3.80 \(\int \frac{\log (d (a+b x+c x^2)^n)}{x^5} \, dx\)

Optimal. Leaf size=190 \[ \frac{n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 a^4}-\frac{n \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{4 a^4}+\frac{n \left (b^2-2 a c\right )}{8 a^2 x^2}-\frac{b n \left (b^2-3 a c\right )}{4 a^3 x}-\frac{b n \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac{b n}{12 a x^3} \]

[Out]

-(b*n)/(12*a*x^3) + ((b^2 - 2*a*c)*n)/(8*a^2*x^2) - (b*(b^2 - 3*a*c)*n)/(4*a^3*x) - (b*Sqrt[b^2 - 4*a*c]*(b^2
- 2*a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*a^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[x])/(4*a^4)
+ ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + b*x + c*x^2])/(8*a^4) - Log[d*(a + b*x + c*x^2)^n]/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.222803, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac{n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 a^4}-\frac{n \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{4 a^4}+\frac{n \left (b^2-2 a c\right )}{8 a^2 x^2}-\frac{b n \left (b^2-3 a c\right )}{4 a^3 x}-\frac{b n \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac{b n}{12 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/x^5,x]

[Out]

-(b*n)/(12*a*x^3) + ((b^2 - 2*a*c)*n)/(8*a^2*x^2) - (b*(b^2 - 3*a*c)*n)/(4*a^3*x) - (b*Sqrt[b^2 - 4*a*c]*(b^2
- 2*a*c)*n*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(4*a^4) - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[x])/(4*a^4)
+ ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*Log[a + b*x + c*x^2])/(8*a^4) - Log[d*(a + b*x + c*x^2)^n]/(4*x^4)

Rule 2525

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 634

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 618

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 206

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

Rule 628

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]

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^5} \, dx &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac{1}{4} n \int \frac{b+2 c x}{x^4 \left (a+b x+c x^2\right )} \, dx\\ &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac{1}{4} n \int \left (\frac{b}{a x^4}+\frac{-b^2+2 a c}{a^2 x^3}+\frac{b^3-3 a b c}{a^3 x^2}+\frac{-b^4+4 a b^2 c-2 a^2 c^2}{a^4 x}+\frac{b \left (b^4-5 a b^2 c+5 a^2 c^2\right )+c \left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{b n}{12 a x^3}+\frac{\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac{b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac{n \int \frac{b \left (b^4-5 a b^2 c+5 a^2 c^2\right )+c \left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{4 a^4}\\ &=-\frac{b n}{12 a x^3}+\frac{\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac{b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}+\frac{\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{8 a^4}+\frac{\left (\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{8 a^4}\\ &=-\frac{b n}{12 a x^3}+\frac{\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac{b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}+\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}-\frac{\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 a^4}\\ &=-\frac{b n}{12 a x^3}+\frac{\left (b^2-2 a c\right ) n}{8 a^2 x^2}-\frac{b \left (b^2-3 a c\right ) n}{4 a^3 x}-\frac{b \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 a^4}-\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log (x)}{4 a^4}+\frac{\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 a^4}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.450394, size = 172, normalized size = 0.91 \[ -\frac{\frac{n x \left (6 x^3 \log (x) \left (2 a^2 c^2-4 a b^2 c+b^4\right )-3 x^3 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log (a+x (b+c x))-3 a^2 x \left (b^2-2 a c\right )+2 a^3 b+6 a b x^2 \left (b^2-3 a c\right )+6 b x^3 \sqrt{b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\right )}{a^4}+6 \log \left (d (a+x (b+c x))^n\right )}{24 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[d*(a + b*x + c*x^2)^n]/x^5,x]

[Out]

-((n*x*(2*a^3*b - 3*a^2*(b^2 - 2*a*c)*x + 6*a*b*(b^2 - 3*a*c)*x^2 + 6*b*Sqrt[b^2 - 4*a*c]*(b^2 - 2*a*c)*x^3*Ar
cTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]] + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^3*Log[x] - 3*(b^4 - 4*a*b^2*c + 2*a^2
*c^2)*x^3*Log[a + x*(b + c*x)]))/a^4 + 6*Log[d*(a + x*(b + c*x))^n])/(24*x^4)

________________________________________________________________________________________

Maple [C]  time = 0.114, size = 3583, normalized size = 18.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(c*x^2+b*x+a)^n)/x^5,x)

[Out]

-1/4/x^4*ln((c*x^2+b*x+a)^n)-1/24*(-6*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*b^6*c^2+20*a*b^8*c-2*b^10+6*
(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3-19*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)
^(1/2)*a^2*b^2*c^2+12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c-2*(-16*a^3*b^2*c^3+20*a^2*b
^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b^7*c*a^2-2*a*b^9-7*(-16*a^3*
b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2+9*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a
^2*b^3*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*c^2*a^2*x^4+12*n*ln((-40*a^4*b^2*c^4+94
*a^3*b^4*c^3-69*a^2*b^6*c^2+20*a*b^8*c-2*b^10+6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3-1
9*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2+12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*
c+b^8)^(1/2)*a*b^4*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3
-61*a^3*b^5*c^2+19*b^7*c*a^2-2*a*b^9-7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2+9*(-16*a
^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2
)*a*b^5)*b^2*c*a*x^4-3*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*b^6*c^2+20*a*b^8*c-2*b^10+6*(-16*a^3*b^2*c^
3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3-19*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*
c^2+12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*
c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b^7*c*a^2-2*a*b^9-7*(-16*a^3*b^2*c^3+20*a^2*
b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2+9*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c-2*(-16
*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*b^4*x^4-6*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*
b^6*c^2+20*a*b^8*c-2*b^10-6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3+19*(-16*a^3*b^2*c^3+2
0*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2-12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c
+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b
^7*c*a^2-2*a*b^9+7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2-9*(-16*a^3*b^2*c^3+20*a^2*b^
4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*c^2*a^2*x^4
+12*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*b^6*c^2+20*a*b^8*c-2*b^10-6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*
a*b^6*c+b^8)^(1/2)*a^3*c^3+19*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2-12*(-16*a^3*b^2
*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x
-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b^7*c*a^2-2*a*b^9+7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b
^8)^(1/2)*a^3*b*c^2-9*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c+2*(-16*a^3*b^2*c^3+20*a^2
*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*b^2*c*a*x^4-3*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*b^6*c^2+20*a*b^
8*c-2*b^10-6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3+19*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8
*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2-12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c+2*(-16*a^3*b^2
*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b^7*c*a^2-2*a*b^
9+7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2-9*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c
+b^8)^(1/2)*a^2*b^3*c+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*b^4*x^4+12*n*ln(x)*c^2*a^2
*x^4-24*n*ln(x)*b^2*c*a*x^4+6*n*ln(x)*b^4*x^4+3*I*Pi*a^4*csgn(I*d)*csgn(I*d*(c*x^2+b*x+a)^n)^2-3*I*Pi*a^4*csgn
(I*d)*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)+3*I*Pi*a^4*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x
+a)^n)^2-3*I*Pi*a^4*csgn(I*d*(c*x^2+b*x+a)^n)^3-18*a^2*b*c*n*x^3+6*a*b^3*n*x^3-3*n*ln((-40*a^4*b^2*c^4+94*a^3*
b^4*c^3-69*a^2*b^6*c^2+20*a*b^8*c-2*b^10+6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3-19*(-1
6*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2+12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8
)^(1/2)*a*b^4*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a
^3*b^5*c^2+19*b^7*c*a^2-2*a*b^9-7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2+9*(-16*a^3*b^
2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c-2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b
^5)*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*x^4+3*n*ln((-40*a^4*b^2*c^4+94*a^3*b^4*c^3-69*a^2*b^6
*c^2+20*a*b^8*c-2*b^10-6*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*c^3+19*(-16*a^3*b^2*c^3+20*a
^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^2*c^2-12*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^4*c+2*
(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*b^6)*x-24*a^5*b*c^4+74*a^4*b^3*c^3-61*a^3*b^5*c^2+19*b^7*
c*a^2-2*a*b^9+7*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a^3*b*c^2-9*(-16*a^3*b^2*c^3+20*a^2*b^4*c
^2-8*a*b^6*c+b^8)^(1/2)*a^2*b^3*c+2*(-16*a^3*b^2*c^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*a*b^5)*(-16*a^3*b^2*c
^3+20*a^2*b^4*c^2-8*a*b^6*c+b^8)^(1/2)*x^4+6*a^3*c*n*x^2-3*a^2*b^2*n*x^2+2*a^3*b*n*x+6*ln(d)*a^4)/a^4/x^4

________________________________________________________________________________________

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(d*(c*x^2+b*x+a)^n)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.80349, size = 932, normalized size = 4.91 \begin{align*} \left [-\frac{3 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} n x^{4} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} \log \left (x\right ) + 2 \, a^{3} b n x + 6 \,{\left (a b^{3} - 3 \, a^{2} b c\right )} n x^{3} + 6 \, a^{4} \log \left (d\right ) - 3 \,{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} n x^{2} - 3 \,{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} - 2 \, a^{4} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, a^{4} x^{4}}, -\frac{6 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} n x^{4} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} \log \left (x\right ) + 2 \, a^{3} b n x + 6 \,{\left (a b^{3} - 3 \, a^{2} b c\right )} n x^{3} + 6 \, a^{4} \log \left (d\right ) - 3 \,{\left (a^{2} b^{2} - 2 \, a^{3} c\right )} n x^{2} - 3 \,{\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n x^{4} - 2 \, a^{4} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, a^{4} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^5,x, algorithm="fricas")

[Out]

[-1/24*(3*(b^3 - 2*a*b*c)*sqrt(b^2 - 4*a*c)*n*x^4*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(
2*c*x + b))/(c*x^2 + b*x + a)) + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4*log(x) + 2*a^3*b*n*x + 6*(a*b^3 - 3*a^2
*b*c)*n*x^3 + 6*a^4*log(d) - 3*(a^2*b^2 - 2*a^3*c)*n*x^2 - 3*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4 - 2*a^4*n)*l
og(c*x^2 + b*x + a))/(a^4*x^4), -1/24*(6*(b^3 - 2*a*b*c)*sqrt(-b^2 + 4*a*c)*n*x^4*arctan(-sqrt(-b^2 + 4*a*c)*(
2*c*x + b)/(b^2 - 4*a*c)) + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4*log(x) + 2*a^3*b*n*x + 6*(a*b^3 - 3*a^2*b*c)
*n*x^3 + 6*a^4*log(d) - 3*(a^2*b^2 - 2*a^3*c)*n*x^2 - 3*((b^4 - 4*a*b^2*c + 2*a^2*c^2)*n*x^4 - 2*a^4*n)*log(c*
x^2 + b*x + a))/(a^4*x^4)]

________________________________________________________________________________________

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(d*(c*x**2+b*x+a)**n)/x**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.23513, size = 284, normalized size = 1.49 \begin{align*} \frac{{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{8 \, a^{4}} - \frac{n \log \left (c x^{2} + b x + a\right )}{4 \, x^{4}} - \frac{{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (x\right )}{4 \, a^{4}} + \frac{{\left (b^{5} n - 6 \, a b^{3} c n + 8 \, a^{2} b c^{2} n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} a^{4}} - \frac{6 \, b^{3} n x^{3} - 18 \, a b c n x^{3} - 3 \, a b^{2} n x^{2} + 6 \, a^{2} c n x^{2} + 2 \, a^{2} b n x + 6 \, a^{3} \log \left (d\right )}{24 \, a^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^5,x, algorithm="giac")

[Out]

1/8*(b^4*n - 4*a*b^2*c*n + 2*a^2*c^2*n)*log(c*x^2 + b*x + a)/a^4 - 1/4*n*log(c*x^2 + b*x + a)/x^4 - 1/4*(b^4*n
- 4*a*b^2*c*n + 2*a^2*c^2*n)*log(x)/a^4 + 1/4*(b^5*n - 6*a*b^3*c*n + 8*a^2*b*c^2*n)*arctan((2*c*x + b)/sqrt(-
b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) - 1/24*(6*b^3*n*x^3 - 18*a*b*c*n*x^3 - 3*a*b^2*n*x^2 + 6*a^2*c*n*x^2 +
2*a^2*b*n*x + 6*a^3*log(d))/(a^3*x^4)