3.2.57 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^3 (c i+d i x)^3} \, dx\) [157]

Optimal. Leaf size=483 \[ -\frac {B d^4 n (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {4 b B d^3 n (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 B d n (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B n (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3}-\frac {3 b^2 B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 483, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2561, 45, 2372, 2338} \begin {gather*} -\frac {b^4 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac {6 b^2 d^2 \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^3 i^3 (b c-a d)^5}+\frac {d^4 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac {4 b d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac {b^4 B n (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 B d n (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac {3 b^2 B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac {B d^4 n (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5}+\frac {4 b B d^3 n (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )}{(157 c+157 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (a+b x)^3}-\frac {3 b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)^2}+\frac {6 b^3 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (c+d x)^3}-\frac {3 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)^2}-\frac {6 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (6 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} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\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)^2} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3869893 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 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}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 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}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {\left (b^2 B n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d 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}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 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}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B 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}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 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}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 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 )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 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 )}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.81, size = 561, normalized size = 1.16 \begin {gather*} -\frac {\frac {b^2 B (b c-a d)^2 n}{(a+b x)^2}-\frac {12 b^3 B c d n}{a+b x}+\frac {12 a b^2 B d^2 n}{a+b x}-\frac {2 b^2 B d (b c-a d) n}{a+b x}+\frac {B d^2 (b c-a d)^2 n}{(c+d x)^2}+\frac {12 b^2 B c d^2 n}{c+d x}-\frac {12 a b B d^3 n}{c+d x}+\frac {2 b B d^2 (b c-a d) n}{c+d x}+\frac {2 b^2 (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 {12 b^2 d (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 {2 d^2 (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 {12 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 )}{c+d x}-24 b^2 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+12 b^2 B d^2 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 )-12 b^2 B d^2 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 )}{4 (b c-a d)^5 g^3 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)^3*(c*i + d*i*x)^3),x]

[Out]

-1/4*((b^2*B*(b*c - a*d)^2*n)/(a + b*x)^2 - (12*b^3*B*c*d*n)/(a + b*x) + (12*a*b^2*B*d^2*n)/(a + b*x) - (2*b^2
*B*d*(b*c - a*d)*n)/(a + b*x) + (B*d^2*(b*c - a*d)^2*n)/(c + d*x)^2 + (12*b^2*B*c*d^2*n)/(c + d*x) - (12*a*b*B
*d^3*n)/(c + d*x) + (2*b*B*d^2*(b*c - a*d)*n)/(c + d*x) + (2*b^2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*
x))^n]))/(a + b*x)^2 - (12*b^2*d*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (2*d^2*(b*c -
a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 - (12*b*d^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]))/(c + d*x) - 24*b^2*d^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b^2*d^2*(A + B*L
og[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 12*b^2*B*d^2*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)]) - 12*b^2*B*d^2*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^3*i^3)

________________________________________________________________________________________

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

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)^3/(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)^3/(d*i*x+c*i)^3,x)

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2300 vs. \(2 (450) = 900\).
time = 0.55, size = 2300, normalized size = 4.76 \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)^3/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/2*B*(12*b^2*d^2*log(b*x + a)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*d^2 + 10*I*a^3*b^2*c^2*d^3 -
5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) - 12*b^2*d^2*log(d*x + c)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*
d^2 + 10*I*a^3*b^2*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) + (12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7
*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/((-I*b^
6*c^4*d^2 + 4*I*a*b^5*c^3*d^3 - 6*I*a^2*b^4*c^2*d^4 + 4*I*a^3*b^3*c*d^5 - I*a^4*b^2*d^6)*g^3*x^4 + 2*(-I*b^6*c
^5*d + 3*I*a*b^5*c^4*d^2 - 2*I*a^2*b^4*c^3*d^3 - 2*I*a^3*b^3*c^2*d^4 + 3*I*a^4*b^2*c*d^5 - I*a^5*b*d^6)*g^3*x^
3 + (-I*b^6*c^6 + 9*I*a^2*b^4*c^4*d^2 - 16*I*a^3*b^3*c^3*d^3 + 9*I*a^4*b^2*c^2*d^4 - I*a^6*d^6)*g^3*x^2 + 2*(-
I*a*b^5*c^6 + 3*I*a^2*b^4*c^5*d - 2*I*a^3*b^3*c^4*d^2 - 2*I*a^4*b^2*c^3*d^3 + 3*I*a^5*b*c^2*d^4 - I*a^6*c*d^5)
*g^3*x + (-I*a^2*b^4*c^6 + 4*I*a^3*b^3*c^5*d - 6*I*a^4*b^2*c^4*d^2 + 4*I*a^5*b*c^3*d^3 - I*a^6*c^2*d^4)*g^3))*
log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/4*(I*b^4*c^4 - 16*I*a*b^3*c^3*d + 30*I*a^2*b^2*c^2*d^2 - 16*I*a^3*b
*c*d^3 + I*a^4*d^4 - 12*(I*b^4*c^2*d^2 - 2*I*a*b^3*c*d^3 + I*a^2*b^2*d^4)*x^2 - 12*(-I*b^4*d^4*x^4 - I*a^2*b^2
*c^2*d^2 + 2*(-I*b^4*c*d^3 - I*a*b^3*d^4)*x^3 + (-I*b^4*c^2*d^2 - 4*I*a*b^3*c*d^3 - I*a^2*b^2*d^4)*x^2 + 2*(-I
*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3)*x)*log(b*x + a)^2 - 24*(I*b^4*d^4*x^4 + I*a^2*b^2*c^2*d^2 + 2*(I*b^4*c*d^3 +
I*a*b^3*d^4)*x^3 + (I*b^4*c^2*d^2 + 4*I*a*b^3*c*d^3 + I*a^2*b^2*d^4)*x^2 + 2*(I*a*b^3*c^2*d^2 + I*a^2*b^2*c*d
^3)*x)*log(b*x + a)*log(d*x + c) - 12*(-I*b^4*d^4*x^4 - I*a^2*b^2*c^2*d^2 + 2*(-I*b^4*c*d^3 - I*a*b^3*d^4)*x^3
+ (-I*b^4*c^2*d^2 - 4*I*a*b^3*c*d^3 - I*a^2*b^2*d^4)*x^2 + 2*(-I*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3)*x)*log(d*x
+ c)^2 - 12*(I*b^4*c^3*d - I*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3 + I*a^3*b*d^4)*x)*B*n/(a^2*b^5*c^7*g^3 - 5*a^3*b^
4*c^6*d*g^3 + 10*a^4*b^3*c^5*d^2*g^3 - 10*a^5*b^2*c^4*d^3*g^3 + 5*a^6*b*c^3*d^4*g^3 - a^7*c^2*d^5*g^3 + (b^7*c
^5*d^2*g^3 - 5*a*b^6*c^4*d^3*g^3 + 10*a^2*b^5*c^3*d^4*g^3 - 10*a^3*b^4*c^2*d^5*g^3 + 5*a^4*b^3*c*d^6*g^3 - a^5
*b^2*d^7*g^3)*x^4 + 2*(b^7*c^6*d*g^3 - 4*a*b^6*c^5*d^2*g^3 + 5*a^2*b^5*c^4*d^3*g^3 - 5*a^4*b^3*c^2*d^5*g^3 + 4
*a^5*b^2*c*d^6*g^3 - a^6*b*d^7*g^3)*x^3 + (b^7*c^7*g^3 - a*b^6*c^6*d*g^3 - 9*a^2*b^5*c^5*d^2*g^3 + 25*a^3*b^4*
c^4*d^3*g^3 - 25*a^4*b^3*c^3*d^4*g^3 + 9*a^5*b^2*c^2*d^5*g^3 + a^6*b*c*d^6*g^3 - a^7*d^7*g^3)*x^2 + 2*(a*b^6*c
^7*g^3 - 4*a^2*b^5*c^6*d*g^3 + 5*a^3*b^4*c^5*d^2*g^3 - 5*a^5*b^2*c^3*d^4*g^3 + 4*a^6*b*c^2*d^5*g^3 - a^7*c*d^6
*g^3)*x) + 1/2*A*(12*b^2*d^2*log(b*x + a)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*d^2 + 10*I*a^3*b^2
*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) - 12*b^2*d^2*log(d*x + c)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a
^2*b^3*c^3*d^2 + 10*I*a^3*b^2*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) + (12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^
2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)
*x)/((-I*b^6*c^4*d^2 + 4*I*a*b^5*c^3*d^3 - 6*I*a^2*b^4*c^2*d^4 + 4*I*a^3*b^3*c*d^5 - I*a^4*b^2*d^6)*g^3*x^4 +
2*(-I*b^6*c^5*d + 3*I*a*b^5*c^4*d^2 - 2*I*a^2*b^4*c^3*d^3 - 2*I*a^3*b^3*c^2*d^4 + 3*I*a^4*b^2*c*d^5 - I*a^5*b*
d^6)*g^3*x^3 + (-I*b^6*c^6 + 9*I*a^2*b^4*c^4*d^2 - 16*I*a^3*b^3*c^3*d^3 + 9*I*a^4*b^2*c^2*d^4 - I*a^6*d^6)*g^3
*x^2 + 2*(-I*a*b^5*c^6 + 3*I*a^2*b^4*c^5*d - 2*I*a^3*b^3*c^4*d^2 - 2*I*a^4*b^2*c^3*d^3 + 3*I*a^5*b*c^2*d^4 - I
*a^6*c*d^5)*g^3*x + (-I*a^2*b^4*c^6 + 4*I*a^3*b^3*c^5*d - 6*I*a^4*b^2*c^4*d^2 + 4*I*a^5*b*c^3*d^3 - I*a^6*c^2*
d^4)*g^3))

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (450) = 900\).
time = 0.41, size = 1265, normalized size = 2.62 \begin {gather*} \frac {2 \, {\left (-i \, A - i \, B\right )} b^{4} c^{4} + 16 \, {\left (i \, A + i \, B\right )} a b^{3} c^{3} d + 16 \, {\left (-i \, A - i \, B\right )} a^{3} b c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a^{4} d^{4} + 24 \, {\left ({\left (i \, A + i \, B\right )} b^{4} c d^{3} + {\left (-i \, A - i \, B\right )} a b^{3} d^{4}\right )} x^{3} + 12 \, {\left (3 \, {\left (i \, A + i \, B\right )} b^{4} c^{2} d^{2} + 3 \, {\left (-i \, A - i \, B\right )} a^{2} b^{2} d^{4} + {\left (i \, B b^{4} c^{2} d^{2} - 2 i \, B a b^{3} c d^{3} + i \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + 12 \, {\left (i \, B b^{4} d^{4} n x^{4} + i \, B a^{2} b^{2} c^{2} d^{2} n + 2 \, {\left (i \, B b^{4} c d^{3} + i \, B a b^{3} d^{4}\right )} n x^{3} + {\left (i \, B b^{4} c^{2} d^{2} + 4 i \, B a b^{3} c d^{3} + i \, B a^{2} b^{2} d^{4}\right )} n x^{2} + 2 \, {\left (i \, B a b^{3} c^{2} d^{2} + i \, B a^{2} b^{2} c d^{3}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (i \, B b^{4} c^{4} - 16 i \, B a b^{3} c^{3} d + 30 i \, B a^{2} b^{2} c^{2} d^{2} - 16 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4}\right )} n + 4 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c^{3} d + 12 \, {\left (i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 12 \, {\left (-i \, A - i \, B\right )} a^{2} b^{2} c d^{3} + 2 \, {\left (-i \, A - i \, B\right )} a^{3} b d^{4} + 3 \, {\left (i \, B b^{4} c^{3} d - i \, B a b^{3} c^{2} d^{2} - i \, B a^{2} b^{2} c d^{3} + i \, B a^{3} b d^{4}\right )} n\right )} x + 2 \, {\left (12 \, {\left (i \, A + i \, B\right )} b^{4} d^{4} x^{4} + 12 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} c^{2} d^{2} + 12 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a b^{3} d^{4} + {\left (i \, B b^{4} c d^{3} - i \, B a b^{3} d^{4}\right )} n\right )} x^{3} + 6 \, {\left (2 \, {\left (i \, A + i \, B\right )} b^{4} c^{2} d^{2} + 8 \, {\left (i \, A + i \, B\right )} a b^{3} c d^{3} + 2 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} d^{4} + 3 \, {\left (i \, B b^{4} c^{2} d^{2} - i \, B a^{2} b^{2} d^{4}\right )} n\right )} x^{2} + {\left (-i \, B b^{4} c^{4} + 8 i \, B a b^{3} c^{3} d - 8 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4}\right )} n + 4 \, {\left (6 \, {\left (i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 6 \, {\left (i \, A + i \, B\right )} a^{2} b^{2} c d^{3} + {\left (i \, B b^{4} c^{3} d + 6 i \, B a b^{3} c^{2} d^{2} - 6 i \, B a^{2} b^{2} c d^{3} - i \, B a^{3} b d^{4}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} g^{3} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} g^{3} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} g^{3} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} g^{3} x + {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5}\right )} g^{3}\right )}} \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)^3/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

1/4*(2*(-I*A - I*B)*b^4*c^4 + 16*(I*A + I*B)*a*b^3*c^3*d + 16*(-I*A - I*B)*a^3*b*c*d^3 + 2*(I*A + I*B)*a^4*d^4
+ 24*((I*A + I*B)*b^4*c*d^3 + (-I*A - I*B)*a*b^3*d^4)*x^3 + 12*(3*(I*A + I*B)*b^4*c^2*d^2 + 3*(-I*A - I*B)*a^
2*b^2*d^4 + (I*B*b^4*c^2*d^2 - 2*I*B*a*b^3*c*d^3 + I*B*a^2*b^2*d^4)*n)*x^2 + 12*(I*B*b^4*d^4*n*x^4 + I*B*a^2*b
^2*c^2*d^2*n + 2*(I*B*b^4*c*d^3 + I*B*a*b^3*d^4)*n*x^3 + (I*B*b^4*c^2*d^2 + 4*I*B*a*b^3*c*d^3 + I*B*a^2*b^2*d^
4)*n*x^2 + 2*(I*B*a*b^3*c^2*d^2 + I*B*a^2*b^2*c*d^3)*n*x)*log((b*x + a)/(d*x + c))^2 - (I*B*b^4*c^4 - 16*I*B*a
*b^3*c^3*d + 30*I*B*a^2*b^2*c^2*d^2 - 16*I*B*a^3*b*c*d^3 + I*B*a^4*d^4)*n + 4*(2*(I*A + I*B)*b^4*c^3*d + 12*(I
*A + I*B)*a*b^3*c^2*d^2 + 12*(-I*A - I*B)*a^2*b^2*c*d^3 + 2*(-I*A - I*B)*a^3*b*d^4 + 3*(I*B*b^4*c^3*d - I*B*a*
b^3*c^2*d^2 - I*B*a^2*b^2*c*d^3 + I*B*a^3*b*d^4)*n)*x + 2*(12*(I*A + I*B)*b^4*d^4*x^4 + 12*(I*A + I*B)*a^2*b^2
*c^2*d^2 + 12*(2*(I*A + I*B)*b^4*c*d^3 + 2*(I*A + I*B)*a*b^3*d^4 + (I*B*b^4*c*d^3 - I*B*a*b^3*d^4)*n)*x^3 + 6*
(2*(I*A + I*B)*b^4*c^2*d^2 + 8*(I*A + I*B)*a*b^3*c*d^3 + 2*(I*A + I*B)*a^2*b^2*d^4 + 3*(I*B*b^4*c^2*d^2 - I*B*
a^2*b^2*d^4)*n)*x^2 + (-I*B*b^4*c^4 + 8*I*B*a*b^3*c^3*d - 8*I*B*a^3*b*c*d^3 + I*B*a^4*d^4)*n + 4*(6*(I*A + I*B
)*a*b^3*c^2*d^2 + 6*(I*A + I*B)*a^2*b^2*c*d^3 + (I*B*b^4*c^3*d + 6*I*B*a*b^3*c^2*d^2 - 6*I*B*a^2*b^2*c*d^3 - I
*B*a^3*b*d^4)*n)*x)*log((b*x + a)/(d*x + c)))/((b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^
4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*g^3*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^
4*b^3*c^2*d^5 + 4*a^5*b^2*c*d^6 - a^6*b*d^7)*g^3*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4
*c^4*d^3 - 25*a^4*b^3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*g^3*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*
c^6*d + 5*a^3*b^4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*g^3*x + (a^2*b^5*c^7 - 5*a^3*b^4*
c^6*d + 10*a^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*g^3)

________________________________________________________________________________________

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)**3/(d*i*x+c*i)**3,x)

[Out]

Timed out

________________________________________________________________________________________

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)^3/(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)^3*(I*d*x + I*c)^3), x)

________________________________________________________________________________________

Mupad [B]
time = 7.93, size = 1341, normalized size = 2.78 \begin {gather*} \frac {\frac {2\,x\,\left (2\,A\,a^2\,b\,d^3+2\,A\,b^3\,c^2\,d+14\,A\,a\,b^2\,c\,d^2-3\,B\,a^2\,b\,d^3\,n+3\,B\,b^3\,c^2\,d\,n\right )}{a\,d-b\,c}-\frac {2\,A\,a^3\,d^3+2\,A\,b^3\,c^3-B\,a^3\,d^3\,n+B\,b^3\,c^3\,n-14\,A\,a\,b^2\,c^2\,d-14\,A\,a^2\,b\,c\,d^2-15\,B\,a\,b^2\,c^2\,d\,n+15\,B\,a^2\,b\,c\,d^2\,n}{2\,\left (a\,d-b\,c\right )}+\frac {6\,x^2\,\left (3\,A\,a\,b^2\,d^3+3\,A\,b^3\,c\,d^2-B\,a\,b^2\,d^3\,n+B\,b^3\,c\,d^2\,n\right )}{a\,d-b\,c}+\frac {12\,A\,b^3\,d^3\,x^3}{a\,d-b\,c}}{x^4\,\left (2\,a^3\,b^2\,d^5\,g^3\,i^3-6\,a^2\,b^3\,c\,d^4\,g^3\,i^3+6\,a\,b^4\,c^2\,d^3\,g^3\,i^3-2\,b^5\,c^3\,d^2\,g^3\,i^3\right )-x\,\left (-4\,a^5\,c\,d^4\,g^3\,i^3+8\,a^4\,b\,c^2\,d^3\,g^3\,i^3-8\,a^2\,b^3\,c^4\,d\,g^3\,i^3+4\,a\,b^4\,c^5\,g^3\,i^3\right )+x^3\,\left (4\,a^4\,b\,d^5\,g^3\,i^3-8\,a^3\,b^2\,c\,d^4\,g^3\,i^3+8\,a\,b^4\,c^3\,d^2\,g^3\,i^3-4\,b^5\,c^4\,d\,g^3\,i^3\right )+x^2\,\left (2\,a^5\,d^5\,g^3\,i^3+2\,a^4\,b\,c\,d^4\,g^3\,i^3-16\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+16\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-2\,a\,b^4\,c^4\,d\,g^3\,i^3-2\,b^5\,c^5\,g^3\,i^3\right )-2\,a^2\,b^3\,c^5\,g^3\,i^3+2\,a^5\,c^2\,d^3\,g^3\,i^3+6\,a^3\,b^2\,c^4\,d\,g^3\,i^3-6\,a^4\,b\,c^3\,d^2\,g^3\,i^3}+\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (x\,\left (\frac {3\,B\,b\,d\,{\left (a\,d+b\,c\right )}^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}-\frac {B\,b\,d}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {6\,B\,a\,b^2\,c\,d^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )-\frac {B\,\left (a\,d+b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {6\,B\,b^3\,d^3\,x^3}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {9\,B\,b^2\,d^2\,x^2\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )}{x\,\left (2\,d\,a^2\,c\,g^3\,i^3+2\,b\,a\,c^2\,g^3\,i^3\right )+x^3\,\left (2\,c\,b^2\,d\,g^3\,i^3+2\,a\,b\,d^2\,g^3\,i^3\right )+x^2\,\left (a^2\,d^2\,g^3\,i^3+4\,a\,b\,c\,d\,g^3\,i^3+b^2\,c^2\,g^3\,i^3\right )+a^2\,c^2\,g^3\,i^3+b^2\,d^2\,g^3\,i^3\,x^4}-\frac {3\,B\,b^2\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^3\,i^3\,n\,{\left (a\,d-b\,c\right )}^5}+\frac {A\,b^2\,d^2\,\mathrm {atan}\left (\frac {\left (a^5\,d^5\,g^3\,i^3-3\,a^4\,b\,c\,d^4\,g^3\,i^3+2\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+2\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-3\,a\,b^4\,c^4\,d\,g^3\,i^3+b^5\,c^5\,g^3\,i^3\right )\,1{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}+\frac {b\,d\,x\,\left (a^4\,d^4\,g^3\,i^3-4\,a^3\,b\,c\,d^3\,g^3\,i^3+6\,a^2\,b^2\,c^2\,d^2\,g^3\,i^3-4\,a\,b^3\,c^3\,d\,g^3\,i^3+b^4\,c^4\,g^3\,i^3\right )\,2{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}\right )\,12{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5} \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)^3*(c*i + d*i*x)^3),x)

[Out]

((2*x*(2*A*a^2*b*d^3 + 2*A*b^3*c^2*d + 14*A*a*b^2*c*d^2 - 3*B*a^2*b*d^3*n + 3*B*b^3*c^2*d*n))/(a*d - b*c) - (2
*A*a^3*d^3 + 2*A*b^3*c^3 - B*a^3*d^3*n + B*b^3*c^3*n - 14*A*a*b^2*c^2*d - 14*A*a^2*b*c*d^2 - 15*B*a*b^2*c^2*d*
n + 15*B*a^2*b*c*d^2*n)/(2*(a*d - b*c)) + (6*x^2*(3*A*a*b^2*d^3 + 3*A*b^3*c*d^2 - B*a*b^2*d^3*n + B*b^3*c*d^2*
n))/(a*d - b*c) + (12*A*b^3*d^3*x^3)/(a*d - b*c))/(x^4*(2*a^3*b^2*d^5*g^3*i^3 - 2*b^5*c^3*d^2*g^3*i^3 + 6*a*b^
4*c^2*d^3*g^3*i^3 - 6*a^2*b^3*c*d^4*g^3*i^3) - x*(4*a*b^4*c^5*g^3*i^3 - 4*a^5*c*d^4*g^3*i^3 - 8*a^2*b^3*c^4*d*
g^3*i^3 + 8*a^4*b*c^2*d^3*g^3*i^3) + x^3*(4*a^4*b*d^5*g^3*i^3 - 4*b^5*c^4*d*g^3*i^3 + 8*a*b^4*c^3*d^2*g^3*i^3
- 8*a^3*b^2*c*d^4*g^3*i^3) + x^2*(2*a^5*d^5*g^3*i^3 - 2*b^5*c^5*g^3*i^3 - 2*a*b^4*c^4*d*g^3*i^3 + 2*a^4*b*c*d^
4*g^3*i^3 + 16*a^2*b^3*c^3*d^2*g^3*i^3 - 16*a^3*b^2*c^2*d^3*g^3*i^3) - 2*a^2*b^3*c^5*g^3*i^3 + 2*a^5*c^2*d^3*g
^3*i^3 + 6*a^3*b^2*c^4*d*g^3*i^3 - 6*a^4*b*c^3*d^2*g^3*i^3) + (log(e*((a + b*x)/(c + d*x))^n)*(x*((3*B*b*d*(a*
d + b*c)^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 - (B*b*d)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (6*B*a*b^2*c*d^2)/(a
^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (B*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (6*B*b^3*d^3*x^3)/(a^
2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 + (9*B*b^2*d^2*x^2*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 + (3*B*a*b*c*
d*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2))/(x*(2*a*b*c^2*g^3*i^3 + 2*a^2*c*d*g^3*i^3) + x^3*(2*a*b*d^2
*g^3*i^3 + 2*b^2*c*d*g^3*i^3) + x^2*(a^2*d^2*g^3*i^3 + b^2*c^2*g^3*i^3 + 4*a*b*c*d*g^3*i^3) + a^2*c^2*g^3*i^3
+ b^2*d^2*g^3*i^3*x^4) + (A*b^2*d^2*atan(((a^5*d^5*g^3*i^3 + b^5*c^5*g^3*i^3 - 3*a*b^4*c^4*d*g^3*i^3 - 3*a^4*b
*c*d^4*g^3*i^3 + 2*a^2*b^3*c^3*d^2*g^3*i^3 + 2*a^3*b^2*c^2*d^3*g^3*i^3)*1i)/(g^3*i^3*(a*d - b*c)^5) + (b*d*x*(
a^4*d^4*g^3*i^3 + b^4*c^4*g^3*i^3 - 4*a*b^3*c^3*d*g^3*i^3 - 4*a^3*b*c*d^3*g^3*i^3 + 6*a^2*b^2*c^2*d^2*g^3*i^3)
*2i)/(g^3*i^3*(a*d - b*c)^5))*12i)/(g^3*i^3*(a*d - b*c)^5) - (3*B*b^2*d^2*log(e*((a + b*x)/(c + d*x))^n)^2)/(g
^3*i^3*n*(a*d - b*c)^5)

________________________________________________________________________________________