3.76 \(\int \frac{x^3 \log (e (\frac{a+b x}{c+d x})^n)}{f-g x^2} \, dx\)

Optimal. Leaf size=560 \[ -\frac{f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^2}-\frac{f n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^2}+\frac{f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^2}+\frac{f n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^2}+\frac{a^2 n \log (a+b x)}{2 b^2 g}+\frac{f \log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g^2}+\frac{x^2 \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac{f n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^2}-\frac{f n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^2}-\frac{n x^2 \log (a+b x)}{2 g}-\frac{a n x}{2 b g}-\frac{c^2 n \log (c+d x)}{2 d^2 g}+\frac{f n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^2}+\frac{f n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^2}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{c n x}{2 d g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.734765, antiderivative size = 560, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 9, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.281, Rules used = {2513, 266, 43, 2416, 2395, 260, 2394, 2393, 2391} \[ -\frac{f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^2}-\frac{f n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^2}+\frac{f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^2}+\frac{f n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^2}+\frac{a^2 n \log (a+b x)}{2 b^2 g}+\frac{f \log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g^2}+\frac{x^2 \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac{f n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^2}-\frac{f n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^2}-\frac{n x^2 \log (a+b x)}{2 g}-\frac{a n x}{2 b g}-\frac{c^2 n \log (c+d x)}{2 d^2 g}+\frac{f n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^2}+\frac{f n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^2}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{c n x}{2 d g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*n*x)/(2*b*g) + (c*n*x)/(2*d*g) + (a^2*n*Log[a + b*x])/(2*b^2*g) - (n*x^2*Log[a + b*x])/(2*g) - (c^2*n*Log[
c + d*x])/(2*d^2*g) + (n*x^2*Log[c + d*x])/(2*g) + (x^2*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*L
og[c + d*x]))/(2*g) - (f*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f*n
*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^2) - (f*n*Log[a + b*x]*Log[(b*(Sqrt
[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^2) + (f*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[
f] - c*Sqrt[g])])/(2*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*g^2) - (f*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^2) - (f*n*PolyLog[2, (Sqrt
[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g
]))])/(2*g^2) + (f*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^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 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 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 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 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^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx &=n \int \frac{x^3 \log (a+b x)}{f-g x^2} \, dx-n \int \frac{x^3 \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^3}{f-g x^2} \, dx\\ &=n \int \left (-\frac{x \log (a+b x)}{g}+\frac{f x \log (a+b x)}{g \left (f-g x^2\right )}\right ) \, dx-n \int \left (-\frac{x \log (c+d x)}{g}+\frac{f x \log (c+d x)}{g \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{x}{f-g x} \, dx,x,x^2\right )\\ &=-\frac{n \int x \log (a+b x) \, dx}{g}+\frac{n \int x \log (c+d x) \, dx}{g}+\frac{(f n) \int \frac{x \log (a+b x)}{f-g x^2} \, dx}{g}-\frac{(f n) \int \frac{x \log (c+d x)}{f-g x^2} \, dx}{g}-\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 \left (-\frac{1}{g}-\frac{f}{g (-f+g x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{n x^2 \log (a+b x)}{2 g}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{x^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 )}{2 g}+\frac{f \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^2}+\frac{(b n) \int \frac{x^2}{a+b x} \, dx}{2 g}-\frac{(d n) \int \frac{x^2}{c+d x} \, dx}{2 g}+\frac{(f 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}{g}-\frac{(f 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}{g}\\ &=-\frac{n x^2 \log (a+b x)}{2 g}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{x^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 )}{2 g}+\frac{f \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^2}+\frac{(f n) \int \frac{\log (a+b x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 g^{3/2}}-\frac{(f n) \int \frac{\log (a+b x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 g^{3/2}}-\frac{(f n) \int \frac{\log (c+d x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 g^{3/2}}+\frac{(f n) \int \frac{\log (c+d x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 g^{3/2}}+\frac{(b n) \int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx}{2 g}-\frac{(d n) \int \left (-\frac{c}{d^2}+\frac{x}{d}+\frac{c^2}{d^2 (c+d x)}\right ) \, dx}{2 g}\\ &=-\frac{a n x}{2 b g}+\frac{c n x}{2 d g}+\frac{a^2 n \log (a+b x)}{2 b^2 g}-\frac{n x^2 \log (a+b x)}{2 g}-\frac{c^2 n \log (c+d x)}{2 d^2 g}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{x^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 )}{2 g}-\frac{f 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^2}+\frac{f 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^2}-\frac{f 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^2}+\frac{f 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^2}+\frac{f \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^2}+\frac{(b f 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^2}+\frac{(b f 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^2}-\frac{(d f 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^2}-\frac{(d f 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^2}\\ &=-\frac{a n x}{2 b g}+\frac{c n x}{2 d g}+\frac{a^2 n \log (a+b x)}{2 b^2 g}-\frac{n x^2 \log (a+b x)}{2 g}-\frac{c^2 n \log (c+d x)}{2 d^2 g}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{x^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 )}{2 g}-\frac{f 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^2}+\frac{f 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^2}-\frac{f 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^2}+\frac{f 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^2}+\frac{f \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^2}+\frac{(f 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^2}+\frac{(f 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^2}-\frac{(f 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^2}-\frac{(f 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^2}\\ &=-\frac{a n x}{2 b g}+\frac{c n x}{2 d g}+\frac{a^2 n \log (a+b x)}{2 b^2 g}-\frac{n x^2 \log (a+b x)}{2 g}-\frac{c^2 n \log (c+d x)}{2 d^2 g}+\frac{n x^2 \log (c+d x)}{2 g}+\frac{x^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 )}{2 g}-\frac{f 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^2}+\frac{f 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^2}-\frac{f 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^2}+\frac{f 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^2}+\frac{f \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^2}-\frac{f n \text{Li}_2\left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^2}-\frac{f n \text{Li}_2\left (\frac{\sqrt{g} (a+b x)}{b \sqrt{f}+a \sqrt{g}}\right )}{2 g^2}+\frac{f n \text{Li}_2\left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^2}+\frac{f n \text{Li}_2\left (\frac{\sqrt{g} (c+d x)}{d \sqrt{f}+c \sqrt{g}}\right )}{2 g^2}\\ \end{align*}

Mathematica [A]  time = 0.387355, size = 461, normalized size = 0.82 \[ \frac{f n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+\log \left (\sqrt{f}-\sqrt{g} x\right ) \left (\log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-\log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )\right )\right )+f n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \left (\log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )-\log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )\right )\right )+\frac{g n \left (a^2 d^2 \log (a+b x)-b \left (d x (a d-b c)+b c^2 \log (c+d x)\right )\right )}{b^2 d^2}-f \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-f \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-g x^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{2 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(g*x^2*Log[e*((a + b*x)/(c + d*x))^n]) + (g*n*(a^2*d^2*Log[a + b*x] - b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c +
d*x])))/(b^2*d^2) - f*Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] - Sqrt[g]*x] - f*Log[e*((a + b*x)/(c + d*x))^
n]*Log[Sqrt[f] + Sqrt[g]*x] + f*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])]) + f*n*((Log[-((Sqrt[g]*(a + b*x))/(b*Sq
rt[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*S
qrt[g])]))/(2*g^2)

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Maple [F]  time = 0.493, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{-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^3*ln(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{3} \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^3*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x^{3} \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^3*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="fricas")

[Out]

integral(-x^3*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), 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(x**3*ln(e*((b*x+a)/(d*x+c))**n)/(-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{x^{3} \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^3*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="giac")

[Out]

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