3.2.42 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)} \, dx\) [142]

Optimal. Leaf size=389 \[ -\frac {3 b B d^2 n (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d n (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 B n (c+d x)^3}{9 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^4 i}+\frac {B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^4 i} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 389, 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^3 (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 (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 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 (a+b x)^2 (b c-a d)^4}-\frac {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 (b c-a d)^4}-\frac {3 b 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 (a+b x) (b c-a d)^4}-\frac {b^3 B n (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B d n (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4}-\frac {3 b B d^2 n (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \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)),x]

[Out]

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

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 )}{(142 c+142 d x) (a g+b g x)^4} \, dx &=\int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d) g^4 (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 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 )}{142 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b 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}{142 (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} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (b 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}{142 (b c-a d)^3 g^4}-\frac {(b d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{142 (b c-a d)^2 g^4}+\frac {b \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{142 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {\left (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}{142 (b c-a d)^4 g^4}-\frac {\left (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}{142 (b c-a d)^4 g^4}+\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^3 g^4}-\frac {(B d n) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d)^2 g^4}+\frac {(B n) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{426 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {(B n) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{426 g^4}+\frac {\left (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}{142 (b c-a d)^4 g^4}-\frac {\left (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}{142 (b c-a d)^4 g^4}+\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^2 g^4}-\frac {(B d n) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {(B n) \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}{426 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac {\left (b B d^3 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac {\left (B d^4 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^4 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (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}{142 (b c-a d)^2 g^4}-\frac {(B d n) \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}{284 (b c-a d) g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{142 (b c-a d)^4 g^4}+\frac {\left (B d^3 n\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}+\frac {\left (b B d^3 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac {\left (B d^4 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac {\left (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 )}{142 (b c-a d)^4 g^4}+\frac {\left (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 )}{142 (b c-a d)^4 g^4}\\ &=-\frac {B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac {11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac {B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac {B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac {B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac {B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 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 = 0.46, size = 518, normalized size = 1.33 \begin {gather*} \frac {-\frac {12 A (b c-a d)^3}{(a+b x)^3}-\frac {4 B (b c-a d)^3 n}{(a+b x)^3}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}+\frac {15 B d (b c-a d)^2 n}{(a+b x)^2}+\frac {36 A d^2 (-b c+a d)}{a+b x}+\frac {66 B d^2 (-b c+a d) n}{a+b x}-36 A d^3 \log (a+b x)-66 B d^3 n \log (a+b x)+18 B d^3 n \log ^2(a+b x)-\frac {12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {18 B d (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2}+\frac {36 B d^2 (-b c+a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x}-36 B d^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 A d^3 \log (c+d x)+66 B d^3 n \log (c+d x)-36 B d^3 n \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+36 B d^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)+18 B d^3 n \log ^2(c+d x)-36 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 n \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^4 g^4 i} \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)),x]

[Out]

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3*n)/(a + b*x)^3 + (18*A*d*(b*c - a*d)^2)/(a + b*x)^2 +
(15*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (36*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d)*n)/(a +
b*x) - 36*A*d^3*Log[a + b*x] - 66*B*d^3*n*Log[a + b*x] + 18*B*d^3*n*Log[a + b*x]^2 - (12*B*(b*c - a*d)^3*Log[e
*((a + b*x)/(c + d*x))^n])/(a + b*x)^3 + (18*B*d*(b*c - a*d)^2*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^2 + (
36*B*d^2*(-(b*c) + a*d)*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x) - 36*B*d^3*Log[a + b*x]*Log[e*((a + b*x)/(c
+ d*x))^n] + 36*A*d^3*Log[c + d*x] + 66*B*d^3*n*Log[c + d*x] - 36*B*d^3*n*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Lo
g[c + d*x] + 36*B*d^3*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x] + 18*B*d^3*n*Log[c + d*x]^2 - 36*B*d^3*n*Log
[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 36*B*d^3*n*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 36*B*d^3*n*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)])/(36*(b*c - a*d)^4*g^4*i)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (b g x +a g \right )^{4} \left (d i x +c i \right )}\, dx\]

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),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),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. 1458 vs. \(2 (359) = 718\).
time = 0.52, size = 1458, normalized size = 3.75 \begin {gather*} -\frac {1}{6} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (i \, b^{6} c^{3} - 3 i \, a b^{5} c^{2} d + 3 i \, a^{2} b^{4} c d^{2} - i \, a^{3} b^{3} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (i \, a b^{5} c^{3} - 3 i \, a^{2} b^{4} c^{2} d + 3 i \, a^{3} b^{3} c d^{2} - i \, a^{4} b^{2} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (i \, a^{2} b^{4} c^{3} - 3 i \, a^{3} b^{3} c^{2} d + 3 i \, a^{4} b^{2} c d^{2} - i \, a^{5} b d^{3}\right )} g^{4} x + {\left (i \, a^{3} b^{3} c^{3} - 3 i \, a^{4} b^{2} c^{2} d + 3 i \, a^{5} b c d^{2} - i \, a^{6} d^{3}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}}\right )} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {{\left (-4 i \, b^{3} c^{3} + 27 i \, a b^{2} c^{2} d - 108 i \, a^{2} b c d^{2} + 85 i \, a^{3} d^{3} - 66 \, {\left (i \, b^{3} c d^{2} - i \, a b^{2} d^{3}\right )} x^{2} - 18 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, a b^{2} d^{3} x^{2} - 3 i \, a^{2} b d^{3} x - i \, a^{3} d^{3}\right )} \log \left (b x + a\right )^{2} - 18 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, a b^{2} d^{3} x^{2} - 3 i \, a^{2} b d^{3} x - i \, a^{3} d^{3}\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (-5 i \, b^{3} c^{2} d + 54 i \, a b^{2} c d^{2} - 49 i \, a^{2} b d^{3}\right )} x - 66 \, {\left (i \, b^{3} d^{3} x^{3} + 3 i \, a b^{2} d^{3} x^{2} + 3 i \, a^{2} b d^{3} x + i \, a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (-11 i \, b^{3} d^{3} x^{3} - 33 i \, a b^{2} d^{3} x^{2} - 33 i \, a^{2} b d^{3} x - 11 i \, a^{3} d^{3} + 6 \, {\left (i \, b^{3} d^{3} x^{3} + 3 i \, a b^{2} d^{3} x^{2} + 3 i \, a^{2} b d^{3} x + i \, a^{3} d^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{36 \, {\left (a^{3} b^{4} c^{4} g^{4} - 4 \, a^{4} b^{3} c^{3} d g^{4} + 6 \, a^{5} b^{2} c^{2} d^{2} g^{4} - 4 \, a^{6} b c d^{3} g^{4} + a^{7} d^{4} g^{4} + {\left (b^{7} c^{4} g^{4} - 4 \, a b^{6} c^{3} d g^{4} + 6 \, a^{2} b^{5} c^{2} d^{2} g^{4} - 4 \, a^{3} b^{4} c d^{3} g^{4} + a^{4} b^{3} d^{4} g^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} g^{4} - 4 \, a^{2} b^{5} c^{3} d g^{4} + 6 \, a^{3} b^{4} c^{2} d^{2} g^{4} - 4 \, a^{4} b^{3} c d^{3} g^{4} + a^{5} b^{2} d^{4} g^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} g^{4} - 4 \, a^{3} b^{4} c^{3} d g^{4} + 6 \, a^{4} b^{3} c^{2} d^{2} g^{4} - 4 \, a^{5} b^{2} c d^{3} g^{4} + a^{6} b d^{4} g^{4}\right )} x\right )}} - \frac {1}{6} \, A {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (i \, b^{6} c^{3} - 3 i \, a b^{5} c^{2} d + 3 i \, a^{2} b^{4} c d^{2} - i \, a^{3} b^{3} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (i \, a b^{5} c^{3} - 3 i \, a^{2} b^{4} c^{2} d + 3 i \, a^{3} b^{3} c d^{2} - i \, a^{4} b^{2} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (i \, a^{2} b^{4} c^{3} - 3 i \, a^{3} b^{3} c^{2} d + 3 i \, a^{4} b^{2} c d^{2} - i \, a^{5} b d^{3}\right )} g^{4} x + {\left (i \, a^{3} b^{3} c^{3} - 3 i \, a^{4} b^{2} c^{2} d + 3 i \, a^{5} b c d^{2} - i \, a^{6} d^{3}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}}\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)^4/(d*i*x+c*i),x, algorithm="maxima")

[Out]

-1/6*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((I*b^6*c^3 - 3*I*a*b
^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d^2
- I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x + (I
*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*a*
b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b
^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1
/36*(-4*I*b^3*c^3 + 27*I*a*b^2*c^2*d - 108*I*a^2*b*c*d^2 + 85*I*a^3*d^3 - 66*(I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 -
18*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x + a)^2 - 18*(-I*b^3*d^3*x^3 - 3
*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*d^3)*log(d*x + c)^2 - 3*(-5*I*b^3*c^2*d + 54*I*a*b^2*c*d^2 - 49*I*a
^2*b*d^3)*x - 66*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a) - 6*(-11*I*b^3
*d^3*x^3 - 33*I*a*b^2*d^3*x^2 - 33*I*a^2*b*d^3*x - 11*I*a^3*d^3 + 6*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a
^2*b*d^3*x + I*a^3*d^3)*log(b*x + a))*log(d*x + c))*B*n/(a^3*b^4*c^4*g^4 - 4*a^4*b^3*c^3*d*g^4 + 6*a^5*b^2*c^2
*d^2*g^4 - 4*a^6*b*c*d^3*g^4 + a^7*d^4*g^4 + (b^7*c^4*g^4 - 4*a*b^6*c^3*d*g^4 + 6*a^2*b^5*c^2*d^2*g^4 - 4*a^3*
b^4*c*d^3*g^4 + a^4*b^3*d^4*g^4)*x^3 + 3*(a*b^6*c^4*g^4 - 4*a^2*b^5*c^3*d*g^4 + 6*a^3*b^4*c^2*d^2*g^4 - 4*a^4*
b^3*c*d^3*g^4 + a^5*b^2*d^4*g^4)*x^2 + 3*(a^2*b^5*c^4*g^4 - 4*a^3*b^4*c^3*d*g^4 + 6*a^4*b^3*c^2*d^2*g^4 - 4*a^
5*b^2*c*d^3*g^4 + a^6*b*d^4*g^4)*x) - 1/6*A*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d
- 5*a*b*d^2)*x)/((I*b^6*c^3 - 3*I*a*b^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 -
3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d^2 - I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^
4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x + (I*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*
d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d
^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^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. 773 vs. \(2 (359) = 718\).
time = 0.44, size = 773, normalized size = 1.99 \begin {gather*} -\frac {12 \, {\left (-i \, A - i \, B\right )} b^{3} c^{3} + 54 \, {\left (i \, A + i \, B\right )} a b^{2} c^{2} d + 108 \, {\left (-i \, A - i \, B\right )} a^{2} b c d^{2} + 66 \, {\left (i \, A + i \, B\right )} a^{3} d^{3} + 6 \, {\left (6 \, {\left (-i \, A - i \, B\right )} b^{3} c d^{2} + 6 \, {\left (i \, A + i \, B\right )} a b^{2} d^{3} + 11 \, {\left (-i \, B b^{3} c d^{2} + i \, B a b^{2} d^{3}\right )} n\right )} x^{2} + 18 \, {\left (-i \, B b^{3} d^{3} n x^{3} - 3 i \, B a b^{2} d^{3} n x^{2} - 3 i \, B a^{2} b d^{3} n x - i \, B a^{3} d^{3} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - {\left (4 i \, B b^{3} c^{3} - 27 i \, B a b^{2} c^{2} d + 108 i \, B a^{2} b c d^{2} - 85 i \, B a^{3} d^{3}\right )} n + 3 \, {\left (6 \, {\left (i \, A + i \, B\right )} b^{3} c^{2} d + 36 \, {\left (-i \, A - i \, B\right )} a b^{2} c d^{2} + 30 \, {\left (i \, A + i \, B\right )} a^{2} b d^{3} + {\left (5 i \, B b^{3} c^{2} d - 54 i \, B a b^{2} c d^{2} + 49 i \, B a^{2} b d^{3}\right )} n\right )} x + 6 \, {\left (6 \, {\left (-i \, A - i \, B\right )} a^{3} d^{3} + {\left (-11 i \, B b^{3} d^{3} n + 6 \, {\left (-i \, A - i \, B\right )} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (6 \, {\left (-i \, A - i \, B\right )} a b^{2} d^{3} + {\left (-2 i \, B b^{3} c d^{2} - 9 i \, B a b^{2} d^{3}\right )} n\right )} x^{2} + {\left (-2 i \, B b^{3} c^{3} + 9 i \, B a b^{2} c^{2} d - 18 i \, B a^{2} b c d^{2}\right )} n + 3 \, {\left (6 \, {\left (-i \, A - i \, B\right )} a^{2} b d^{3} + {\left (i \, B b^{3} c^{2} d - 6 i \, B a b^{2} c d^{2} - 6 i \, B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4}\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)^4/(d*i*x+c*i),x, algorithm="fricas")

[Out]

-1/36*(12*(-I*A - I*B)*b^3*c^3 + 54*(I*A + I*B)*a*b^2*c^2*d + 108*(-I*A - I*B)*a^2*b*c*d^2 + 66*(I*A + I*B)*a^
3*d^3 + 6*(6*(-I*A - I*B)*b^3*c*d^2 + 6*(I*A + I*B)*a*b^2*d^3 + 11*(-I*B*b^3*c*d^2 + I*B*a*b^2*d^3)*n)*x^2 + 1
8*(-I*B*b^3*d^3*n*x^3 - 3*I*B*a*b^2*d^3*n*x^2 - 3*I*B*a^2*b*d^3*n*x - I*B*a^3*d^3*n)*log((b*x + a)/(d*x + c))^
2 - (4*I*B*b^3*c^3 - 27*I*B*a*b^2*c^2*d + 108*I*B*a^2*b*c*d^2 - 85*I*B*a^3*d^3)*n + 3*(6*(I*A + I*B)*b^3*c^2*d
+ 36*(-I*A - I*B)*a*b^2*c*d^2 + 30*(I*A + I*B)*a^2*b*d^3 + (5*I*B*b^3*c^2*d - 54*I*B*a*b^2*c*d^2 + 49*I*B*a^2
*b*d^3)*n)*x + 6*(6*(-I*A - I*B)*a^3*d^3 + (-11*I*B*b^3*d^3*n + 6*(-I*A - I*B)*b^3*d^3)*x^3 + 3*(6*(-I*A - I*B
)*a*b^2*d^3 + (-2*I*B*b^3*c*d^2 - 9*I*B*a*b^2*d^3)*n)*x^2 + (-2*I*B*b^3*c^3 + 9*I*B*a*b^2*c^2*d - 18*I*B*a^2*b
*c*d^2)*n + 3*(6*(-I*A - I*B)*a^2*b*d^3 + (I*B*b^3*c^2*d - 6*I*B*a*b^2*c*d^2 - 6*I*B*a^2*b*d^3)*n)*x)*log((b*x
+ a)/(d*x + c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^4*x^3 + 3*(
a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*x^2 + 3*(a^2*b^5*c^4 - 4*
a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*x + (a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^
5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*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),x)

[Out]

Timed out

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Giac [A]
time = 139.65, size = 236, normalized size = 0.61 \begin {gather*} -\frac {1}{36} \, {\left (\frac {6 \, {\left (-2 i \, B b n - \frac {3 \, {\left (-i \, b x - i \, a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {-4 i \, B b n - \frac {9 \, {\left (-i \, b x - i \, a\right )} B d n}{d x + c} - 12 i \, A b - 12 i \, B b - \frac {18 \, {\left (-i \, b x - i \, a\right )} A d}{d x + c} - \frac {18 \, {\left (-i \, b x - i \, a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\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} \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),x, algorithm="giac")

[Out]

-1/36*(6*(-2*I*B*b*n - 3*(-I*b*x - I*a)*B*d*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x +
c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3) + (-4*I*B*b*n - 9*(-I*b*x - I*a)*B*d*n/(d*x + c) - 12*I*A*b - 12*I*B*b
- 18*(-I*b*x - I*a)*A*d/(d*x + c) - 18*(-I*b*x - I*a)*B*d/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x
+ a)^3*a*d*g^4/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)^2

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Mupad [B]
time = 7.23, size = 986, normalized size = 2.53 \begin {gather*} \frac {\frac {66\,A\,a^2\,d^2+12\,A\,b^2\,c^2+85\,B\,a^2\,d^2\,n+4\,B\,b^2\,c^2\,n-42\,A\,a\,b\,c\,d-23\,B\,a\,b\,c\,d\,n}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (30\,A\,a\,b\,d^2-6\,A\,b^2\,c\,d+49\,B\,a\,b\,d^2\,n-5\,B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (6\,A\,b^2\,d+11\,B\,b^2\,d\,n\right )}{a\,d-b\,c}}{x\,\left (18\,i\,a^4\,b\,d^2\,g^4-36\,i\,a^3\,b^2\,c\,d\,g^4+18\,i\,a^2\,b^3\,c^2\,g^4\right )+x^2\,\left (18\,i\,a^3\,b^2\,d^2\,g^4-36\,i\,a^2\,b^3\,c\,d\,g^4+18\,i\,a\,b^4\,c^2\,g^4\right )+x^3\,\left (6\,i\,a^2\,b^3\,d^2\,g^4-12\,i\,a\,b^4\,c\,d\,g^4+6\,i\,b^5\,c^2\,g^4\right )+6\,a^5\,d^2\,g^4\,i+6\,a^3\,b^2\,c^2\,g^4\,i-12\,a^4\,b\,c\,d\,g^4\,i}-\frac {B\,d^3\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,d^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (x\,\left (b\,\left (\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}\right )+\frac {2\,a\,b\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}+\frac {b\,g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{3\,d^2}\right )+a\,\left (\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{6\,d^2}+\frac {a\,g^4\,i\,n\,\left (a\,d-b\,c\right )}{3\,d}\right )+\frac {g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{3\,d^3}+\frac {b^2\,g^4\,i\,n\,x^2\,\left (a\,d-b\,c\right )}{d}\right )}{g^4\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (i\,a^3\,g^4+3\,i\,a^2\,b\,g^4\,x+3\,i\,a\,b^2\,g^4\,x^2+i\,b^3\,g^4\,x^3\right )}+\frac {d^3\,\mathrm {atan}\left (\frac {d^3\,\left (\frac {i\,a^4\,d^4\,g^4-2\,i\,a^3\,b\,c\,d^3\,g^4+2\,i\,a\,b^3\,c^3\,d\,g^4-i\,b^4\,c^4\,g^4}{i\,a^3\,d^3\,g^4-3\,i\,a^2\,b\,c\,d^2\,g^4+3\,i\,a\,b^2\,c^2\,d\,g^4-i\,b^3\,c^3\,g^4}+2\,b\,d\,x\right )\,\left (A+\frac {11\,B\,n}{6}\right )\,\left (i\,a^3\,d^3\,g^4-3\,i\,a^2\,b\,c\,d^2\,g^4+3\,i\,a\,b^2\,c^2\,d\,g^4-i\,b^3\,c^3\,g^4\right )\,6{}\mathrm {i}}{g^4\,i\,\left (6\,A\,d^3+11\,B\,d^3\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (A+\frac {11\,B\,n}{6}\right )\,2{}\mathrm {i}}{g^4\,i\,{\left (a\,d-b\,c\right )}^4} \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)),x)

[Out]

((66*A*a^2*d^2 + 12*A*b^2*c^2 + 85*B*a^2*d^2*n + 4*B*b^2*c^2*n - 42*A*a*b*c*d - 23*B*a*b*c*d*n)/(6*(a*d - b*c)
) + (x*(30*A*a*b*d^2 - 6*A*b^2*c*d + 49*B*a*b*d^2*n - 5*B*b^2*c*d*n))/(2*(a*d - b*c)) + (d*x^2*(6*A*b^2*d + 11
*B*b^2*d*n))/(a*d - b*c))/(x*(18*a^4*b*d^2*g^4*i + 18*a^2*b^3*c^2*g^4*i - 36*a^3*b^2*c*d*g^4*i) + x^2*(18*a*b^
4*c^2*g^4*i + 18*a^3*b^2*d^2*g^4*i - 36*a^2*b^3*c*d*g^4*i) + x^3*(6*b^5*c^2*g^4*i + 6*a^2*b^3*d^2*g^4*i - 12*a
*b^4*c*d*g^4*i) + 6*a^5*d^2*g^4*i + 6*a^3*b^2*c^2*g^4*i - 12*a^4*b*c*d*g^4*i) + (d^3*atan((d^3*((a^4*d^4*g^4*i
- b^4*c^4*g^4*i + 2*a*b^3*c^3*d*g^4*i - 2*a^3*b*c*d^3*g^4*i)/(a^3*d^3*g^4*i - b^3*c^3*g^4*i + 3*a*b^2*c^2*d*g
^4*i - 3*a^2*b*c*d^2*g^4*i) + 2*b*d*x)*(A + (11*B*n)/6)*(a^3*d^3*g^4*i - b^3*c^3*g^4*i + 3*a*b^2*c^2*d*g^4*i -
3*a^2*b*c*d^2*g^4*i)*6i)/(g^4*i*(6*A*d^3 + 11*B*d^3*n)*(a*d - b*c)^4))*(A + (11*B*n)/6)*2i)/(g^4*i*(a*d - b*c
)^4) - (B*d^3*log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^4*i*n*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*
a^2*b*c*d^2)) + (B*d^3*log(e*((a + b*x)/(c + d*x))^n)*(x*(b*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*
g^4*i*n*(a*d - b*c))/(3*d)) + (2*a*b*g^4*i*n*(a*d - b*c))/(3*d) + (b*g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(3*d^2
)) + a*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*g^4*i*n*(a*d - b*c))/(3*d)) + (g^4*i*n*(a*d - b*c)*(3
*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*d^3) + (b^2*g^4*i*n*x^2*(a*d - b*c))/d))/(g^4*i*n*(a*d - b*c)*(a^3*d^3 - b
^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*g^4*i + b^3*g^4*i*x^3 + 3*a^2*b*g^4*i*x + 3*a*b^2*g^4*i*x^2))

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