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

Optimal. Leaf size=762 \[ -\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}} \]

[Out]

-(n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-
d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b +
Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (n*Log[-((Sqrt[e]*(b - Sqrt[b^2
- 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*
Sqrt[e]) + (n*Log[-((Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])
)]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (Log[Sqrt[-d] - Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2
*Sqrt[-d]*Sqrt[e]) - (Log[Sqrt[-d] + Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[
2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*P
olyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])
+ (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*S
qrt[e]) + (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqr
t[-d]*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 1.45202, antiderivative size = 762, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2528, 2524, 2418, 2394, 2393, 2391} \[ -\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-
d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) - (n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b +
Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (n*Log[-((Sqrt[e]*(b - Sqrt[b^2
- 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*
Sqrt[e]) + (n*Log[-((Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])
)]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e]) + (Log[Sqrt[-d] - Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2
*Sqrt[-d]*Sqrt[e]) - (Log[Sqrt[-d] + Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[
2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*P
olyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])
+ (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*S
qrt[e]) + (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqr
t[-d]*Sqrt[e])

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx &=\int \left (\frac{\sqrt{-d} \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\sqrt{-d} \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 d \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx\\ &=-\frac{\int \frac{\log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{\log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \int \frac{(b+2 c x) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{a+b x+c x^2} \, dx}{2 \sqrt{-d} \sqrt{e}}+\frac{n \int \frac{(b+2 c x) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{a+b x+c x^2} \, dx}{2 \sqrt{-d} \sqrt{e}}\\ &=\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \int \left (\frac{2 c \log \left (\sqrt{-d}-\sqrt{e} x\right )}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{2 c \log \left (\sqrt{-d}-\sqrt{e} x\right )}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{2 \sqrt{-d} \sqrt{e}}+\frac{n \int \left (\frac{2 c \log \left (\sqrt{-d}+\sqrt{e} x\right )}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{2 c \log \left (\sqrt{-d}+\sqrt{e} x\right )}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{2 \sqrt{-d} \sqrt{e}}\\ &=\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{(c n) \int \frac{\log \left (\sqrt{-d}-\sqrt{e} x\right )}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{-d} \sqrt{e}}-\frac{(c n) \int \frac{\log \left (\sqrt{-d}-\sqrt{e} x\right )}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{-d} \sqrt{e}}+\frac{(c n) \int \frac{\log \left (\sqrt{-d}+\sqrt{e} x\right )}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{-d} \sqrt{e}}+\frac{(c n) \int \frac{\log \left (\sqrt{-d}+\sqrt{e} x\right )}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{\sqrt{-d} \sqrt{e}}\\ &=-\frac{n \log \left (\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \int \frac{\log \left (-\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{-2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{n \int \frac{\log \left (\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{-2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{n \int \frac{\log \left (-\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{-2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 \sqrt{-d}}-\frac{n \int \frac{\log \left (\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{-2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \sqrt{-d}}\\ &=-\frac{n \log \left (\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{-2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{x} \, dx,x,\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{-2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{x} \, dx,x,\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{-2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{x} \, dx,x,\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 c x}{-2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{x} \, dx,x,\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}\\ &=-\frac{n \log \left (\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b-\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (-\frac{\sqrt{e} \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \text{Li}_2\left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{2 c \sqrt{-d}+\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \text{Li}_2\left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{2 c \sqrt{-d}+\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{Li}_2\left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\left (b-\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{Li}_2\left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\left (b+\sqrt{b^2-4 a c}\right ) \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.940713, size = 626, normalized size = 0.82 \[ \frac{-n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )-n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )+n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}-b\right )+2 c \sqrt{-d}}\right )+n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )-n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )-n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )+n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}-b\right )+2 c \sqrt{-d}}\right )+n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )-2 c \sqrt{-d}}\right )+\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g (a+x (b+c x))^n\right )}{2 \sqrt{-d} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[
-d] - Sqrt[e]*x]) - n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sq
rt[e])]*Log[Sqrt[-d] - Sqrt[e]*x] + n*Log[(Sqrt[e]*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x))/(2*c*Sqrt[-d] + (-b + Sqr
t[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] + Sqrt[e]*x] + n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*Sq
rt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] + Sqrt[e]*x] + Log[Sqrt[-d] - Sqrt[e]*x]*Log[g*(a + x*
(b + c*x))^n] - Log[Sqrt[-d] + Sqrt[e]*x]*Log[g*(a + x*(b + c*x))^n] - n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x
))/(2*c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])] - n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d]
+ (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])] + n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] + (-b + Sqrt[b^
2 - 4*a*c])*Sqrt[e])] + n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt
[e])])/(2*Sqrt[-d]*Sqrt[e])

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Maple [C]  time = 0.135, size = 610, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(ln((c*x^2+b*x+a)^n)-n*ln(c*x^2+b*x+a))/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/2*n/e*sum(1/_alpha*(ln(x-_alpha)
*ln(c*x^2+b*x+a)-ln(x-_alpha)*ln((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1)-x+_alpha)/
RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1))-ln(x-_alpha)*ln((RootOf(_Z^2*c*e+(2*_alpha*
c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=2)-x+_alpha)/RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,ind
ex=2))-dilog((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1)-x+_alpha)/RootOf(_Z^2*c*e+(2*_
alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1))-dilog((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d
,index=2)-x+_alpha)/RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=2))),_alpha=RootOf(_Z^2*e+d
))-1/2*I/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*Pi*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*g)*csgn(I*g*(c*x^2+b*x+a)^n)+1/
2*I/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*Pi*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*g*(c*x^2+b*x+a)^n)^2+1/2*I/(d*e)^(1/
2)*arctan(e*x/(d*e)^(1/2))*Pi*csgn(I*g)*csgn(I*g*(c*x^2+b*x+a)^n)^2-1/2*I/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*
Pi*csgn(I*g*(c*x^2+b*x+a)^n)^3+1/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))*ln(g)

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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(g*(c*x^2+b*x+a)^n)/(e*x^2+d),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 ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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