3.1.81 \(\int \frac {\log (e (\frac {a+b x}{c+d x})^n)}{x^2 (f-g x^2)} \, dx\) [81]

Optimal. Leaf size=596 \[ \frac {b n \log (x)}{a f}-\frac {d n \log (x)}{c f}-\frac {b n \log (a+b x)}{a f}-\frac {n \log (a+b x)}{f x}+\frac {d n \log (c+d x)}{c f}+\frac {n \log (c+d x)}{f x}+\frac {n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)}{f x}-\frac {\sqrt {g} \tanh ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{f^{3/2}}-\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 f^{3/2}}-\frac {\sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 f^{3/2}}+\frac {\sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 f^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.41, antiderivative size = 596, normalized size of antiderivative = 1.00, 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 = {2593, 331, 214, 2463, 2442, 36, 29, 31, 2456, 2441, 2440, 2438} \begin {gather*} \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} \end {gather*}

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 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 214

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

Rule 331

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 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 2442

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 2456

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 2463

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 2593

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^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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.22, size = 479, normalized size = 0.80 \begin {gather*} \frac {-\frac {2 \sqrt {f} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x}+\frac {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 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\sqrt {g} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\sqrt {g} n \left (\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 ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\text {Li}_2\left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )-\text {Li}_2\left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )\right )-\sqrt {g} n \left (\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 ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\text {Li}_2\left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )-\text {Li}_2\left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )\right )}{2 f^{3/2}} \end {gather*}

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 = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{x^{2} \left (-g \,x^{2}+f \right )}\, dx\]

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 [A]
time = 0.60, size = 970, normalized size = 1.63 \begin {gather*} \frac {1}{2} \, {\left (2 \, a c d {\left (\frac {b^{2} \log \left (b x + a\right )}{{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} f} + \frac {d}{{\left (b c^{2} d - a c d^{2}\right )} f x + {\left (b c^{3} - a c^{2} d\right )} f} - \frac {{\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} f} - \frac {\log \left (x\right )}{a c^{2} f}\right )} + 2 \, b d^{2} {\left (\frac {c}{{\left (b c d^{2} - a d^{3}\right )} f x + {\left (b c^{2} d - a c d^{2}\right )} f} + \frac {a \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {a \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f}\right )} - 2 \, b c d {\left (\frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} + \frac {1}{{\left (b c d - a d^{2}\right )} f x + {\left (b c^{2} - a c d\right )} f}\right )} - 2 \, a d^{2} {\left (\frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} f} + \frac {1}{{\left (b c d - a d^{2}\right )} f x + {\left (b c^{2} - a c d\right )} f}\right )} - 2 \, b c {\left (\frac {b \log \left (b x + a\right )}{{\left (a b c - a^{2} d\right )} f} - \frac {d \log \left (d x + c\right )}{{\left (b c^{2} - a c d\right )} f} - \frac {\log \left (x\right )}{a c f}\right )} + 2 \, b d {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} f} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} f}\right )} + \frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {b \sqrt {g} x - b \sqrt {f}}{b \sqrt {f} + a \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} - \frac {{\left (\log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}} + 1\right ) + {\rm Li}_2\left (\frac {b \sqrt {g} x + b \sqrt {f}}{b \sqrt {f} - a \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} - \frac {{\left (\log \left (\sqrt {g} x - \sqrt {f}\right ) \log \left (\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (-\frac {d \sqrt {g} x - d \sqrt {f}}{d \sqrt {f} + c \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}} + \frac {{\left (\log \left (\sqrt {g} x + \sqrt {f}\right ) \log \left (-\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}} + 1\right ) + {\rm Li}_2\left (\frac {d \sqrt {g} x + d \sqrt {f}}{d \sqrt {f} - c \sqrt {g}}\right )\right )} \sqrt {g}}{f^{\frac {3}{2}}}\right )} n - \frac {1}{2} \, {\left (\frac {g \log \left (\frac {g x - \sqrt {f g}}{g x + \sqrt {f g}}\right )}{\sqrt {f g} f} + \frac {2}{f x}\right )} \log \left (\left (\frac {b x + a}{d x + c}\right )^{n} e\right ) \end {gather*}

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]

1/2*(2*a*c*d*(b^2*log(b*x + a)/((a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*f) + d/((b*c^2*d - a*c*d^2)*f*x + (b*c^3 -
a*c^2*d)*f) - (2*b*c*d - a*d^2)*log(d*x + c)/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f) - log(x)/(a*c^2*f)) +
2*b*d^2*(c/((b*c*d^2 - a*d^3)*f*x + (b*c^2*d - a*c*d^2)*f) + a*log(b*x + a)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f
) - a*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f)) - 2*b*c*d*(b*log(b*x + a)/((b^2*c^2 - 2*a*b*c*d + a^2*
d^2)*f) - b*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f) + 1/((b*c*d - a*d^2)*f*x + (b*c^2 - a*c*d)*f)) -
2*a*d^2*(b*log(b*x + a)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*f) - b*log(d*x + c)/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*
f) + 1/((b*c*d - a*d^2)*f*x + (b*c^2 - a*c*d)*f)) - 2*b*c*(b*log(b*x + a)/((a*b*c - a^2*d)*f) - d*log(d*x + c)
/((b*c^2 - a*c*d)*f) - log(x)/(a*c*f)) + 2*b*d*(log(b*x + a)/((b*c - a*d)*f) - log(d*x + c)/((b*c - a*d)*f)) +
(log(sqrt(g)*x - sqrt(f))*log((b*sqrt(g)*x - b*sqrt(f))/(b*sqrt(f) + a*sqrt(g)) + 1) + dilog(-(b*sqrt(g)*x -
b*sqrt(f))/(b*sqrt(f) + a*sqrt(g))))*sqrt(g)/f^(3/2) - (log(sqrt(g)*x + sqrt(f))*log(-(b*sqrt(g)*x + b*sqrt(f)
)/(b*sqrt(f) - a*sqrt(g)) + 1) + dilog((b*sqrt(g)*x + b*sqrt(f))/(b*sqrt(f) - a*sqrt(g))))*sqrt(g)/f^(3/2) - (
log(sqrt(g)*x - sqrt(f))*log((d*sqrt(g)*x - d*sqrt(f))/(d*sqrt(f) + c*sqrt(g)) + 1) + dilog(-(d*sqrt(g)*x - d*
sqrt(f))/(d*sqrt(f) + c*sqrt(g))))*sqrt(g)/f^(3/2) + (log(sqrt(g)*x + sqrt(f))*log(-(d*sqrt(g)*x + d*sqrt(f))/
(d*sqrt(f) - c*sqrt(g)) + 1) + dilog((d*sqrt(g)*x + d*sqrt(f))/(d*sqrt(f) - c*sqrt(g))))*sqrt(g)/f^(3/2))*n -
1/2*(g*log((g*x - sqrt(f*g))/(g*x + sqrt(f*g)))/(sqrt(f*g)*f) + 2/(f*x))*log(((b*x + a)/(d*x + c))^n*e)

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

[Out]

integral(-log(((b*x + a)/(d*x + c))^n*e)/(g*x^4 - f*x^2), 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(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 = 0.00, 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(log(e*((b*x+a)/(d*x+c))^n)/x^2/(-g*x^2+f),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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