3.2.58 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)^3} \, dx\) [158]

Optimal. Leaf size=587 \[ \frac {B d^5 n (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {5 b B d^4 n (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 B d^2 n (c+d x)}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B d n (c+d x)^2}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 B n (c+d x)^3}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {d^5 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^6 g^4 i^3}+\frac {5 b^2 B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^6 g^4 i^3} \]

[Out]

1/4*B*d^5*n*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-5*b*B*d^4*n*(b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*B
*d^2*n*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/4*b^4*B*d*n*(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-1/9*b^5*B*n
*(d*x+c)^3/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-1/2*d^5*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i
^3/(d*x+c)^2+5*b*d^4*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*d^2*(d*x+c)*(
A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^4*d*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n)
)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-1/3*b^5*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+a
)^3-10*b^2*d^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+b*c)^6/g^4/i^3+5*b^2*B*d^3*n*ln((b*x+
a)/(d*x+c))^2/(-a*d+b*c)^6/g^4/i^3

________________________________________________________________________________________

Rubi [A]
time = 0.26, antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2561, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {b^5 (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^3 (a+b x)^3 (b c-a d)^6}+\frac {5 b^4 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i^3 (a+b x)^2 (b c-a d)^6}-\frac {10 b^3 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^3 (a+b x) (b c-a d)^6}-\frac {10 b^2 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^3 (b c-a d)^6}-\frac {d^5 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i^3 (c+d x)^2 (b c-a d)^6}+\frac {5 b 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^3 (c+d x) (b c-a d)^6}-\frac {b^5 B n (c+d x)^3}{9 g^4 i^3 (a+b x)^3 (b c-a d)^6}+\frac {5 b^4 B d n (c+d x)^2}{4 g^4 i^3 (a+b x)^2 (b c-a d)^6}-\frac {10 b^3 B d^2 n (c+d x)}{g^4 i^3 (a+b x) (b c-a d)^6}+\frac {5 b^2 B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^4 i^3 (b c-a d)^6}+\frac {B d^5 n (a+b x)^2}{4 g^4 i^3 (c+d x)^2 (b c-a d)^6}-\frac {5 b B d^4 n (a+b x)}{g^4 i^3 (c+d x) (b c-a d)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*d^5*n*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (5*b*B*d^4*n*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(
c + d*x)) - (10*b^3*B*d^2*n*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*n*(c + d*x)^2)/(4*(b*c -
a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*B*n*(c + d*x)^3)/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*
(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)
/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c -
a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((
b*c - a*d)^6*g^4*i^3) + (5*b^2*B*d^3*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(158 c+158 d x)^3 (a g+b g x)^4} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3944312 (b c-a d)^3 g^4 (a+b x)^4}-\frac {3 b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3944312 (b c-a d)^4 g^4 (a+b x)^3}+\frac {3 b^3 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)^2}-\frac {5 b^3 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3944312 (b c-a d)^4 g^4 (c+d x)^3}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)^2}+\frac {5 b^2 d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (5 b^3 d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 d^4\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (3 b^3 d^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{1972156 (b c-a d)^5 g^4}+\frac {\left (b d^4\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{986078 (b c-a d)^5 g^4}-\frac {\left (3 b^3 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3944312 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3944312 (b c-a d)^4 g^4}+\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3944312 (b c-a d)^3 g^4}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}-\frac {\left (5 b^2 B d^3 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (3 b^2 B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{1972156 (b c-a d)^5 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{986078 (b c-a d)^5 g^4}-\frac {\left (3 b^2 B d n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7888624 (b c-a d)^4 g^4}+\frac {\left (B d^3 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7888624 (b c-a d)^4 g^4}+\frac {\left (b^2 B n\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{11832936 (b c-a d)^3 g^4}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{1972156 (b c-a d)^6 g^4}-\frac {\left (5 b^2 B d^3 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (3 b^2 B d^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{1972156 (b c-a d)^4 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{986078 (b c-a d)^4 g^4}-\frac {\left (3 b^2 B d n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{7888624 (b c-a d)^3 g^4}+\frac {\left (B d^3 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7888624 (b c-a d)^3 g^4}+\frac {\left (b^2 B n\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{11832936 (b c-a d)^2 g^4}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^3 B d^3 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}-\frac {\left (5 b^3 B d^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}-\frac {\left (5 b^2 B d^4 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^4 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (3 b^2 B d^2 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{1972156 (b c-a d)^4 g^4}+\frac {\left (b B d^3 n\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{986078 (b c-a d)^4 g^4}-\frac {\left (3 b^2 B d n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7888624 (b c-a d)^3 g^4}+\frac {\left (B d^3 n\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7888624 (b c-a d)^3 g^4}+\frac {\left (b^2 B n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{11832936 (b c-a d)^2 g^4}\\ &=-\frac {b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac {11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac {47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac {B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac {9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^3 B d^3 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^4 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}\\ &=-\frac {b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac {11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac {47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac {B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac {9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log ^2(a+b x)}{3944312 (b c-a d)^6 g^4}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log ^2(c+d x)}{3944312 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{1972156 (b c-a d)^6 g^4}+\frac {\left (5 b^2 B d^3 n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{1972156 (b c-a d)^6 g^4}\\ &=-\frac {b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac {11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac {47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac {B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac {9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log ^2(a+b x)}{3944312 (b c-a d)^6 g^4}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac {3 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac {5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac {5 b^2 B d^3 n \log ^2(c+d x)}{3944312 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}-\frac {5 b^2 B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.19, size = 671, normalized size = 1.14 \begin {gather*} -\frac {\frac {4 b^2 B (b c-a d)^3 n}{(a+b x)^3}-\frac {33 b^2 B d (b c-a d)^2 n}{(a+b x)^2}+\frac {216 b^3 B c d^2 n}{a+b x}-\frac {216 a b^2 B d^3 n}{a+b x}+\frac {66 b^2 B d^2 (b c-a d) n}{a+b x}-\frac {9 B d^3 (b c-a d)^2 n}{(c+d x)^2}-\frac {144 b^2 B c d^3 n}{c+d x}+\frac {144 a b B d^4 n}{c+d x}-\frac {18 b B d^3 (b c-a d) n}{c+d x}+120 b^2 B d^3 n \log (a+b x)+\frac {12 b^2 (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 {54 b^2 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 {216 b^2 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 {18 d^3 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}+\frac {144 b 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}+360 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-120 b^2 B d^3 n \log (c+d x)-360 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-180 b^2 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 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+180 b^2 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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{36 (b c-a d)^6 g^4 i^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*((4*b^2*B*(b*c - a*d)^3*n)/(a + b*x)^3 - (33*b^2*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (216*b^3*B*c*d^2*n)/
(a + b*x) - (216*a*b^2*B*d^3*n)/(a + b*x) + (66*b^2*B*d^2*(b*c - a*d)*n)/(a + b*x) - (9*B*d^3*(b*c - a*d)^2*n)
/(c + d*x)^2 - (144*b^2*B*c*d^3*n)/(c + d*x) + (144*a*b*B*d^4*n)/(c + d*x) - (18*b*B*d^3*(b*c - a*d)*n)/(c + d
*x) + 120*b^2*B*d^3*n*Log[a + b*x] + (12*b^2*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^3
- (54*b^2*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (216*b^2*d^2*(b*c - a*d)*(A +
B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (18*d^3*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/
(c + d*x)^2 + (144*b*d^3*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 360*b^2*d^3*Log[a + b
*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 120*b^2*B*d^3*n*Log[c + d*x] - 360*b^2*d^3*(A + B*Log[e*((a + b*x
)/(c + d*x))^n])*Log[c + d*x] - 180*b^2*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)]) + 180*b^2*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)^6*g^4*i^3)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right )^{4} \left (d i x +c i \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3673 vs. \(2 (545) = 1090\).
time = 1.11, size = 3673, normalized size = 6.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/6*B*(60*b^2*d^3*log(b*x + a)/((I*b^6*c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 1
5*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) - 60*b^2*d^3*log(d*x + c)/((I*b^6*c^6 - 6*I*a*b^5*c^5*
d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) + (
60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^
3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a*b^3*c^2*
d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((I*b^8*c^5*d^2 - 5*I*a*b^7*c^4*d^3 + 10*I*a^2*b^6*c^3*d^4 - 10*I*a^3
*b^5*c^2*d^5 + 5*I*a^4*b^4*c*d^6 - I*a^5*b^3*d^7)*g^4*x^5 + (2*I*b^8*c^6*d - 7*I*a*b^7*c^5*d^2 + 5*I*a^2*b^6*c
^4*d^3 + 10*I*a^3*b^5*c^3*d^4 - 20*I*a^4*b^4*c^2*d^5 + 13*I*a^5*b^3*c*d^6 - 3*I*a^6*b^2*d^7)*g^4*x^4 + (I*b^8*
c^7 + I*a*b^7*c^6*d - 17*I*a^2*b^6*c^5*d^2 + 35*I*a^3*b^5*c^4*d^3 - 25*I*a^4*b^4*c^3*d^4 - I*a^5*b^3*c^2*d^5 +
9*I*a^6*b^2*c*d^6 - 3*I*a^7*b*d^7)*g^4*x^3 + (3*I*a*b^7*c^7 - 9*I*a^2*b^6*c^6*d + I*a^3*b^5*c^5*d^2 + 25*I*a^
4*b^4*c^4*d^3 - 35*I*a^5*b^3*c^3*d^4 + 17*I*a^6*b^2*c^2*d^5 - I*a^7*b*c*d^6 - I*a^8*d^7)*g^4*x^2 + (3*I*a^2*b^
6*c^7 - 13*I*a^3*b^5*c^6*d + 20*I*a^4*b^4*c^5*d^2 - 10*I*a^5*b^3*c^4*d^3 - 5*I*a^6*b^2*c^3*d^4 + 7*I*a^7*b*c^2
*d^5 - 2*I*a^8*c*d^6)*g^4*x + (I*a^3*b^5*c^7 - 5*I*a^4*b^4*c^6*d + 10*I*a^5*b^3*c^5*d^2 - 10*I*a^6*b^2*c^4*d^3
+ 5*I*a^7*b*c^3*d^4 - I*a^8*c^2*d^5)*g^4))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/36*(-4*I*b^5*c^5 + 45*I
*a*b^4*c^4*d - 360*I*a^2*b^3*c^3*d^2 + 490*I*a^3*b^2*c^2*d^3 - 180*I*a^4*b*c*d^4 + 9*I*a^5*d^5 - 120*(I*b^5*c*
d^4 - I*a*b^4*d^5)*x^4 - 120*(3*I*b^5*c^2*d^3 - 2*I*a*b^4*c*d^4 - I*a^2*b^3*d^5)*x^3 - 20*(11*I*b^5*c^3*d^2 +
21*I*a*b^4*c^2*d^3 - 39*I*a^2*b^3*c*d^4 + 7*I*a^3*b^2*d^5)*x^2 - 180*(-I*b^5*d^5*x^5 - I*a^3*b^2*c^2*d^3 + (-2
*I*b^5*c*d^4 - 3*I*a*b^4*d^5)*x^4 + (-I*b^5*c^2*d^3 - 6*I*a*b^4*c*d^4 - 3*I*a^2*b^3*d^5)*x^3 + (-3*I*a*b^4*c^2
*d^3 - 6*I*a^2*b^3*c*d^4 - I*a^3*b^2*d^5)*x^2 + (-3*I*a^2*b^3*c^2*d^3 - 2*I*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 -
180*(-I*b^5*d^5*x^5 - I*a^3*b^2*c^2*d^3 + (-2*I*b^5*c*d^4 - 3*I*a*b^4*d^5)*x^4 + (-I*b^5*c^2*d^3 - 6*I*a*b^4*
c*d^4 - 3*I*a^2*b^3*d^5)*x^3 + (-3*I*a*b^4*c^2*d^3 - 6*I*a^2*b^3*c*d^4 - I*a^3*b^2*d^5)*x^2 + (-3*I*a^2*b^3*c^
2*d^3 - 2*I*a^3*b^2*c*d^4)*x)*log(d*x + c)^2 - 5*(-5*I*b^5*c^4*d + 108*I*a*b^4*c^3*d^2 - 78*I*a^2*b^3*c^2*d^3
- 52*I*a^3*b^2*c*d^4 + 27*I*a^4*b*d^5)*x - 120*(I*b^5*d^5*x^5 + I*a^3*b^2*c^2*d^3 + (2*I*b^5*c*d^4 + 3*I*a*b^4
*d^5)*x^4 + (I*b^5*c^2*d^3 + 6*I*a*b^4*c*d^4 + 3*I*a^2*b^3*d^5)*x^3 + (3*I*a*b^4*c^2*d^3 + 6*I*a^2*b^3*c*d^4 +
I*a^3*b^2*d^5)*x^2 + (3*I*a^2*b^3*c^2*d^3 + 2*I*a^3*b^2*c*d^4)*x)*log(b*x + a) - 120*(-I*b^5*d^5*x^5 - I*a^3*
b^2*c^2*d^3 + (-2*I*b^5*c*d^4 - 3*I*a*b^4*d^5)*x^4 + (-I*b^5*c^2*d^3 - 6*I*a*b^4*c*d^4 - 3*I*a^2*b^3*d^5)*x^3
+ (-3*I*a*b^4*c^2*d^3 - 6*I*a^2*b^3*c*d^4 - I*a^3*b^2*d^5)*x^2 + (-3*I*a^2*b^3*c^2*d^3 - 2*I*a^3*b^2*c*d^4)*x
+ 3*(I*b^5*d^5*x^5 + I*a^3*b^2*c^2*d^3 + (2*I*b^5*c*d^4 + 3*I*a*b^4*d^5)*x^4 + (I*b^5*c^2*d^3 + 6*I*a*b^4*c*d^
4 + 3*I*a^2*b^3*d^5)*x^3 + (3*I*a*b^4*c^2*d^3 + 6*I*a^2*b^3*c*d^4 + I*a^3*b^2*d^5)*x^2 + (3*I*a^2*b^3*c^2*d^3
+ 2*I*a^3*b^2*c*d^4)*x)*log(b*x + a))*log(d*x + c))*B*n/(a^3*b^6*c^8*g^4 - 6*a^4*b^5*c^7*d*g^4 + 15*a^5*b^4*c^
6*d^2*g^4 - 20*a^6*b^3*c^5*d^3*g^4 + 15*a^7*b^2*c^4*d^4*g^4 - 6*a^8*b*c^3*d^5*g^4 + a^9*c^2*d^6*g^4 + (b^9*c^6
*d^2*g^4 - 6*a*b^8*c^5*d^3*g^4 + 15*a^2*b^7*c^4*d^4*g^4 - 20*a^3*b^6*c^3*d^5*g^4 + 15*a^4*b^5*c^2*d^6*g^4 - 6*
a^5*b^4*c*d^7*g^4 + a^6*b^3*d^8*g^4)*x^5 + (2*b^9*c^7*d*g^4 - 9*a*b^8*c^6*d^2*g^4 + 12*a^2*b^7*c^5*d^3*g^4 + 5
*a^3*b^6*c^4*d^4*g^4 - 30*a^4*b^5*c^3*d^5*g^4 + 33*a^5*b^4*c^2*d^6*g^4 - 16*a^6*b^3*c*d^7*g^4 + 3*a^7*b^2*d^8*
g^4)*x^4 + (b^9*c^8*g^4 - 18*a^2*b^7*c^6*d^2*g^4 + 52*a^3*b^6*c^5*d^3*g^4 - 60*a^4*b^5*c^4*d^4*g^4 + 24*a^5*b^
4*c^3*d^5*g^4 + 10*a^6*b^3*c^2*d^6*g^4 - 12*a^7*b^2*c*d^7*g^4 + 3*a^8*b*d^8*g^4)*x^3 + (3*a*b^8*c^8*g^4 - 12*a
^2*b^7*c^7*d*g^4 + 10*a^3*b^6*c^6*d^2*g^4 + 24*a^4*b^5*c^5*d^3*g^4 - 60*a^5*b^4*c^4*d^4*g^4 + 52*a^6*b^3*c^3*d
^5*g^4 - 18*a^7*b^2*c^2*d^6*g^4 + a^9*d^8*g^4)*x^2 + (3*a^2*b^7*c^8*g^4 - 16*a^3*b^6*c^7*d*g^4 + 33*a^4*b^5*c^
6*d^2*g^4 - 30*a^5*b^4*c^5*d^3*g^4 + 5*a^6*b^3*c^4*d^4*g^4 + 12*a^7*b^2*c^3*d^5*g^4 - 9*a^8*b*c^2*d^6*g^4 + 2*
a^9*c*d^7*g^4)*x) + 1/6*A*(60*b^2*d^3*log(b*x + a)/((I*b^6*c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I
*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) - 60*b^2*d^3*log(d*x + c)/((I*b^6*
c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 +
I*a^6*d^6)*g^4) + (60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*
d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c
^3*d - 11*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((I*b^8*c^5*d^2 - 5*I*a*b^7*c^4*d^3 + 10*I*a^2*b^
6*c^3*d^4 - 10*I*a^3*b^5*c^2*d^5 + 5*I*a^4*b^4*...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1922 vs. \(2 (545) = 1090\).
time = 0.60, size = 1922, normalized size = 3.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/36*(12*(I*A + I*B)*b^5*c^5 + 90*(-I*A - I*B)*a*b^4*c^4*d + 360*(I*A + I*B)*a^2*b^3*c^3*d^2 + 120*(-I*A - I*
B)*a^3*b^2*c^2*d^3 + 180*(-I*A - I*B)*a^4*b*c*d^4 + 18*(I*A + I*B)*a^5*d^5 + 120*(3*(I*A + I*B)*b^5*c*d^4 + 3*
(-I*A - I*B)*a*b^4*d^5 + (I*B*b^5*c*d^4 - I*B*a*b^4*d^5)*n)*x^4 + 60*(9*(I*A + I*B)*b^5*c^2*d^3 + 6*(I*A + I*B
)*a*b^4*c*d^4 + 15*(-I*A - I*B)*a^2*b^3*d^5 + 2*(3*I*B*b^5*c^2*d^3 - 2*I*B*a*b^4*c*d^4 - I*B*a^2*b^3*d^5)*n)*x
^3 + 20*(6*(I*A + I*B)*b^5*c^3*d^2 + 63*(I*A + I*B)*a*b^4*c^2*d^3 + 36*(-I*A - I*B)*a^2*b^3*c*d^4 + 33*(-I*A -
I*B)*a^3*b^2*d^5 + (11*I*B*b^5*c^3*d^2 + 21*I*B*a*b^4*c^2*d^3 - 39*I*B*a^2*b^3*c*d^4 + 7*I*B*a^3*b^2*d^5)*n)*
x^2 + 180*(I*B*b^5*d^5*n*x^5 + I*B*a^3*b^2*c^2*d^3*n + (2*I*B*b^5*c*d^4 + 3*I*B*a*b^4*d^5)*n*x^4 + (I*B*b^5*c^
2*d^3 + 6*I*B*a*b^4*c*d^4 + 3*I*B*a^2*b^3*d^5)*n*x^3 + (3*I*B*a*b^4*c^2*d^3 + 6*I*B*a^2*b^3*c*d^4 + I*B*a^3*b^
2*d^5)*n*x^2 + (3*I*B*a^2*b^3*c^2*d^3 + 2*I*B*a^3*b^2*c*d^4)*n*x)*log((b*x + a)/(d*x + c))^2 - (-4*I*B*b^5*c^5
+ 45*I*B*a*b^4*c^4*d - 360*I*B*a^2*b^3*c^3*d^2 + 490*I*B*a^3*b^2*c^2*d^3 - 180*I*B*a^4*b*c*d^4 + 9*I*B*a^5*d^
5)*n + 5*(6*(-I*A - I*B)*b^5*c^4*d + 72*(I*A + I*B)*a*b^4*c^3*d^2 + 144*(I*A + I*B)*a^2*b^3*c^2*d^3 + 192*(-I*
A - I*B)*a^3*b^2*c*d^4 + 18*(-I*A - I*B)*a^4*b*d^5 + (-5*I*B*b^5*c^4*d + 108*I*B*a*b^4*c^3*d^2 - 78*I*B*a^2*b^
3*c^2*d^3 - 52*I*B*a^3*b^2*c*d^4 + 27*I*B*a^4*b*d^5)*n)*x + 6*(60*(I*A + I*B)*a^3*b^2*c^2*d^3 + 20*(I*B*b^5*d^
5*n + 3*(I*A + I*B)*b^5*d^5)*x^5 + 20*(5*I*B*b^5*c*d^4*n + 6*(I*A + I*B)*b^5*c*d^4 + 9*(I*A + I*B)*a*b^4*d^5)*
x^4 + 10*(6*(I*A + I*B)*b^5*c^2*d^3 + 36*(I*A + I*B)*a*b^4*c*d^4 + 18*(I*A + I*B)*a^2*b^3*d^5 + (11*I*B*b^5*c^
2*d^3 + 18*I*B*a*b^4*c*d^4 - 9*I*B*a^2*b^3*d^5)*n)*x^3 + 10*(18*(I*A + I*B)*a*b^4*c^2*d^3 + 36*(I*A + I*B)*a^2
*b^3*c*d^4 + 6*(I*A + I*B)*a^3*b^2*d^5 + (2*I*B*b^5*c^3*d^2 + 27*I*B*a*b^4*c^2*d^3 - 9*I*B*a^3*b^2*d^5)*n)*x^2
+ (2*I*B*b^5*c^5 - 15*I*B*a*b^4*c^4*d + 60*I*B*a^2*b^3*c^3*d^2 - 30*I*B*a^4*b*c*d^4 + 3*I*B*a^5*d^5)*n + 5*(3
6*(I*A + I*B)*a^2*b^3*c^2*d^3 + 24*(I*A + I*B)*a^3*b^2*c*d^4 + (-I*B*b^5*c^4*d + 12*I*B*a*b^4*c^3*d^2 + 36*I*B
*a^2*b^3*c^2*d^3 - 24*I*B*a^3*b^2*c*d^4 - 3*I*B*a^4*b*d^5)*n)*x)*log((b*x + a)/(d*x + c)))/((b^9*c^6*d^2 - 6*a
*b^8*c^5*d^3 + 15*a^2*b^7*c^4*d^4 - 20*a^3*b^6*c^3*d^5 + 15*a^4*b^5*c^2*d^6 - 6*a^5*b^4*c*d^7 + a^6*b^3*d^8)*g
^4*x^5 + (2*b^9*c^7*d - 9*a*b^8*c^6*d^2 + 12*a^2*b^7*c^5*d^3 + 5*a^3*b^6*c^4*d^4 - 30*a^4*b^5*c^3*d^5 + 33*a^5
*b^4*c^2*d^6 - 16*a^6*b^3*c*d^7 + 3*a^7*b^2*d^8)*g^4*x^4 + (b^9*c^8 - 18*a^2*b^7*c^6*d^2 + 52*a^3*b^6*c^5*d^3
- 60*a^4*b^5*c^4*d^4 + 24*a^5*b^4*c^3*d^5 + 10*a^6*b^3*c^2*d^6 - 12*a^7*b^2*c*d^7 + 3*a^8*b*d^8)*g^4*x^3 + (3*
a*b^8*c^8 - 12*a^2*b^7*c^7*d + 10*a^3*b^6*c^6*d^2 + 24*a^4*b^5*c^5*d^3 - 60*a^5*b^4*c^4*d^4 + 52*a^6*b^3*c^3*d
^5 - 18*a^7*b^2*c^2*d^6 + a^9*d^8)*g^4*x^2 + (3*a^2*b^7*c^8 - 16*a^3*b^6*c^7*d + 33*a^4*b^5*c^6*d^2 - 30*a^5*b
^4*c^5*d^3 + 5*a^6*b^3*c^4*d^4 + 12*a^7*b^2*c^3*d^5 - 9*a^8*b*c^2*d^6 + 2*a^9*c*d^7)*g^4*x + (a^3*b^6*c^8 - 6*
a^4*b^5*c^7*d + 15*a^5*b^4*c^6*d^2 - 20*a^6*b^3*c^5*d^3 + 15*a^7*b^2*c^4*d^4 - 6*a^8*b*c^3*d^5 + a^9*c^2*d^6)*
g^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4/(d*i*x+c*i)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

integrate((B*log(((b*x + a)/(d*x + c))^n*e) + A)/((b*g*x + a*g)^4*(I*d*x + I*c)^3), x)

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Mupad [B]
time = 10.22, size = 2400, normalized size = 4.09 \begin {gather*} \ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {x\,\left (\frac {5\,B\,\left (c\,b^2\,d+2\,a\,b\,d^2\right )\,\left (a\,d+b\,c\right )}{3\,{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}-\frac {5\,B\,b\,d}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {5\,B\,a\,b^2\,c\,d^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )+x^2\,\left (\frac {5\,B\,b\,d\,\left (c\,b^2\,d+2\,a\,b\,d^2\right )}{3\,{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {5\,B\,b^2\,d^2\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )-\frac {B\,\left (3\,a\,d+2\,b\,c\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {5\,B\,a\,c\,\left (c\,b^2\,d+2\,a\,b\,d^2\right )}{3\,{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {5\,B\,b^3\,d^3\,x^3}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}}{x\,\left (2\,d\,a^3\,c\,g^4\,i^3+3\,b\,a^2\,c^2\,g^4\,i^3\right )+x^2\,\left (a^3\,d^2\,g^4\,i^3+6\,a^2\,b\,c\,d\,g^4\,i^3+3\,a\,b^2\,c^2\,g^4\,i^3\right )+x^3\,\left (3\,a^2\,b\,d^2\,g^4\,i^3+6\,a\,b^2\,c\,d\,g^4\,i^3+b^3\,c^2\,g^4\,i^3\right )+x^4\,\left (2\,c\,b^3\,d\,g^4\,i^3+3\,a\,b^2\,d^2\,g^4\,i^3\right )+a^3\,c^2\,g^4\,i^3+b^3\,d^2\,g^4\,i^3\,x^5}+\frac {10\,B\,b^2\,d^3\,\left (x^2\,\left (\frac {g^4\,i^3\,n\,{\left (a\,d+b\,c\right )}^2\,\left (a\,d-b\,c\right )}{d}+2\,a\,b\,c\,g^4\,i^3\,n\,\left (a\,d-b\,c\right )\right )+b^2\,d\,g^4\,i^3\,n\,x^4\,\left (a\,d-b\,c\right )+\frac {a^2\,c^2\,g^4\,i^3\,n\,\left (a\,d-b\,c\right )}{d}+2\,b\,g^4\,i^3\,n\,x^3\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )+\frac {2\,a\,c\,g^4\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{g^4\,i^3\,n\,{\left (a\,d-b\,c\right )}^6\,\left (x\,\left (2\,d\,a^3\,c\,g^4\,i^3+3\,b\,a^2\,c^2\,g^4\,i^3\right )+x^2\,\left (a^3\,d^2\,g^4\,i^3+6\,a^2\,b\,c\,d\,g^4\,i^3+3\,a\,b^2\,c^2\,g^4\,i^3\right )+x^3\,\left (3\,a^2\,b\,d^2\,g^4\,i^3+6\,a\,b^2\,c\,d\,g^4\,i^3+b^3\,c^2\,g^4\,i^3\right )+x^4\,\left (2\,c\,b^3\,d\,g^4\,i^3+3\,a\,b^2\,d^2\,g^4\,i^3\right )+a^3\,c^2\,g^4\,i^3+b^3\,d^2\,g^4\,i^3\,x^5\right )}\right )+\frac {\frac {12\,A\,b^4\,c^4-18\,A\,a^4\,d^4+9\,B\,a^4\,d^4\,n+4\,B\,b^4\,c^4\,n+282\,A\,a^2\,b^2\,c^2\,d^2-78\,A\,a\,b^3\,c^3\,d+162\,A\,a^3\,b\,c\,d^3+319\,B\,a^2\,b^2\,c^2\,d^2\,n-41\,B\,a\,b^3\,c^3\,d\,n-171\,B\,a^3\,b\,c\,d^3\,n}{6\,\left (a\,d-b\,c\right )}+\frac {5\,x\,\left (18\,A\,a^3\,b\,d^4-6\,A\,b^4\,c^3\,d+66\,A\,a\,b^3\,c^2\,d^2+210\,A\,a^2\,b^2\,c\,d^3-27\,B\,a^3\,b\,d^4\,n-5\,B\,b^4\,c^3\,d\,n+103\,B\,a\,b^3\,c^2\,d^2\,n+25\,B\,a^2\,b^2\,c\,d^3\,n\right )}{6\,\left (a\,d-b\,c\right )}+\frac {20\,x^4\,\left (3\,A\,b^4\,d^4+B\,b^4\,d^4\,n\right )}{a\,d-b\,c}+\frac {10\,x^2\,\left (33\,A\,a^2\,b^2\,d^4+6\,A\,b^4\,c^2\,d^2-7\,B\,a^2\,b^2\,d^4\,n+11\,B\,b^4\,c^2\,d^2\,n+69\,A\,a\,b^3\,c\,d^3+32\,B\,a\,b^3\,c\,d^3\,n\right )}{3\,\left (a\,d-b\,c\right )}+\frac {10\,x^3\,\left (15\,A\,a\,b^3\,d^4+9\,A\,b^4\,c\,d^3+2\,B\,a\,b^3\,d^4\,n+6\,B\,b^4\,c\,d^3\,n\right )}{a\,d-b\,c}}{x^5\,\left (6\,a^4\,b^3\,d^6\,g^4\,i^3-24\,a^3\,b^4\,c\,d^5\,g^4\,i^3+36\,a^2\,b^5\,c^2\,d^4\,g^4\,i^3-24\,a\,b^6\,c^3\,d^3\,g^4\,i^3+6\,b^7\,c^4\,d^2\,g^4\,i^3\right )+x\,\left (12\,a^7\,c\,d^5\,g^4\,i^3-30\,a^6\,b\,c^2\,d^4\,g^4\,i^3+60\,a^4\,b^3\,c^4\,d^2\,g^4\,i^3-60\,a^3\,b^4\,c^5\,d\,g^4\,i^3+18\,a^2\,b^5\,c^6\,g^4\,i^3\right )+x^2\,\left (6\,a^7\,d^6\,g^4\,i^3+12\,a^6\,b\,c\,d^5\,g^4\,i^3-90\,a^5\,b^2\,c^2\,d^4\,g^4\,i^3+120\,a^4\,b^3\,c^3\,d^3\,g^4\,i^3-30\,a^3\,b^4\,c^4\,d^2\,g^4\,i^3-36\,a^2\,b^5\,c^5\,d\,g^4\,i^3+18\,a\,b^6\,c^6\,g^4\,i^3\right )+x^3\,\left (18\,a^6\,b\,d^6\,g^4\,i^3-36\,a^5\,b^2\,c\,d^5\,g^4\,i^3-30\,a^4\,b^3\,c^2\,d^4\,g^4\,i^3+120\,a^3\,b^4\,c^3\,d^3\,g^4\,i^3-90\,a^2\,b^5\,c^4\,d^2\,g^4\,i^3+12\,a\,b^6\,c^5\,d\,g^4\,i^3+6\,b^7\,c^6\,g^4\,i^3\right )+x^4\,\left (18\,a^5\,b^2\,d^6\,g^4\,i^3-60\,a^4\,b^3\,c\,d^5\,g^4\,i^3+60\,a^3\,b^4\,c^2\,d^4\,g^4\,i^3-30\,a\,b^6\,c^4\,d^2\,g^4\,i^3+12\,b^7\,c^5\,d\,g^4\,i^3\right )+6\,a^3\,b^4\,c^6\,g^4\,i^3+6\,a^7\,c^2\,d^4\,g^4\,i^3-24\,a^4\,b^3\,c^5\,d\,g^4\,i^3-24\,a^6\,b\,c^3\,d^3\,g^4\,i^3+36\,a^5\,b^2\,c^4\,d^2\,g^4\,i^3}-\frac {5\,B\,b^2\,d^3\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^4\,i^3\,n\,{\left (a\,d-b\,c\right )}^6}+\frac {b^2\,d^3\,\mathrm {atan}\left (\frac {b^2\,d^3\,\left (3\,A+B\,n\right )\,\left (\frac {a^6\,d^6\,g^4\,i^3-4\,a^5\,b\,c\,d^5\,g^4\,i^3+5\,a^4\,b^2\,c^2\,d^4\,g^4\,i^3-5\,a^2\,b^4\,c^4\,d^2\,g^4\,i^3+4\,a\,b^5\,c^5\,d\,g^4\,i^3-b^6\,c^6\,g^4\,i^3}{a^5\,d^5\,g^4\,i^3-5\,a^4\,b\,c\,d^4\,g^4\,i^3+10\,a^3\,b^2\,c^2\,d^3\,g^4\,i^3-10\,a^2\,b^3\,c^3\,d^2\,g^4\,i^3+5\,a\,b^4\,c^4\,d\,g^4\,i^3-b^5\,c^5\,g^4\,i^3}+2\,b\,d\,x\right )\,\left (a^5\,d^5\,g^4\,i^3-5\,a^4\,b\,c\,d^4\,g^4\,i^3+10\,a^3\,b^2\,c^2\,d^3\,g^4\,i^3-10\,a^2\,b^3\,c^3\,d^2\,g^4\,i^3+5\,a\,b^4\,c^4\,d\,g^4\,i^3-b^5\,c^5\,g^4\,i^3\right )\,10{}\mathrm {i}}{g^4\,i^3\,\left (30\,A\,b^2\,d^3+10\,B\,b^2\,d^3\,n\right )\,{\left (a\,d-b\,c\right )}^6}\right )\,\left (3\,A+B\,n\right )\,20{}\mathrm {i}}{3\,g^4\,i^3\,{\left (a\,d-b\,c\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log(e*((a + b*x)/(c + d*x))^n))/((a*g + b*g*x)^4*(c*i + d*i*x)^3),x)

[Out]

log(e*((a + b*x)/(c + d*x))^n)*((x*((5*B*(2*a*b*d^2 + b^2*c*d)*(a*d + b*c))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)
^2) - (5*B*b*d)/(6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (5*B*a*b^2*c*d^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + x
^2*((5*B*b*d*(2*a*b*d^2 + b^2*c*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (5*B*b^2*d^2*(a*d + b*c))/(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)^2) - (B*(3*a*d + 2*b*c))/(6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (5*B*a*c*(2*a*b*d^2 + b
^2*c*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (5*B*b^3*d^3*x^3)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2)/(x*(2*a^
3*c*d*g^4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2*(a^3*d^2*g^4*i^3 + 3*a*b^2*c^2*g^4*i^3 + 6*a^2*b*c*d*g^4*i^3) + x^3
*(b^3*c^2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6*a*b^2*c*d*g^4*i^3) + x^4*(2*b^3*c*d*g^4*i^3 + 3*a*b^2*d^2*g^4*i^3)
+ a^3*c^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5) + (10*B*b^2*d^3*(x^2*((g^4*i^3*n*(a*d + b*c)^2*(a*d - b*c))/d + 2*a*
b*c*g^4*i^3*n*(a*d - b*c)) + b^2*d*g^4*i^3*n*x^4*(a*d - b*c) + (a^2*c^2*g^4*i^3*n*(a*d - b*c))/d + 2*b*g^4*i^3
*n*x^3*(a*d + b*c)*(a*d - b*c) + (2*a*c*g^4*i^3*n*x*(a*d + b*c)*(a*d - b*c))/d))/(g^4*i^3*n*(a*d - b*c)^6*(x*(
2*a^3*c*d*g^4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2*(a^3*d^2*g^4*i^3 + 3*a*b^2*c^2*g^4*i^3 + 6*a^2*b*c*d*g^4*i^3) +
x^3*(b^3*c^2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6*a*b^2*c*d*g^4*i^3) + x^4*(2*b^3*c*d*g^4*i^3 + 3*a*b^2*d^2*g^4*
i^3) + a^3*c^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5))) + ((12*A*b^4*c^4 - 18*A*a^4*d^4 + 9*B*a^4*d^4*n + 4*B*b^4*c^4*
n + 282*A*a^2*b^2*c^2*d^2 - 78*A*a*b^3*c^3*d + 162*A*a^3*b*c*d^3 + 319*B*a^2*b^2*c^2*d^2*n - 41*B*a*b^3*c^3*d*
n - 171*B*a^3*b*c*d^3*n)/(6*(a*d - b*c)) + (5*x*(18*A*a^3*b*d^4 - 6*A*b^4*c^3*d + 66*A*a*b^3*c^2*d^2 + 210*A*a
^2*b^2*c*d^3 - 27*B*a^3*b*d^4*n - 5*B*b^4*c^3*d*n + 103*B*a*b^3*c^2*d^2*n + 25*B*a^2*b^2*c*d^3*n))/(6*(a*d - b
*c)) + (20*x^4*(3*A*b^4*d^4 + B*b^4*d^4*n))/(a*d - b*c) + (10*x^2*(33*A*a^2*b^2*d^4 + 6*A*b^4*c^2*d^2 - 7*B*a^
2*b^2*d^4*n + 11*B*b^4*c^2*d^2*n + 69*A*a*b^3*c*d^3 + 32*B*a*b^3*c*d^3*n))/(3*(a*d - b*c)) + (10*x^3*(15*A*a*b
^3*d^4 + 9*A*b^4*c*d^3 + 2*B*a*b^3*d^4*n + 6*B*b^4*c*d^3*n))/(a*d - b*c))/(x^5*(6*a^4*b^3*d^6*g^4*i^3 + 6*b^7*
c^4*d^2*g^4*i^3 - 24*a*b^6*c^3*d^3*g^4*i^3 - 24*a^3*b^4*c*d^5*g^4*i^3 + 36*a^2*b^5*c^2*d^4*g^4*i^3) + x*(18*a^
2*b^5*c^6*g^4*i^3 + 12*a^7*c*d^5*g^4*i^3 - 60*a^3*b^4*c^5*d*g^4*i^3 - 30*a^6*b*c^2*d^4*g^4*i^3 + 60*a^4*b^3*c^
4*d^2*g^4*i^3) + x^2*(6*a^7*d^6*g^4*i^3 + 18*a*b^6*c^6*g^4*i^3 + 12*a^6*b*c*d^5*g^4*i^3 - 36*a^2*b^5*c^5*d*g^4
*i^3 - 30*a^3*b^4*c^4*d^2*g^4*i^3 + 120*a^4*b^3*c^3*d^3*g^4*i^3 - 90*a^5*b^2*c^2*d^4*g^4*i^3) + x^3*(6*b^7*c^6
*g^4*i^3 + 18*a^6*b*d^6*g^4*i^3 + 12*a*b^6*c^5*d*g^4*i^3 - 36*a^5*b^2*c*d^5*g^4*i^3 - 90*a^2*b^5*c^4*d^2*g^4*i
^3 + 120*a^3*b^4*c^3*d^3*g^4*i^3 - 30*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(18*a^5*b^2*d^6*g^4*i^3 + 12*b^7*c^5*d*g^
4*i^3 - 30*a*b^6*c^4*d^2*g^4*i^3 - 60*a^4*b^3*c*d^5*g^4*i^3 + 60*a^3*b^4*c^2*d^4*g^4*i^3) + 6*a^3*b^4*c^6*g^4*
i^3 + 6*a^7*c^2*d^4*g^4*i^3 - 24*a^4*b^3*c^5*d*g^4*i^3 - 24*a^6*b*c^3*d^3*g^4*i^3 + 36*a^5*b^2*c^4*d^2*g^4*i^3
) + (b^2*d^3*atan((b^2*d^3*(3*A + B*n)*((a^6*d^6*g^4*i^3 - b^6*c^6*g^4*i^3 + 4*a*b^5*c^5*d*g^4*i^3 - 4*a^5*b*c
*d^5*g^4*i^3 - 5*a^2*b^4*c^4*d^2*g^4*i^3 + 5*a^4*b^2*c^2*d^4*g^4*i^3)/(a^5*d^5*g^4*i^3 - b^5*c^5*g^4*i^3 + 5*a
*b^4*c^4*d*g^4*i^3 - 5*a^4*b*c*d^4*g^4*i^3 - 10*a^2*b^3*c^3*d^2*g^4*i^3 + 10*a^3*b^2*c^2*d^3*g^4*i^3) + 2*b*d*
x)*(a^5*d^5*g^4*i^3 - b^5*c^5*g^4*i^3 + 5*a*b^4*c^4*d*g^4*i^3 - 5*a^4*b*c*d^4*g^4*i^3 - 10*a^2*b^3*c^3*d^2*g^4
*i^3 + 10*a^3*b^2*c^2*d^3*g^4*i^3)*10i)/(g^4*i^3*(30*A*b^2*d^3 + 10*B*b^2*d^3*n)*(a*d - b*c)^6))*(3*A + B*n)*2
0i)/(3*g^4*i^3*(a*d - b*c)^6) - (5*B*b^2*d^3*log(e*((a + b*x)/(c + d*x))^n)^2)/(g^4*i^3*n*(a*d - b*c)^6)

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