3.1.6 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(f+\frac {g}{x})^3} \, dx\) [6]

Optimal. Leaf size=531 \[ \frac {A x}{f^3}+\frac {B (b c-a d) g^3 n}{2 f^3 (a f-b g) (c f-d g) (g+f x)}-\frac {b^2 B g^3 n \log (a+b x)}{2 f^4 (a f-b g)^2}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f^3}+\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 f^4 (g+f x)^2}-\frac {3 g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{f^3 (a f-b g) (g+f x)}-\frac {B (b c-a d) n \log (c+d x)}{b d f^3}+\frac {B d^2 g^3 n \log (c+d x)}{2 f^4 (c f-d g)^2}+\frac {B (b c-a d) g^3 (b c f+a d f-2 b d g) n \log (g+f x)}{2 f^3 (a f-b g)^2 (c f-d g)^2}+\frac {3 B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^4}-\frac {3 g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^4}-\frac {3 B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^4}+\frac {3 B (b c-a d) g^2 n \log \left (\frac {g+f x}{c+d x}\right )}{f^3 (a f-b g) (c f-d g)}+\frac {3 B g n \text {Li}_2\left (-\frac {b (g+f x)}{a f-b g}\right )}{f^4}-\frac {3 B g n \text {Li}_2\left (-\frac {d (g+f x)}{c f-d g}\right )}{f^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.46, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2608, 2535, 31, 2547, 84, 2553, 2351, 2545, 2441, 2440, 2438} \begin {gather*} \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 \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 {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 {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} \end {gather*}

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) + (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

Rule 31

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

Rule 84

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 2351

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

Rule 2438

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]

Rule 2440

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 2441

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 2535

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

Rule 2545

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

Rule 2547

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x
_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Dist[B*n*((b*c -
a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m
, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]

Rule 2553

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

Rule 2608

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]

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) \text {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) \text {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*}

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Mathematica [A]
time = 0.81, size = 470, normalized size = 0.89 \begin {gather*} \frac {2 A f x+\frac {2 B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(g+f x)^2}-\frac {6 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g+f x}-\frac {2 B (b c-a d) f n \log (c+d x)}{b d}-6 g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)+\frac {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 \left (-\frac {b^2 \log (a+b x)}{(b c-a d) (a f-b g)^2}+\frac {\frac {d^2 \log (c+d x)}{b c-a d}+\frac {f \left (\frac {(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)\right )}{(a f-b g)^2}}{(c f-d g)^2}\right )+6 B g n \left (\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 ) \log (g+f x)+\text {Li}_2\left (\frac {b (g+f x)}{-a f+b g}\right )-\text {Li}_2\left (\frac {d (g+f x)}{-c f+d g}\right )\right )}{2 f^4} \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (f +\frac {g}{x}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to 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))/(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)/(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 = 0.00, 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))/(f+g/x)^3,x, algorithm="fricas")

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{{\left (f+\frac {g}{x}\right )}^3} \,d x \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))/(f + g/x)^3,x)

[Out]

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

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