\(\int (f+\frac {g}{x})^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [1]

Optimal result

Integrand size = 32, antiderivative size = 404 \[ \int \left (f+\frac {g}{x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d) g^3 n}{2 a c x}+A f^3 x-\frac {1}{2} B \left (\frac {b^2}{a^2}-\frac {d^2}{c^2}\right ) g^3 n \log (x)+\frac {b^2 B g^3 n \log (a+b x)}{2 a^2}-3 B f^2 g n \log (x) \log \left (1+\frac {b x}{a}\right )+\frac {B f^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}+\frac {3 (b c-a d) f g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a (c+d x) \left (a-\frac {c (a+b x)}{c+d x}\right )}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {B (b c-a d) f^3 n \log (c+d x)}{b d}-\frac {B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (1+\frac {d x}{c}\right )+\frac {3 B (b c-a d) f g^2 n \log \left (a-\frac {c (a+b x)}{c+d x}\right )}{a c}-3 B f^2 g n \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )+3 B f^2 g n \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2608, 2535, 31, 2547, 84, 2553, 2351, 2545, 2354, 2438} \[ \int \left (f+\frac {g}{x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {1}{2} B g^3 n \log (x) \left (\frac {b^2}{a^2}-\frac {d^2}{c^2}\right )+\frac {b^2 B g^3 n \log (a+b x)}{2 a^2}+3 f^2 g \log (x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {3 f g^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a (c+d x) \left (a-\frac {c (a+b x)}{c+d x}\right )}-\frac {g^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 x^2}+\frac {B f^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {B f^3 n (b c-a d) \log (c+d x)}{b d}+\frac {3 B f g^2 n (b c-a d) \log \left (a-\frac {c (a+b x)}{c+d x}\right )}{a c}-\frac {B g^3 n (b c-a d)}{2 a c x}-3 B f^2 g n \operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-3 B f^2 g n \log (x) \log \left (\frac {b x}{a}+1\right )+A f^3 x-\frac {B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )+3 B f^2 g n \log (x) \log \left (\frac {d x}{c}+1\right ) \]

[In]

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

[Out]

-1/2*(B*(b*c - a*d)*g^3*n)/(a*c*x) + A*f^3*x - (B*(b^2/a^2 - d^2/c^2)*g^3*n*Log[x])/2 + (b^2*B*g^3*n*Log[a + b
*x])/(2*a^2) - 3*B*f^2*g*n*Log[x]*Log[1 + (b*x)/a] + (B*f^3*(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]))/(2*x^2) + (3*(b*c - a*d)*f*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(a*(c + d*x)*(a - (c*(a + b*x))/(c + d*x))) + 3*f^2*g*Log[x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
) - (B*(b*c - a*d)*f^3*n*Log[c + d*x])/(b*d) - (B*d^2*g^3*n*Log[c + d*x])/(2*c^2) + 3*B*f^2*g*n*Log[x]*Log[1 +
(d*x)/c] + (3*B*(b*c - a*d)*f*g^2*n*Log[a - (c*(a + b*x))/(c + d*x)])/(a*c) - 3*B*f^2*g*n*PolyLog[2, -((b*x)/
a)] + 3*B*f^2*g*n*PolyLog[2, -((d*x)/c)]

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 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 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*} \text {integral}& = \int \left (f^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{x^3}+\frac {3 f g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{x^2}+\frac {3 f^2 g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{x}\right ) \, dx \\ & = f^3 \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx+\left (3 f^2 g\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x} \, dx+\left (3 f g^2\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^2} \, dx+g^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x^3} \, dx \\ & = A f^3 x-\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\left (B f^3\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+\left (3 (b c-a d) f g^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(-a+c x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )-\left (3 b B f^2 g n\right ) \int \frac {\log (x)}{a+b x} \, dx+\left (3 B d f^2 g n\right ) \int \frac {\log (x)}{c+d x} \, dx+\frac {1}{2} \left (B (b c-a d) g^3 n\right ) \int \frac {1}{x^2 (a+b x) (c+d x)} \, dx \\ & = A f^3 x-3 B f^2 g n \log (x) \log \left (1+\frac {b x}{a}\right )+\frac {B f^3 (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 )}{2 x^2}+\frac {3 (b c-a d) f g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a (c+d x) \left (a-\frac {c (a+b x)}{c+d x}\right )}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+3 B f^2 g n \log (x) \log \left (1+\frac {d x}{c}\right )-\frac {\left (B (b c-a d) f^3 n\right ) \int \frac {1}{c+d x} \, dx}{b}+\left (3 B f^2 g n\right ) \int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx-\left (3 B f^2 g n\right ) \int \frac {\log \left (1+\frac {d x}{c}\right )}{x} \, dx+\frac {\left (3 B (b c-a d) f g^2 n\right ) \text {Subst}\left (\int \frac {1}{-a+c x} \, dx,x,\frac {a+b x}{c+d x}\right )}{a}+\frac {1}{2} \left (B (b c-a d) g^3 n\right ) \int \left (\frac {1}{a c x^2}+\frac {-b c-a d}{a^2 c^2 x}-\frac {b^3}{a^2 (-b c+a d) (a+b x)}-\frac {d^3}{c^2 (b c-a d) (c+d x)}\right ) \, dx \\ & = -\frac {B (b c-a d) g^3 n}{2 a c x}+A f^3 x-\frac {B (b c-a d) (b c+a d) g^3 n \log (x)}{2 a^2 c^2}+\frac {b^2 B g^3 n \log (a+b x)}{2 a^2}-3 B f^2 g n \log (x) \log \left (1+\frac {b x}{a}\right )+\frac {B f^3 (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 )}{2 x^2}+\frac {3 (b c-a d) f g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a (c+d x) \left (a-\frac {c (a+b x)}{c+d x}\right )}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {B (b c-a d) f^3 n \log (c+d x)}{b d}-\frac {B d^2 g^3 n \log (c+d x)}{2 c^2}+3 B f^2 g n \log (x) \log \left (1+\frac {d x}{c}\right )+\frac {3 B (b c-a d) f g^2 n \log \left (a-\frac {c (a+b x)}{c+d x}\right )}{a c}-3 B f^2 g n \text {Li}_2\left (-\frac {b x}{a}\right )+3 B f^2 g n \text {Li}_2\left (-\frac {d x}{c}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.83 \[ \int \left (f+\frac {g}{x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=A f^3 x+\frac {B f^3 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {g^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 x^2}-\frac {3 f g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{x}+3 f^2 g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {B (b c-a d) f^3 n \log (c+d x)}{b d}+\frac {3 B f g^2 n ((b c-a d) \log (x)-b c \log (a+b x)+a d \log (c+d x))}{a c}+\frac {B g^3 n \left (\left (-b^2 c^2 x+a^2 d^2 x\right ) \log (x)+b^2 c^2 x \log (a+b x)+a \left (-b c^2+a c d-a d^2 x \log (c+d x)\right )\right )}{2 a^2 c^2 x}-3 B f^2 g n \left (\log (x) \left (\log \left (1+\frac {b x}{a}\right )-\log \left (1+\frac {d x}{c}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,-\frac {d x}{c}\right )\right ) \]

[In]

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

[Out]

A*f^3*x + (B*f^3*(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]))/(2
*x^2) - (3*f*g^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/x + 3*f^2*g*Log[x]*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]) - (B*(b*c - a*d)*f^3*n*Log[c + d*x])/(b*d) + (3*B*f*g^2*n*((b*c - a*d)*Log[x] - b*c*Log[a + b*x] + a*d*Lo
g[c + d*x]))/(a*c) + (B*g^3*n*((-(b^2*c^2*x) + a^2*d^2*x)*Log[x] + b^2*c^2*x*Log[a + b*x] + a*(-(b*c^2) + a*c*
d - a*d^2*x*Log[c + d*x])))/(2*a^2*c^2*x) - 3*B*f^2*g*n*(Log[x]*(Log[1 + (b*x)/a] - Log[1 + (d*x)/c]) + PolyLo
g[2, -((b*x)/a)] - PolyLog[2, -((d*x)/c)])

Maple [F]

\[\int \left (f +\frac {g}{x}\right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (f+\frac {g}{x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\int {\left (f+\frac {g}{x}\right )}^3\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right ) \,d x \]

[In]

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

[Out]

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