\(\int \log ^2(d (a+b x+c x^2)^n) \, dx\) [97]

Optimal result

Integrand size = 17, antiderivative size = 587 \[ \int \log ^2\left (d \left (a+b x+c x^2\right )^n\right ) \, dx=8 n^2 x-\frac {4 \sqrt {b^2-4 a c} n^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \operatorname {PolyLog}\left (2,-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \operatorname {PolyLog}\left (2,\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {2603, 2608, 787, 648, 632, 212, 642, 2604, 2465, 2437, 2338, 2441, 2440, 2438} \[ \int \log ^2\left (d \left (a+b x+c x^2\right )^n\right ) \, dx=-\frac {4 n^2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}+\frac {n \left (b-\sqrt {b^2-4 a c}\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {n \left (\sqrt {b^2-4 a c}+b\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}-\frac {n^2 \left (b-\sqrt {b^2-4 a c}\right ) \operatorname {PolyLog}\left (2,-\frac {b+2 c x-\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {n^2 \left (\sqrt {b^2-4 a c}+b\right ) \operatorname {PolyLog}\left (2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {n^2 \left (b-\sqrt {b^2-4 a c}\right ) \log ^2\left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c}-\frac {n^2 \left (\sqrt {b^2-4 a c}+b\right ) \log ^2\left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c}-\frac {n^2 \left (\sqrt {b^2-4 a c}+b\right ) \log \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c}-\frac {n^2 \left (b-\sqrt {b^2-4 a c}\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}+8 n^2 x \]

[In]

Int[Log[d*(a + b*x + c*x^2)^n]^2,x]

[Out]

8*n^2*x - (4*Sqrt[b^2 - 4*a*c]*n^2*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c - ((b - Sqrt[b^2 - 4*a*c])*n^2*Lo
g[b - Sqrt[b^2 - 4*a*c] + 2*c*x]^2)/(2*c) - ((b + Sqrt[b^2 - 4*a*c])*n^2*Log[-1/2*(b - Sqrt[b^2 - 4*a*c] + 2*c
*x)/Sqrt[b^2 - 4*a*c]]*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x])/c - ((b + Sqrt[b^2 - 4*a*c])*n^2*Log[b + Sqrt[b^2 -
4*a*c] + 2*c*x]^2)/(2*c) - ((b - Sqrt[b^2 - 4*a*c])*n^2*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[(b + Sqrt[b^2
- 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c - (2*b*n^2*Log[a + b*x + c*x^2])/c - 4*n*x*Log[d*(a + b*x + c*x^2)
^n] + ((b - Sqrt[b^2 - 4*a*c])*n*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[d*(a + b*x + c*x^2)^n])/c + ((b + Sqrt
[b^2 - 4*a*c])*n*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[d*(a + b*x + c*x^2)^n])/c + x*Log[d*(a + b*x + c*x^2)^
n]^2 - ((b - Sqrt[b^2 - 4*a*c])*n^2*PolyLog[2, -1/2*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c - ((
b + Sqrt[b^2 - 4*a*c])*n^2*PolyLog[2, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/c

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 787

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*g*(x/c
), x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

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 2603

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

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}& = x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-(2 n) \int \frac {x (b+2 c x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{a+b x+c x^2} \, dx \\ & = x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-(2 n) \int \left (2 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {(2 a+b x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{a+b x+c x^2}\right ) \, dx \\ & = x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )+(2 n) \int \frac {(2 a+b x) \log \left (d \left (a+b x+c x^2\right )^n\right )}{a+b x+c x^2} \, dx-(4 n) \int \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx \\ & = -4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )+(2 n) \int \left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx+\left (4 n^2\right ) \int \frac {x (b+2 c x)}{a+b x+c x^2} \, dx \\ & = 8 n^2 x-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )+\left (2 \left (b-\sqrt {b^2-4 a c}\right ) n\right ) \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx+\left (2 \left (b+\sqrt {b^2-4 a c}\right ) n\right ) \int \frac {\log \left (d \left (a+b x+c x^2\right )^n\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx+\frac {\left (4 n^2\right ) \int \frac {-2 a c-b c x}{a+b x+c x^2} \, dx}{c} \\ & = 8 n^2 x-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-\frac {\left (2 b n^2\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c}+\frac {\left (2 \left (b^2-4 a c\right ) n^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{c}-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {(b+2 c x) \log \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{a+b x+c x^2} \, dx}{c}-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {(b+2 c x) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{a+b x+c x^2} \, dx}{c} \\ & = 8 n^2 x-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-\frac {\left (4 \left (b^2-4 a c\right ) n^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c}-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \int \left (\frac {2 c \log \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c}-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \int \left (\frac {2 c \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{c} \\ & = 8 n^2 x-\frac {4 \sqrt {b^2-4 a c} n^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-\left (2 \left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx-\left (2 \left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx-\left (2 \left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx-\left (2 \left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx \\ & = 8 n^2 x-\frac {4 \sqrt {b^2-4 a c} n^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )+\left (2 \left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (\frac {2 c \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \left (b-\sqrt {b^2-4 a c}\right )+2 c \left (b+\sqrt {b^2-4 a c}\right )}\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,b-\sqrt {b^2-4 a c}+2 c x\right )}{c}+\left (2 \left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \int \frac {\log \left (\frac {2 c \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \left (b-\sqrt {b^2-4 a c}\right )-2 c \left (b+\sqrt {b^2-4 a c}\right )}\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,b+\sqrt {b^2-4 a c}+2 c x\right )}{c} \\ & = 8 n^2 x-\frac {4 \sqrt {b^2-4 a c} n^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c \left (b-\sqrt {b^2-4 a c}\right )+2 c \left (b+\sqrt {b^2-4 a c}\right )}\right )}{x} \, dx,x,b-\sqrt {b^2-4 a c}+2 c x\right )}{c}+\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{2 c \left (b-\sqrt {b^2-4 a c}\right )-2 c \left (b+\sqrt {b^2-4 a c}\right )}\right )}{x} \, dx,x,b+\sqrt {b^2-4 a c}+2 c x\right )}{c} \\ & = 8 n^2 x-\frac {4 \sqrt {b^2-4 a c} n^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \log ^2\left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c}-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {2 b n^2 \log \left (a+b x+c x^2\right )}{c}-4 n x \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b-\sqrt {b^2-4 a c}\right ) n \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+\frac {\left (b+\sqrt {b^2-4 a c}\right ) n \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d \left (a+b x+c x^2\right )^n\right )}{c}+x \log ^2\left (d \left (a+b x+c x^2\right )^n\right )-\frac {\left (b-\sqrt {b^2-4 a c}\right ) n^2 \text {Li}_2\left (-\frac {b-\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c}-\frac {\left (b+\sqrt {b^2-4 a c}\right ) n^2 \text {Li}_2\left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 478, normalized size of antiderivative = 0.81 \[ \int \log ^2\left (d \left (a+b x+c x^2\right )^n\right ) \, dx=x \log ^2\left (d (a+x (b+c x))^n\right )+\frac {n \left (4 n \left (4 c x-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )-b \log (a+x (b+c x))\right )-8 c x \log \left (d (a+x (b+c x))^n\right )+2 \left (b-\sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )+2 \left (b+\sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \log \left (d (a+x (b+c x))^n\right )+\left (-b+\sqrt {b^2-4 a c}\right ) n \left (\log \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\log \left (b-\sqrt {b^2-4 a c}+2 c x\right )+2 \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )\right )-\left (b+\sqrt {b^2-4 a c}\right ) n \left (\log \left (b+\sqrt {b^2-4 a c}+2 c x\right ) \left (2 \log \left (\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )+\log \left (b+\sqrt {b^2-4 a c}+2 c x\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {b+\sqrt {b^2-4 a c}+2 c x}{2 \sqrt {b^2-4 a c}}\right )\right )\right )}{2 c} \]

[In]

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

[Out]

x*Log[d*(a + x*(b + c*x))^n]^2 + (n*(4*n*(4*c*x - 2*Sqrt[b^2 - 4*a*c]*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] -
b*Log[a + x*(b + c*x)]) - 8*c*x*Log[d*(a + x*(b + c*x))^n] + 2*(b - Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b^2 - 4*a
*c] + 2*c*x]*Log[d*(a + x*(b + c*x))^n] + 2*(b + Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Log[d*(
a + x*(b + c*x))^n] + (-b + Sqrt[b^2 - 4*a*c])*n*(Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*(Log[b - Sqrt[b^2 - 4*a*c
] + 2*c*x] + 2*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]) + 2*PolyLog[2, (-b + Sqrt[b^2 - 4*a
*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])]) - (b + Sqrt[b^2 - 4*a*c])*n*(Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*(2*Log[(-
b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a*c])] + Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x]) + 2*PolyLog[2, (b
+ Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])))/(2*c)

Maple [F]

\[\int {\ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(log(d*(a + b*x + c*x**2)**n)**2, x)

Maxima [F(-2)]

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

[In]

integrate(log(d*(c*x^2+b*x+a)^n)^2,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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(log(d*(a + b*x + c*x^2)^n)^2,x)

[Out]

int(log(d*(a + b*x + c*x^2)^n)^2, x)