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

Optimal. Leaf size=596 \[ \frac{\sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f^{3/2}}+\frac{\sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f^{3/2}}+\frac{-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)}{f x}-\frac{\sqrt{g} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f^{3/2}}+\frac{\sqrt{g} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f^{3/2}}+\frac{b n \log (x)}{a f}-\frac{b n \log (a+b x)}{a f}-\frac{n \log (a+b x)}{f x}+\frac{\sqrt{g} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f^{3/2}}-\frac{d n \log (x)}{c f}+\frac{d n \log (c+d x)}{c f}+\frac{n \log (c+d x)}{f x} \]

[Out]

(b*n*Log[x])/(a*f) - (d*n*Log[x])/(c*f) - (b*n*Log[a + b*x])/(a*f) - (n*Log[a + b*x])/(f*x) + (d*n*Log[c + d*x
])/(c*f) + (n*Log[c + d*x])/(f*x) + (n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])/(f*x) -
(Sqrt[g]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f^(
3/2) - (Sqrt[g]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*
n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*Log[a + b*x]*L
og[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*f^(3/2)) - (Sqrt[g]*n*Log[c + d*x]*Log[(d*(Sqrt[f] +
Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] -
a*Sqrt[g]))])/(2*f^(3/2)) - (Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) -
(Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, (Sq
rt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.583434, antiderivative size = 596, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2513, 325, 208, 2416, 2395, 36, 29, 31, 2409, 2394, 2393, 2391} \[ \frac{\sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f^{3/2}}+\frac{\sqrt{g} n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{f^{3/2}}+\frac{-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)}{f x}-\frac{\sqrt{g} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 f^{3/2}}+\frac{\sqrt{g} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f^{3/2}}+\frac{b n \log (x)}{a f}-\frac{b n \log (a+b x)}{a f}-\frac{n \log (a+b x)}{f x}+\frac{\sqrt{g} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f^{3/2}}-\frac{d n \log (x)}{c f}+\frac{d n \log (c+d x)}{c f}+\frac{n \log (c+d x)}{f x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*n*Log[x])/(a*f) - (d*n*Log[x])/(c*f) - (b*n*Log[a + b*x])/(a*f) - (n*Log[a + b*x])/(f*x) + (d*n*Log[c + d*x
])/(c*f) + (n*Log[c + d*x])/(f*x) + (n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])/(f*x) -
(Sqrt[g]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x]))/f^(
3/2) - (Sqrt[g]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*
n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*Log[a + b*x]*L
og[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*f^(3/2)) - (Sqrt[g]*n*Log[c + d*x]*Log[(d*(Sqrt[f] +
Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] -
a*Sqrt[g]))])/(2*f^(3/2)) - (Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*f^(3/2)) -
(Sqrt[g]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2*f^(3/2)) + (Sqrt[g]*n*PolyLog[2, (Sq
rt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*f^(3/2))

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]]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 2395

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

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 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 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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{\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x^2 \left (f-g x^2\right )} \, dx &=n \int \frac{\log (a+b x)}{x^2 \left (f-g x^2\right )} \, dx-n \int \frac{\log (c+d x)}{x^2 \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^2 \left (f-g x^2\right )} \, dx\\ &=\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}+n \int \left (\frac{\log (a+b x)}{f x^2}+\frac{g \log (a+b x)}{f \left (f-g x^2\right )}\right ) \, dx-n \int \left (\frac{\log (c+d x)}{f x^2}+\frac{g \log (c+d x)}{f \left (f-g x^2\right )}\right ) \, dx-\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 ) \int \frac{1}{f-g x^2} \, dx}{f}\\ &=\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}+\frac{n \int \frac{\log (a+b x)}{x^2} \, dx}{f}-\frac{n \int \frac{\log (c+d x)}{x^2} \, dx}{f}+\frac{(g n) \int \frac{\log (a+b x)}{f-g x^2} \, dx}{f}-\frac{(g n) \int \frac{\log (c+d x)}{f-g x^2} \, dx}{f}\\ &=-\frac{n \log (a+b x)}{f x}+\frac{n \log (c+d x)}{f x}+\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}+\frac{(b n) \int \frac{1}{x (a+b x)} \, dx}{f}-\frac{(d n) \int \frac{1}{x (c+d x)} \, dx}{f}+\frac{(g n) \int \left (\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{f}-\frac{(g n) \int \left (\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{f}\\ &=-\frac{n \log (a+b x)}{f x}+\frac{n \log (c+d x)}{f x}+\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}+\frac{(b n) \int \frac{1}{x} \, dx}{a f}-\frac{\left (b^2 n\right ) \int \frac{1}{a+b x} \, dx}{a f}-\frac{(d n) \int \frac{1}{x} \, dx}{c f}+\frac{\left (d^2 n\right ) \int \frac{1}{c+d x} \, dx}{c f}+\frac{(g n) \int \frac{\log (a+b x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 f^{3/2}}+\frac{(g n) \int \frac{\log (a+b x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 f^{3/2}}-\frac{(g n) \int \frac{\log (c+d x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 f^{3/2}}-\frac{(g n) \int \frac{\log (c+d x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 f^{3/2}}\\ &=\frac{b n \log (x)}{a f}-\frac{d n \log (x)}{c f}-\frac{b n \log (a+b x)}{a f}-\frac{n \log (a+b x)}{f x}+\frac{d n \log (c+d x)}{c f}+\frac{n \log (c+d x)}{f x}+\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}+\frac{\left (b \sqrt{g} n\right ) \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^{3/2}}-\frac{\left (b \sqrt{g} n\right ) \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^{3/2}}-\frac{\left (d \sqrt{g} n\right ) \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^{3/2}}+\frac{\left (d \sqrt{g} n\right ) \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^{3/2}}\\ &=\frac{b n \log (x)}{a f}-\frac{d n \log (x)}{c f}-\frac{b n \log (a+b x)}{a f}-\frac{n \log (a+b x)}{f x}+\frac{d n \log (c+d x)}{c f}+\frac{n \log (c+d x)}{f x}+\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}-\frac{\left (\sqrt{g} n\right ) \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^{3/2}}+\frac{\left (\sqrt{g} n\right ) \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^{3/2}}+\frac{\left (\sqrt{g} n\right ) \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^{3/2}}-\frac{\left (\sqrt{g} n\right ) \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^{3/2}}\\ &=\frac{b n \log (x)}{a f}-\frac{d n \log (x)}{c f}-\frac{b n \log (a+b x)}{a f}-\frac{n \log (a+b x)}{f x}+\frac{d n \log (c+d x)}{c f}+\frac{n \log (c+d x)}{f x}+\frac{n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac{\sqrt{g} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} 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^{3/2}}-\frac{\sqrt{g} 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^{3/2}}+\frac{\sqrt{g} n \text{Li}_2\left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{Li}_2\left (\frac{\sqrt{g} (a+b x)}{b \sqrt{f}+a \sqrt{g}}\right )}{2 f^{3/2}}-\frac{\sqrt{g} n \text{Li}_2\left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 f^{3/2}}+\frac{\sqrt{g} n \text{Li}_2\left (\frac{\sqrt{g} (c+d x)}{d \sqrt{f}+c \sqrt{g}}\right )}{2 f^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.30863, size = 479, normalized size = 0.8 \[ \frac{\sqrt{g} n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+\log \left (\sqrt{f}-\sqrt{g} x\right ) \left (\log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-\log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )\right )\right )-\sqrt{g} n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \left (\log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )-\log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )\right )\right )-\sqrt{g} \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\sqrt{g} \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\frac{2 \sqrt{f} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x}+\frac{2 \sqrt{f} n (\log (x) (b c-a d)-b c \log (a+b x)+a d \log (c+d x))}{a c}}{2 f^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.455, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( -g{x}^{2}+f \right ) }\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{g x^{4} - f x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-log(e*((b*x + a)/(d*x + c))^n)/(g*x^4 - f*x^2), 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(ln(e*((b*x+a)/(d*x+c))**n)/x**2/(-g*x**2+f),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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