3.80 \(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{x (f-g x^2)} \, dx\)

Optimal. Leaf size=518 \[ \frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 f}-\frac {\log (x) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f}-\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}-\frac {n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}+\frac {n \text {Li}_2\left (\frac {b x}{a}+1\right )}{f}+\frac {n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f}+\frac {n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}+\frac {n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}-\frac {n \text {Li}_2\left (\frac {d x}{c}+1\right )}{f}-\frac {n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f} \]

[Out]

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

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Rubi [A]  time = 0.60, antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2513, 266, 36, 29, 31, 2416, 2394, 2315, 260, 2393, 2391} \[ -\frac {n \text {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}-\frac {n \text {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )}{2 f}+\frac {n \text {PolyLog}\left (2,\frac {b x}{a}+1\right )}{f}+\frac {n \text {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}+\frac {n \text {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )}{2 f}-\frac {n \text {PolyLog}\left (2,\frac {d x}{c}+1\right )}{f}+\frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 f}-\frac {\log (x) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 f}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f}+\frac {n \log \left (-\frac {b x}{a}\right ) \log (a+b x)}{f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 f}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f}-\frac {n \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{f} \]

Antiderivative was successfully verified.

[In]

Int[Log[e*((a + b*x)/(c + d*x))^n]/(x*(f - g*x^2)),x]

[Out]

(n*Log[-((b*x)/a)]*Log[a + b*x])/f - (n*Log[-((d*x)/c)]*Log[c + d*x])/f - (Log[x]*(n*Log[a + b*x] - Log[e*((a
+ b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f - (n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[
g])])/(2*f) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f) - (n*Log[a + b*x]*
Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*f) + (n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/
(d*Sqrt[f] - c*Sqrt[g])])/(2*f) + ((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f -
g*x^2])/(2*f) - (n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*f) - (n*PolyLog[2, (Sqrt[g]*
(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f) + (n*PolyLog[2, 1 + (b*x)/a])/f + (n*PolyLog[2, -((Sqrt[g]*(c + d*x
))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*f) + (n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f) - (n*P
olyLog[2, 1 + (d*x)/c])/f

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 260

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2315

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

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 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 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 487, normalized size = 0.94 \[ -\frac {\log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 \log (x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \text {Li}_2\left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} a+b \sqrt {f}}\right )-n \text {Li}_2\left (\frac {b \left (\sqrt {g} x+\sqrt {f}\right )}{b \sqrt {f}-a \sqrt {g}}\right )-n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (a+b x)}{a \sqrt {g}+b \sqrt {f}}\right )-n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )+2 n \text {Li}_2\left (-\frac {b x}{a}\right )+2 n \log (x) \log \left (\frac {b x}{a}+1\right )+n \text {Li}_2\left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{\sqrt {g} c+d \sqrt {f}}\right )+n \text {Li}_2\left (\frac {d \left (\sqrt {g} x+\sqrt {f}\right )}{d \sqrt {f}-c \sqrt {g}}\right )+n \log \left (\sqrt {f}-\sqrt {g} x\right ) \log \left (\frac {\sqrt {g} (c+d x)}{c \sqrt {g}+d \sqrt {f}}\right )+n \log \left (\sqrt {f}+\sqrt {g} x\right ) \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )-2 n \text {Li}_2\left (-\frac {d x}{c}\right )-2 n \log (x) \log \left (\frac {d x}{c}+1\right )}{2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[e*((a + b*x)/(c + d*x))^n]/(x*(f - g*x^2)),x]

[Out]

-1/2*(2*n*Log[x]*Log[1 + (b*x)/a] - 2*Log[x]*Log[e*((a + b*x)/(c + d*x))^n] - 2*n*Log[x]*Log[1 + (d*x)/c] - n*
Log[(Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])]*Log[Sqrt[f] - Sqrt[g]*x] + Log[e*((a + b*x)/(c + d*x))^n]*Log
[Sqrt[f] - Sqrt[g]*x] + n*Log[(Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])]*Log[Sqrt[f] - Sqrt[g]*x] - n*Log[-(
(Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))]*Log[Sqrt[f] + Sqrt[g]*x] + Log[e*((a + b*x)/(c + d*x))^n]*Log[Sq
rt[f] + Sqrt[g]*x] + n*Log[-((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))]*Log[Sqrt[f] + Sqrt[g]*x] + 2*n*Poly
Log[2, -((b*x)/a)] - 2*n*PolyLog[2, -((d*x)/c)] - n*PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g
])] + n*PolyLog[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])] - n*PolyLog[2, (b*(Sqrt[f] + Sqrt[g]*x))
/(b*Sqrt[f] - a*Sqrt[g])] + n*PolyLog[2, (d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/f

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fricas [F]  time = 1.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{3} - f x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)/x/(-g*x^2+f),x, algorithm="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)/x/(-g*x^2+f),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{\left (-g \,x^{2}+f \right ) x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*((b*x+a)/(d*x+c))^n)/x/(-g*x^2+f),x)

[Out]

int(ln(e*((b*x+a)/(d*x+c))^n)/x/(-g*x^2+f),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{{\left (g x^{2} - f\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*((b*x+a)/(d*x+c))^n)/x/(-g*x^2+f),x, algorithm="maxima")

[Out]

-integrate(log(e*((b*x + a)/(d*x + c))^n)/((g*x^2 - f)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{x\,\left (f-g\,x^2\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*((a + b*x)/(c + d*x))^n)/(x*(f - g*x^2)),x)

[Out]

int(log(e*((a + b*x)/(c + d*x))^n)/(x*(f - g*x^2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*((b*x+a)/(d*x+c))**n)/x/(-g*x**2+f),x)

[Out]

Timed out

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