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

Optimal. Leaf size=153 \[ \frac{\text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{\text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{\sqrt{b^2-4 a c}}+\frac{\log (x) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{\sqrt{b^2-4 a c}}-\frac{\log (x) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{\sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.137424, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2357, 2317, 2391} \[ \frac{\text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{\text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{\sqrt{b^2-4 a c}}+\frac{\log (x) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{\sqrt{b^2-4 a c}}-\frac{\log (x) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/(a + b*x + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0562416, size = 144, normalized size = 0.94 \[ \frac{\text{PolyLog}\left (2,\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )-\text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )+\log (x) \left (\log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )-\log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/(a + b*x + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.066, size = 177, normalized size = 1.2 \begin{align*}{\ln \left ( x \right ) \left ( \ln \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) -\ln \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}+{{\it dilog} \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}-{{\it dilog} \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*(ln((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-b+(-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))))/(-4*a*c+b^2)^(1/2)+1/(-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)
))-1/(-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)))

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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)/(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^{2} + b x + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(x)/(c*x^2 + b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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