3.6 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(f+\frac{g}{x})^3} \, dx\)

Optimal. Leaf size=531 \[ \frac{3 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^4}+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 f^4 (f x+g)^2}-\frac{3 g^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^3 (f x+g) (a f-b g)}-\frac{3 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{B g^3 n (b c-a d)}{2 f^3 (f x+g) (a f-b g) (c f-d g)}+\frac{B g^3 n (b c-a d) \log (f x+g) (a d f+b c f-2 b d g)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g^2 n (b c-a d) \log \left (\frac{f x+g}{c+d x}\right )}{f^3 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^3}+\frac{3 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^4}+\frac{A x}{f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}-\frac{3 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.796941, antiderivative size = 562, normalized size of antiderivative = 1.06, number of steps used = 22, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2528, 2486, 31, 2525, 12, 72, 2524, 2418, 2394, 2393, 2391} \[ \frac{3 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^4}+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 f^4 (f x+g)^2}-\frac{3 g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4 (f x+g)}-\frac{3 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^4}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{B g^3 n (b c-a d)}{2 f^3 (f x+g) (a f-b g) (c f-d g)}+\frac{B g^3 n (b c-a d) \log (f x+g) (a d f+b c f-2 b d g)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g^2 n (b c-a d) \log (f x+g)}{f^3 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^3}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{3 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^4}+\frac{A x}{f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}-\frac{3 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{\left (f+\frac{g}{x}\right )^3} \, dx &=\int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f^3}-\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)^3}+\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)^2}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}\right ) \, dx\\ &=\frac{\int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{f^3}-\frac{(3 g) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f^3}+\frac{\left (3 g^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(g+f x)^2} \, dx}{f^3}-\frac{g^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(g+f x)^3} \, dx}{f^3}\\ &=\frac{A x}{f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}+\frac{B \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{f^3}+\frac{(3 B g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (g+f x)}{a+b x} \, dx}{f^4}+\frac{\left (3 B g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x) (g+f x)} \, dx}{f^4}-\frac{\left (B g^3 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x) (g+f x)^2} \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b f^3}+\frac{(3 B g n) \int \left (\frac{b \log (g+f x)}{a+b x}-\frac{d \log (g+f x)}{c+d x}\right ) \, dx}{f^4}+\frac{\left (3 B (b c-a d) g^2 n\right ) \int \frac{1}{(a+b x) (c+d x) (g+f x)} \, dx}{f^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \frac{1}{(a+b x) (c+d x) (g+f x)^2} \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}+\frac{(3 b B g n) \int \frac{\log (g+f x)}{a+b x} \, dx}{f^4}-\frac{(3 B d g n) \int \frac{\log (g+f x)}{c+d x} \, dx}{f^4}+\frac{\left (3 B (b c-a d) g^2 n\right ) \int \left (\frac{b^2}{(b c-a d) (-a f+b g) (a+b x)}+\frac{d^2}{(b c-a d) (c f-d g) (c+d x)}+\frac{f^2}{(a f-b g) (c f-d g) (g+f x)}\right ) \, dx}{f^4}-\frac{\left (B (b c-a d) g^3 n\right ) \int \left (\frac{b^3}{(b c-a d) (-a f+b g)^2 (a+b x)}-\frac{d^3}{(b c-a d) (c f-d g)^2 (c+d x)}+\frac{f^2}{(a f-b g) (c f-d g) (g+f x)^2}-\frac{f^2 (b c f+a d f-2 b d g)}{(a f-b g)^2 (c f-d g)^2 (g+f x)}\right ) \, dx}{2 f^4}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}-\frac{(3 B g n) \int \frac{\log \left (\frac{f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f^3}+\frac{(3 B g n) \int \frac{\log \left (\frac{f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f^3}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}-\frac{(3 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a f-b g}\right )}{x} \, dx,x,g+f x\right )}{f^4}+\frac{(3 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{c f-d g}\right )}{x} \, dx,x,g+f x\right )}{f^4}\\ &=\frac{A x}{f^3}+\frac{B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac{b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}-\frac{3 b B g^2 n \log (a+b x)}{f^4 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac{g^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac{3 g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^4 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^3}+\frac{B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac{3 B d g^2 n \log (c+d x)}{f^4 (c f-d g)}+\frac{3 B (b c-a d) g^2 n \log (g+f x)}{f^3 (a f-b g) (c f-d g)}+\frac{B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac{3 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac{3 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac{3 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}+\frac{3 B g n \text{Li}_2\left (-\frac{b (g+f x)}{a f-b g}\right )}{f^4}-\frac{3 B g n \text{Li}_2\left (-\frac{d (g+f x)}{c f-d g}\right )}{f^4}\\ \end{align*}

Mathematica [A]  time = 1.65344, size = 470, normalized size = 0.89 \[ \frac{6 B g n \left (\text{PolyLog}\left (2,\frac{b (f x+g)}{b g-a f}\right )-\text{PolyLog}\left (2,\frac{d (f x+g)}{d g-c f}\right )+\log (f x+g) \left (\log \left (\frac{f (a+b x)}{a f-b g}\right )-\log \left (\frac{f (c+d x)}{c f-d g}\right )\right )\right )+\frac{g^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(f x+g)^2}-\frac{6 g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f x+g}-6 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+B g^3 n (b c-a d) \left (\frac{\frac{d^2 \log (c+d x)}{b c-a d}+\frac{f \left (\frac{(a f-b g) (c f-d g)}{f x+g}+\log (f x+g) (a d f+b c f-2 b d g)\right )}{(a f-b g)^2}}{(c f-d g)^2}-\frac{b^2 \log (a+b x)}{(b c-a d) (a f-b g)^2}\right )+\frac{2 B f (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}+\frac{6 B g^2 n (b \log (a+b x) (d g-c f)+d (a f-b g) \log (c+d x)+f (b c-a d) \log (f x+g))}{(a f-b g) (c f-d g)}-\frac{2 B f n (b c-a d) \log (c+d x)}{b d}+2 A f x}{2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*f*x + (2*B*f*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + (g^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(
g + f*x)^2 - (6*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(g + f*x) - (2*B*(b*c - a*d)*f*n*Log[c + d*x])/(b*
d) - 6*g*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[g + f*x] + (6*B*g^2*n*(b*(-(c*f) + d*g)*Log[a + b*x] + d*(
a*f - b*g)*Log[c + d*x] + (b*c - a*d)*f*Log[g + f*x]))/((a*f - b*g)*(c*f - d*g)) + B*(b*c - a*d)*g^3*n*(-((b^2
*Log[a + b*x])/((b*c - a*d)*(a*f - b*g)^2)) + ((d^2*Log[c + d*x])/(b*c - a*d) + (f*(((a*f - b*g)*(c*f - d*g))/
(g + f*x) + (b*c*f + a*d*f - 2*b*d*g)*Log[g + f*x]))/(a*f - b*g)^2)/(c*f - d*g)^2) + 6*B*g*n*((Log[(f*(a + b*x
))/(a*f - b*g)] - Log[(f*(c + d*x))/(c*f - d*g)])*Log[g + f*x] + PolyLog[2, (b*(g + f*x))/(-(a*f) + b*g)] - Po
lyLog[2, (d*(g + f*x))/(-(c*f) + d*g)]))/(2*f^4)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \left ( f+{\frac{g}{x}} \right ) ^{-3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, A{\left (\frac{6 \, f g^{2} x + 5 \, g^{3}}{f^{6} x^{2} + 2 \, f^{5} g x + f^{4} g^{2}} - \frac{2 \, x}{f^{3}} + \frac{6 \, g \log \left (f x + g\right )}{f^{4}}\right )} - B \int -\frac{x^{3} \log \left ({\left (b x + a\right )}^{n}\right ) - x^{3} \log \left ({\left (d x + c\right )}^{n}\right ) + x^{3} \log \left (e\right )}{f^{3} x^{3} + 3 \, f^{2} g x^{2} + 3 \, f g^{2} x + g^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*A*((6*f*g^2*x + 5*g^3)/(f^6*x^2 + 2*f^5*g*x + f^4*g^2) - 2*x/f^3 + 6*g*log(f*x + g)/f^4) - B*integrate(-(
x^3*log((b*x + a)^n) - x^3*log((d*x + c)^n) + x^3*log(e))/(f^3*x^3 + 3*f^2*g*x^2 + 3*f*g^2*x + g^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B x^{3} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A x^{3}}{f^{3} x^{3} + 3 \, f^{2} g x^{2} + 3 \, f g^{2} x + g^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^3*log(e*((b*x + a)/(d*x + c))^n) + A*x^3)/(f^3*x^3 + 3*f^2*g*x^2 + 3*f*g^2*x + g^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (f + \frac{g}{x}\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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