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

Optimal. Leaf size=403 \[ -\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g}-\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g}+\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g}+\frac{\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.352342, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2513, 260, 2416, 2394, 2393, 2391} \[ -\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g}-\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g}+\frac{n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g}+\frac{n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g}+\frac{\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g}-\frac{n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g}+\frac{n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.121141, size = 413, normalized size = 1.02 \[ -\frac{-n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-n \text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )+n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+n \text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )+\log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )+n \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )+n \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-(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[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
*PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])] + n*PolyLog[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqr
t[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])])/(2*g)

________________________________________________________________________________________

Maple [F]  time = 0.44, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{-g{x}^{2}+f}\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(x*ln(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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