\(\int \frac {\log (g (a+b x+c x^2)^n)}{d+e x^2} \, dx\) [94]

Optimal result

Integrand size = 25, antiderivative size = 762 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=-\frac {n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]

[Out]

1/2*ln(g*(c*x^2+b*x+a)^n)*ln((-d)^(1/2)-x*e^(1/2))/(-d)^(1/2)/e^(1/2)-1/2*ln(g*(c*x^2+b*x+a)^n)*ln((-d)^(1/2)+
x*e^(1/2))/(-d)^(1/2)/e^(1/2)+1/2*n*ln((-d)^(1/2)+x*e^(1/2))*ln(-(b+2*c*x-(-4*a*c+b^2)^(1/2))*e^(1/2)/(2*c*(-d
)^(1/2)-(b-(-4*a*c+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*n*ln((-d)^(1/2)-x*e^(1/2))*ln((b+2*c*x-(-4*a*c
+b^2)^(1/2))*e^(1/2)/(2*c*(-d)^(1/2)+(b-(-4*a*c+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*n*ln((-d)^(1/2)+x
*e^(1/2))*ln(-(b+2*c*x+(-4*a*c+b^2)^(1/2))*e^(1/2)/(2*c*(-d)^(1/2)-(b+(-4*a*c+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)
/e^(1/2)-1/2*n*ln((-d)^(1/2)-x*e^(1/2))*ln((b+2*c*x+(-4*a*c+b^2)^(1/2))*e^(1/2)/(2*c*(-d)^(1/2)+(b+(-4*a*c+b^2
)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*n*polylog(2,2*c*((-d)^(1/2)+x*e^(1/2))/(2*c*(-d)^(1/2)-(b-(-4*a*c+b^
2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*n*polylog(2,2*c*((-d)^(1/2)-x*e^(1/2))/(2*c*(-d)^(1/2)+(b-(-4*a*c+b
^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*n*polylog(2,2*c*((-d)^(1/2)+x*e^(1/2))/(2*c*(-d)^(1/2)-(b+(-4*a*c+
b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2*n*polylog(2,2*c*((-d)^(1/2)-x*e^(1/2))/(2*c*(-d)^(1/2)+(b+(-4*a*c
+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2608, 2604, 2465, 2441, 2440, 2438} \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}} \]

[In]

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

[Out]

-1/2*(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[Sq
rt[-d] - Sqrt[e]*x])/(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*PolyLo
g[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]*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*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*S
qrt[-d]*Sqrt[e])

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 2465

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 2604

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 2608

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]

Rubi steps \begin{align*} \text {integral}& = \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 \text {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 \text {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 \text {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 \text {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] (verified)

Time = 0.64 (sec) , antiderivative size = 626, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\frac {-n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+n \log \left (\frac {\sqrt {e} \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c \sqrt {-d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )+n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )+\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )-n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )-n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )+n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )+n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]

[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])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.88 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.73

method	result	size
risch	\(\text {Expression too large to display}\)	\(555\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\int \frac {\ln \left (g\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{e\,x^2+d} \,d x \]

[In]

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

[Out]

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