*A*x^2*a^3*b^6*c^7*n-9*B*x*a^9*c^2*d^
5*n^2-3*B*x*a^4*b^5*c^7*n^2+54*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b^2*c^3*d^4+54*B*x^2*ln(e*((b*x+a)/(d*x+c
))^n)^2*a^6*b^3*c^4*d^3-27*B*x^2*a^8*b*c^2*d^5*n^2+81*B*x^2*a^7*b^2*c^3*d^4*n^2+27*B*x^2*a^6*b^3*c^4*d^3*n^2-1
02*B*x^2*a^5*b^4*c^5*d^2*n^2+24*B*x^2*a^4*b^5*c^6*d*n^2+108*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c^3*d^4+10
8*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^4*d^3+27*A*x^2*a^8*b*c^2*d^5*n+27*A*x^2*a^7*b^2*c^3*d^4*n-90*A*x^2
*a^5*b^4*c^5*d^2*n+45*A*x^2*a^4*b^5*c^6*d*n+18*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*b*c^3*d^4+54*B*x*ln(e*((b*x
+a)/(d*x+c))^n)^2*a^7*b^2*c^4*d^3+9*B*x*a^8*b*c^3*d^4*n^2+54*B*x*a^7*b^2*c^4*d^3*n^2-72*B*x*a^6*b^3*c^5*d^2*n^
2+21*B*x*a^5*b^4*c^6*d*n^2+18*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b^4*c^3*d^4-9*B*x^4*a^6*b^3*c^2*d^5*n^2+55
*B*x^4*a^5*b^4*c^3*d^4*n^2-54*B*x^4*a^4*b^5*c^4*d^3*n^2+9*B*x^4*a^3*b^6*c^5*d^2*n^2-B*x^4*a^2*b^7*c^6*d*n^2+36
*A*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^3*d^4+9*A*x^4*a^6*b^3*c^2*d^5*n+30*A*x^4*a^5*b^4*c^3*d^4*n-54*A*x^4
*a^4*b^5*c^4*d^3*n+18*A*x^4*a^3*b^6*c^5*d^2*n-3*A*x^4*a^2*b^7*c^6*d*n+54*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6
*b^3*c^3*d^4+36*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^8*b*c^3*d^4+108*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c^4*d^3-
9*A*x*a^8*b*c^3*d^4*n+54*A*x*a^7*b^2*c^4*d^3*n+54*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^3*d^4*n+66*B*x^3*l
n(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^4*d^3*n+162*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^4*d^3*n+18*B*x^2*ln(e
*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^5*d^2*n-36*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^8*b*c^3*d^4*n+108*B*x*ln(e*((b*x+a)
/(d*x+c))^n)*a^7*b^2*c^4*d^3*n+54*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^5*d^2*n-6*B*x*ln(e*((b*x+a)/(d*x+c))
^n)*a^5*b^4*c^6*d*n+30*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^3*d^4*n)/i^2/g^4/(d*x+c)/(b*x+a)^3/(a*d-b*c)^
2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/c^3/a^5/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1458 vs. \(2 (473) = 946\).

Time = 0.43 (sec) , antiderivative size = 1458, normalized size of antiderivative = 3.06 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

-1/9*(3*A*b^4*c^4 - 18*A*a*b^3*c^3*d + 54*A*a^2*b^2*c^2*d^2 - 30*A*a^3*b*c*d^3 - 9*A*a^4*d^4 + 6*(6*A*b^4*c*d^
3 - 6*A*a*b^3*d^4 + 5*(B*b^4*c*d^3 - B*a*b^3*d^4)*n)*x^3 + 3*(6*A*b^4*c^2*d^2 + 24*A*a*b^3*c*d^3 - 30*A*a^2*b^
2*d^4 + (11*B*b^4*c^2*d^2 + 8*B*a*b^3*c*d^3 - 19*B*a^2*b^2*d^4)*n)*x^2 + 18*(B*b^4*d^4*n*x^4 + B*a^3*b*c*d^3*n
+ (B*b^4*c*d^3 + 3*B*a*b^3*d^4)*n*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*n*x^2 + (3*B*a^2*b^2*c*d^3 + B*a^3*
b*d^4)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^4*c^4 - 9*B*a*b^3*c^3*d + 54*B*a^2*b^2*c^2*d^2 - 55*B*a^3*b*c*d^
3 + 9*B*a^4*d^4)*n - (6*A*b^4*c^3*d - 54*A*a*b^3*c^2*d^2 - 18*A*a^2*b^2*c*d^3 + 66*A*a^3*b*d^4 + (5*B*b^4*c^3*
d - 81*B*a*b^3*c^2*d^2 + 57*B*a^2*b^2*c*d^3 + 19*B*a^3*b*d^4)*n)*x + 3*(B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*B*a^2
*b^2*c^2*d^2 - 10*B*a^3*b*c*d^3 - 3*B*a^4*d^4 + 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 + 6*(B*b^4*c^2*d^2 + 4*B*a*
b^3*c*d^3 - 5*B*a^2*b^2*d^4)*x^2 - 2*(B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2 - 3*B*a^2*b^2*c*d^3 + 11*B*a^3*b*d^4)*x
+ 12*(B*b^4*d^4*x^4 + B*a^3*b*c*d^3 + (B*b^4*c*d^3 + 3*B*a*b^3*d^4)*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*x^
2 + (3*B*a^2*b^2*c*d^3 + B*a^3*b*d^4)*x)*log((b*x + a)/(d*x + c)))*log(e) + 3*(12*A*a^3*b*c*d^3 + 2*(5*B*b^4*d
^4*n + 6*A*b^4*d^4)*x^4 + 2*(6*A*b^4*c*d^3 + 18*A*a*b^3*d^4 + (11*B*b^4*c*d^3 + 9*B*a*b^3*d^4)*n)*x^3 + 6*(6*A
*a*b^3*c*d^3 + 6*A*a^2*b^2*d^4 + (B*b^4*c^2*d^2 + 9*B*a*b^3*c*d^3)*n)*x^2 + (B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*
B*a^2*b^2*c^2*d^2 - 3*B*a^4*d^4)*n + 2*(18*A*a^2*b^2*c*d^3 + 6*A*a^3*b*d^4 - (B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2
- 18*B*a^2*b^2*c*d^3 + 6*B*a^3*b*d^4)*n)*x)*log((b*x + a)/(d*x + c)))/((b^8*c^5*d - 5*a*b^7*c^4*d^2 + 10*a^2*b
^6*c^3*d^3 - 10*a^3*b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d^6)*g^4*i^2*x^4 + (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^
2*b^6*c^4*d^2 + 20*a^3*b^5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*g^4*i^2*x^3 + 3*(a
*b^7*c^6 - 4*a^2*b^6*c^5*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c*d^5 - a^7*b*d^6)*g^4*i^2*x^2
+ (3*a^2*b^6*c^6 - 14*a^3*b^5*c^5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*a^6*b^2*c^2*d^4 + 2*a^7*b*c*
d^5 - a^8*d^6)*g^4*i^2*x + (a^3*b^5*c^6 - 5*a^4*b^4*c^5*d + 10*a^5*b^3*c^4*d^2 - 10*a^6*b^2*c^3*d^3 + 5*a^7*b*
c^2*d^4 - a^8*c*d^5)*g^4*i^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4/(d*i*x+c*i)**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2563 vs. \(2 (473) = 946\).

Time = 0.36 (sec) , antiderivative size = 2563, normalized size of antiderivative = 5.37 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

-1/3*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x
^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3
*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a
^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2
*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a
^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6
*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a
^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*
b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) -
1/9*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 - a*b^3*d^4)*x
^3 + 3*(11*b^4*c^2*d^2 + 8*a*b^3*c*d^3 - 19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 - 18*(b^4
*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3
+ a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c^3*d - 81*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 + 19*a^3*b*d^4)*x + 30*(b^
4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3
+ a^3*b*d^4)*x)*log(b*x + a) - 6*(5*b^4*d^4*x^4 + 5*a^3*b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 15*(a*b^3
*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*c*d^3 + a^3*b*d^4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*log(d*x +
c))*B*n/(a^3*b^5*c^6*g^4*i^2 - 5*a^4*b^4*c^5*d*g^4*i^2 + 10*a^5*b^3*c^4*d^2*g^4*i^2 - 10*a^6*b^2*c^3*d^3*g^4*i
^2 + 5*a^7*b*c^2*d^4*g^4*i^2 - a^8*c*d^5*g^4*i^2 + (b^8*c^5*d*g^4*i^2 - 5*a*b^7*c^4*d^2*g^4*i^2 + 10*a^2*b^6*c
^3*d^3*g^4*i^2 - 10*a^3*b^5*c^2*d^4*g^4*i^2 + 5*a^4*b^4*c*d^5*g^4*i^2 - a^5*b^3*d^6*g^4*i^2)*x^4 + (b^8*c^6*g^
4*i^2 - 2*a*b^7*c^5*d*g^4*i^2 - 5*a^2*b^6*c^4*d^2*g^4*i^2 + 20*a^3*b^5*c^3*d^3*g^4*i^2 - 25*a^4*b^4*c^2*d^4*g^
4*i^2 + 14*a^5*b^3*c*d^5*g^4*i^2 - 3*a^6*b^2*d^6*g^4*i^2)*x^3 + 3*(a*b^7*c^6*g^4*i^2 - 4*a^2*b^6*c^5*d*g^4*i^2
+ 5*a^3*b^5*c^4*d^2*g^4*i^2 - 5*a^5*b^3*c^2*d^4*g^4*i^2 + 4*a^6*b^2*c*d^5*g^4*i^2 - a^7*b*d^6*g^4*i^2)*x^2 +
(3*a^2*b^6*c^6*g^4*i^2 - 14*a^3*b^5*c^5*d*g^4*i^2 + 25*a^4*b^4*c^4*d^2*g^4*i^2 - 20*a^5*b^3*c^3*d^3*g^4*i^2 +
5*a^6*b^2*c^2*d^4*g^4*i^2 + 2*a^7*b*c*d^5*g^4*i^2 - a^8*d^6*g^4*i^2)*x) - 1/3*A*((12*b^3*d^3*x^3 + b^3*c^3 - 5
*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 1
1*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4
+ (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2
*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g
^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^
5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12
*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*
d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*
a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))

Giac [A] (verification not implemented)

none

Time = 180.85 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.84 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} n - \frac {3 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, B b^{2} n - \frac {9 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 6 \, B b^{2} \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B b d \log \left (e\right )}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} - \frac {18 \, {\left (b x + a\right )} A b d}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

-1/18*(6*(B*b^2*n - 3*(b*x + a)*B*b*d*n/(d*x + c) + 3*(b*x + a)^2*B*d^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c)
)/((b*x + a)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*x + a)^3*a*b*c*d*g^4*i^2/(d*x + c)^3 + (b*x + a)^3*a^2*d^2*g
^4*i^2/(d*x + c)^3) + (2*B*b^2*n - 9*(b*x + a)*B*b*d*n/(d*x + c) + 18*(b*x + a)^2*B*d^2*n/(d*x + c)^2 + 6*B*b^
2*log(e) - 18*(b*x + a)*B*b*d*log(e)/(d*x + c) + 18*(b*x + a)^2*B*d^2*log(e)/(d*x + c)^2 + 6*A*b^2 - 18*(b*x +
a)*A*b*d/(d*x + c) + 18*(b*x + a)^2*A*d^2/(d*x + c)^2)/((b*x + a)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*x + a)
^3*a*b*c*d*g^4*i^2/(d*x + c)^3 + (b*x + a)^3*a^2*d^2*g^4*i^2/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d
)^2)^2

Mupad [B] (verification not implemented)

Time = 6.58 (sec) , antiderivative size = 1665, normalized size of antiderivative = 3.49 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*B*b*d^3*log(e*((a + b*x)/(c + d*x))^n)^2)/(g^4*i^2*n*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - log(e
*((a + b*x)/(c + d*x))^n)*(((B*(3*a*d + b*c))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B*b*d*x)/(3*(a^2*d^2 +
b^2*c^2 - 2*a*b*c*d)))/(x^3*(b^3*c*g^4*i^2 + 3*a*b^2*d*g^4*i^2) + x^2*(3*a*b^2*c*g^4*i^2 + 3*a^2*b*d*g^4*i^2)
+ x*(a^3*d*g^4*i^2 + 3*a^2*b*c*g^4*i^2) + a^3*c*g^4*i^2 + b^3*d*g^4*i^2*x^4) + (4*B*b*d^3*(x*((a*d + b*c)*((a*
g^4*i^2*n*(a*d - b*c))/(2*d) + (g^4*i^2*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2)) + (a*b*c*g^4*i^2*n*(a*d - b*c))/
d) + x^2*(b*d*((a*g^4*i^2*n*(a*d - b*c))/(2*d) + (g^4*i^2*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2)) + (b*g^4*i^2*n
*(a*d + b*c)*(a*d - b*c))/d) + a*c*((a*g^4*i^2*n*(a*d - b*c))/(2*d) + (g^4*i^2*n*(a*d - b*c)*(2*a*d - b*c))/(2
*d^2)) + b^2*g^4*i^2*n*x^3*(a*d - b*c)))/(g^4*i^2*n*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^3*(b^3*c*
g^4*i^2 + 3*a*b^2*d*g^4*i^2) + x^2*(3*a*b^2*c*g^4*i^2 + 3*a^2*b*d*g^4*i^2) + x*(a^3*d*g^4*i^2 + 3*a^2*b*c*g^4*
i^2) + a^3*c*g^4*i^2 + b^3*d*g^4*i^2*x^4))) - (b*d^3*atan((b*d^3*((a^5*d^5*g^4*i^2 + b^5*c^5*g^4*i^2 - 3*a*b^4
*c^4*d*g^4*i^2 - 3*a^4*b*c*d^4*g^4*i^2 + 2*a^2*b^3*c^3*d^2*g^4*i^2 + 2*a^3*b^2*c^2*d^3*g^4*i^2)/(a^4*d^4*g^4*i
^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4*a^3*b*c*d^3*g^4*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2) + 2*b*d*x)*(
6*A + 5*B*n)*(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4*a^3*b*c*d^3*g^4*i^2 + 6*a^2*b^2*c^
2*d^2*g^4*i^2)*2i)/(g^4*i^2*(12*A*b*d^3 + 10*B*b*d^3*n)*(a*d - b*c)^5))*(6*A + 5*B*n)*4i)/(3*g^4*i^2*(a*d - b*
c)^5) - ((9*A*a^3*d^3 + 3*A*b^3*c^3 - 9*B*a^3*d^3*n + B*b^3*c^3*n - 15*A*a*b^2*c^2*d + 39*A*a^2*b*c*d^2 - 8*B*
a*b^2*c^2*d*n + 46*B*a^2*b*c*d^2*n)/(3*(a*d - b*c)) + (2*x^3*(6*A*b^3*d^3 + 5*B*b^3*d^3*n))/(a*d - b*c) + (x*(
66*A*a^2*b*d^3 - 6*A*b^3*c^2*d + 48*A*a*b^2*c*d^2 + 19*B*a^2*b*d^3*n - 5*B*b^3*c^2*d*n + 76*B*a*b^2*c*d^2*n))/
(3*(a*d - b*c)) + (x^2*(30*A*a*b^2*d^3 + 6*A*b^3*c*d^2 + 19*B*a*b^2*d^3*n + 11*B*b^3*c*d^2*n))/(a*d - b*c))/(x
*(3*a^6*d^4*g^4*i^2 - 9*a^2*b^4*c^4*g^4*i^2 + 24*a^3*b^3*c^3*d*g^4*i^2 - 18*a^4*b^2*c^2*d^2*g^4*i^2) - x^2*(9*
a*b^5*c^4*g^4*i^2 - 9*a^5*b*d^4*g^4*i^2 - 18*a^2*b^4*c^3*d*g^4*i^2 + 18*a^4*b^2*c*d^3*g^4*i^2) - x^3*(3*b^6*c^
4*g^4*i^2 - 9*a^4*b^2*d^4*g^4*i^2 + 24*a^3*b^3*c*d^3*g^4*i^2 - 18*a^2*b^4*c^2*d^2*g^4*i^2) + x^4*(3*a^3*b^3*d^
4*g^4*i^2 - 3*b^6*c^3*d*g^4*i^2 + 9*a*b^5*c^2*d^2*g^4*i^2 - 9*a^2*b^4*c*d^3*g^4*i^2) - 3*a^3*b^3*c^4*g^4*i^2 +
3*a^6*c*d^3*g^4*i^2 + 9*a^4*b^2*c^3*d*g^4*i^2 - 9*a^5*b*c^2*d^2*g^4*i^2)