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

Optimal. Leaf size=308 \[ -\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 a^3}+\frac{\log ^2(x) \left (b^2-a c\right )}{2 a^3}-\frac{\log (x) \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 a^3}-\frac{\log (x) \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 a^3}+\frac{b}{a^2 x}+\frac{b \log (x)}{a^2 x}-\frac{1}{4 a x^2}-\frac{\log (x)}{2 a x^2} \]

[Out]

-1/(4*a*x^2) + b/(a^2*x) - Log[x]/(2*a*x^2) + (b*Log[x])/(a^2*x) + ((b^2 - a*c)*Log[x]^2)/(2*a^3) - ((b^2 - a*
c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*
c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*
c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c -
(b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^3)

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Rubi [A]  time = 0.512413, antiderivative size = 308, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2357, 2304, 2301, 2317, 2391} \[ -\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 a^3}+\frac{\log ^2(x) \left (b^2-a c\right )}{2 a^3}-\frac{\log (x) \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 a^3}-\frac{\log (x) \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 a^3}+\frac{b}{a^2 x}+\frac{b \log (x)}{a^2 x}-\frac{1}{4 a x^2}-\frac{\log (x)}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x^3*(a + b*x + c*x^2)),x]

[Out]

-1/(4*a*x^2) + b/(a^2*x) - Log[x]/(2*a*x^2) + (b*Log[x])/(a^2*x) + ((b^2 - a*c)*Log[x]^2)/(2*a^3) - ((b^2 - a*
c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*
c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*
c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^3) - ((b^2 - a*c -
(b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(2*a^3)

Rule 2357

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2301

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 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 (x)}{x^3 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac{\log (x)}{a x^3}-\frac{b \log (x)}{a^2 x^2}+\frac{\left (b^2-a c\right ) \log (x)}{a^3 x}+\frac{\left (-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x\right ) \log (x)}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x\right ) \log (x)}{a+b x+c x^2} \, dx}{a^3}+\frac{\int \frac{\log (x)}{x^3} \, dx}{a}-\frac{b \int \frac{\log (x)}{x^2} \, dx}{a^2}+\frac{\left (b^2-a c\right ) \int \frac{\log (x)}{x} \, dx}{a^3}\\ &=-\frac{1}{4 a x^2}+\frac{b}{a^2 x}-\frac{\log (x)}{2 a x^2}+\frac{b \log (x)}{a^2 x}+\frac{\left (b^2-a c\right ) \log ^2(x)}{2 a^3}+\frac{\int \left (\frac{\left (-\frac{b c \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-c \left (b^2-a c\right )\right ) \log (x)}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (\frac{b c \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-c \left (b^2-a c\right )\right ) \log (x)}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{a^3}\\ &=-\frac{1}{4 a x^2}+\frac{b}{a^2 x}-\frac{\log (x)}{2 a x^2}+\frac{b \log (x)}{a^2 x}+\frac{\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac{\left (c \left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\log (x)}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{a^3}-\frac{\left (c \left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{\log (x)}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{a^3}\\ &=-\frac{1}{4 a x^2}+\frac{b}{a^2 x}-\frac{\log (x)}{2 a x^2}+\frac{b \log (x)}{a^2 x}+\frac{\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 a^3}+\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx}{2 a^3}+\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{x} \, dx}{2 a^3}\\ &=-\frac{1}{4 a x^2}+\frac{b}{a^2 x}-\frac{\log (x)}{2 a x^2}+\frac{b \log (x)}{a^2 x}+\frac{\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \text{Li}_2\left (-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 a^3}-\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \text{Li}_2\left (-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 a^3}\\ \end{align*}

Mathematica [A]  time = 0.449763, size = 311, normalized size = 1.01 \[ -\frac{2 \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+2 \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )+\frac{a^2}{x^2}+\frac{2 a^2 \log (x)}{x^2}-2 \log ^2(x) \left (b^2-a c\right )+2 \log (x) \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )+2 \log (x) \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )-\frac{4 a b}{x}-\frac{4 a b \log (x)}{x}}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2/x^2 - (4*a*b)/x + (2*a^2*Log[x])/x^2 - (4*a*b*Log[x])/x - 2*(b^2 - a*c)*Log[x]^2 + 2*(b^2 - a*c + (b*(b^
2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[(b - Sqrt[b^2 - 4*a*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c])] + 2*(b^2 -
a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Log[x]*Log[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sqrt[b^2 - 4*a*c])]
+ 2*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + 2*(b^2 -
a*c - (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*PolyLog[2, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(4*a^3)

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Maple [B]  time = 0.067, size = 816, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/x^3/(c*x^2+b*x+a),x)

[Out]

-1/2/a^2*ln(x)^2*c+1/2/a^3*ln(x)^2*b^2+b*ln(x)/x/a^2+b/x/a^2+1/2/a^2*ln(x)*ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-
b+(-4*a*c+b^2)^(1/2)))*c-1/2/a^3*ln(x)*ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^2+3/2/a^2*l
n(x)/(-4*a*c+b^2)^(1/2)*ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b*c-1/2/a^3*ln(x)/(-4*a*c+b^
2)^(1/2)*ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3+1/2/a^2*ln(x)*ln((2*c*x+(-4*a*c+b^2)^(1
/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*c-1/2/a^3*ln(x)*ln((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b^2-3/2
/a^2*ln(x)/(-4*a*c+b^2)^(1/2)*ln((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b*c+1/2/a^3*ln(x)/(-4*a*
c+b^2)^(1/2)*ln((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b^3+1/2/a^2*dilog((-2*c*x+(-4*a*c+b^2)^(1
/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*c-1/2/a^3*dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^2+3/2
/a^2/(-4*a*c+b^2)^(1/2)*dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b*c-1/2/a^3/(-4*a*c+b^2)^
(1/2)*dilog((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-4*a*c+b^2)^(1/2)))*b^3+1/2/a^2*dilog((2*c*x+(-4*a*c+b^2)^(1/2)
+b)/(b+(-4*a*c+b^2)^(1/2)))*c-1/2/a^3*dilog((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b^2-3/2/a^2/(
-4*a*c+b^2)^(1/2)*dilog((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b*c+1/2/a^3/(-4*a*c+b^2)^(1/2)*di
log((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))*b^3-1/2*ln(x)/a/x^2-1/4/a/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(x)/x^3/(c*x^2+b*x+a),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 (x\right )}{c x^{5} + b x^{4} + a x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(x)/(c*x^5 + b*x^4 + a*x^3), 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(x)/x**3/(c*x**2+b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x\right )}{{\left (c x^{2} + b x + a\right )} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(x)/((c*x^2 + b*x + a)*x^3), x)