3.78 \(\int \frac{\log (d (a+b x+c x^2)^n)}{x^3} \, dx\)

Optimal. Leaf size=121 \[ \frac{n \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{4 a^2}-\frac{n \log (x) \left (b^2-2 a c\right )}{2 a^2}-\frac{b n \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{2 a^2}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}-\frac{b n}{2 a x} \]

[Out]

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

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Rubi [A]  time = 0.154217, antiderivative size = 121, 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} \[ \frac{n \left (b^2-2 a c\right ) \log \left (a+b x+c x^2\right )}{4 a^2}-\frac{n \log (x) \left (b^2-2 a c\right )}{2 a^2}-\frac{b n \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{2 a^2}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{2 x^2}-\frac{b n}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Log[d*(a + b*x + c*x^2)^n]/x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.237582, size = 105, normalized size = 0.87 \[ -\frac{\frac{n x \left (2 x \log (x) \left (b^2-2 a c\right )-x \left (b^2-2 a c\right ) \log (a+x (b+c x))+2 b x \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+2 a b\right )}{a^2}+2 \log \left (d (a+x (b+c x))^n\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.089, size = 1178, normalized size = 9.7 \begin{align*} \text{result 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^3,x)

[Out]

-1/2/x^2*ln((c*x^2+b*x+a)^n)-1/4*(-I*Pi*a^2*csgn(I*d)*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)+I*Pi*a
^2*csgn(I*d)*csgn(I*d*(c*x^2+b*x+a)^n)^2+I*Pi*a^2*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*d*(c*x^2+b*x+a)^n)^2-I*Pi*a^2
*csgn(I*d*(c*x^2+b*x+a)^n)^3+2*n*ln((-16*a^2*b^2*c^2+12*a*b^4*c-2*b^6-6*(-4*a*b^2*c+b^4)^(1/2)*a^2*c^2+8*(-4*a
*b^2*c+b^4)^(1/2)*a*b^2*c-2*(-4*a*b^2*c+b^4)^(1/2)*b^4)*x-12*a^3*b*c^2+11*b^3*c*a^2-2*a*b^5+5*(-4*a*b^2*c+b^4)
^(1/2)*a^2*b*c-2*(-4*a*b^2*c+b^4)^(1/2)*a*b^3)*c*a*x^2-n*ln((-16*a^2*b^2*c^2+12*a*b^4*c-2*b^6-6*(-4*a*b^2*c+b^
4)^(1/2)*a^2*c^2+8*(-4*a*b^2*c+b^4)^(1/2)*a*b^2*c-2*(-4*a*b^2*c+b^4)^(1/2)*b^4)*x-12*a^3*b*c^2+11*b^3*c*a^2-2*
a*b^5+5*(-4*a*b^2*c+b^4)^(1/2)*a^2*b*c-2*(-4*a*b^2*c+b^4)^(1/2)*a*b^3)*b^2*x^2+2*n*ln((-16*a^2*b^2*c^2+12*a*b^
4*c-2*b^6+6*(-4*a*b^2*c+b^4)^(1/2)*a^2*c^2-8*(-4*a*b^2*c+b^4)^(1/2)*a*b^2*c+2*(-4*a*b^2*c+b^4)^(1/2)*b^4)*x-12
*a^3*b*c^2+11*b^3*c*a^2-2*a*b^5-5*(-4*a*b^2*c+b^4)^(1/2)*a^2*b*c+2*(-4*a*b^2*c+b^4)^(1/2)*a*b^3)*c*a*x^2-n*ln(
(-16*a^2*b^2*c^2+12*a*b^4*c-2*b^6+6*(-4*a*b^2*c+b^4)^(1/2)*a^2*c^2-8*(-4*a*b^2*c+b^4)^(1/2)*a*b^2*c+2*(-4*a*b^
2*c+b^4)^(1/2)*b^4)*x-12*a^3*b*c^2+11*b^3*c*a^2-2*a*b^5-5*(-4*a*b^2*c+b^4)^(1/2)*a^2*b*c+2*(-4*a*b^2*c+b^4)^(1
/2)*a*b^3)*b^2*x^2-4*n*ln(x)*c*a*x^2+2*n*ln(x)*b^2*x^2-n*ln((-16*a^2*b^2*c^2+12*a*b^4*c-2*b^6-6*(-4*a*b^2*c+b^
4)^(1/2)*a^2*c^2+8*(-4*a*b^2*c+b^4)^(1/2)*a*b^2*c-2*(-4*a*b^2*c+b^4)^(1/2)*b^4)*x-12*a^3*b*c^2+11*b^3*c*a^2-2*
a*b^5+5*(-4*a*b^2*c+b^4)^(1/2)*a^2*b*c-2*(-4*a*b^2*c+b^4)^(1/2)*a*b^3)*(-4*a*b^2*c+b^4)^(1/2)*x^2+n*ln((-16*a^
2*b^2*c^2+12*a*b^4*c-2*b^6+6*(-4*a*b^2*c+b^4)^(1/2)*a^2*c^2-8*(-4*a*b^2*c+b^4)^(1/2)*a*b^2*c+2*(-4*a*b^2*c+b^4
)^(1/2)*b^4)*x-12*a^3*b*c^2+11*b^3*c*a^2-2*a*b^5-5*(-4*a*b^2*c+b^4)^(1/2)*a^2*b*c+2*(-4*a*b^2*c+b^4)^(1/2)*a*b
^3)*(-4*a*b^2*c+b^4)^(1/2)*x^2+2*a*b*n*x+2*ln(d)*a^2)/a^2/x^2

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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(d*(c*x^2+b*x+a)^n)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.35442, size = 628, normalized size = 5.19 \begin{align*} \left [\frac{\sqrt{b^{2} - 4 \, a c} b n x^{2} \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 ) - 2 \,{\left (b^{2} - 2 \, a c\right )} n x^{2} \log \left (x\right ) - 2 \, a b n x - 2 \, a^{2} \log \left (d\right ) +{\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2} x^{2}}, -\frac{2 \, \sqrt{-b^{2} + 4 \, a c} b n x^{2} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \,{\left (b^{2} - 2 \, a c\right )} n x^{2} \log \left (x\right ) + 2 \, a b n x + 2 \, a^{2} \log \left (d\right ) -{\left ({\left (b^{2} - 2 \, a c\right )} n x^{2} - 2 \, a^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2} x^{2}}\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^3,x, algorithm="fricas")

[Out]

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

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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(d*(c*x**2+b*x+a)**n)/x**3,x)

[Out]

Timed out

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Giac [A]  time = 1.26885, size = 174, normalized size = 1.44 \begin{align*} \frac{{\left (b^{2} n - 2 \, a c n\right )} \log \left (c x^{2} + b x + a\right )}{4 \, a^{2}} - \frac{n \log \left (c x^{2} + b x + a\right )}{2 \, x^{2}} - \frac{{\left (b^{2} n - 2 \, a c n\right )} \log \left (x\right )}{2 \, a^{2}} + \frac{{\left (b^{3} n - 4 \, a b c n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{2}} - \frac{b n x + a \log \left (d\right )}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(d*(c*x^2+b*x+a)^n)/x^3,x, algorithm="giac")

[Out]

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