3.4.56 \(\int \frac {\log (x)}{x^2 (a+b x+c x^2)} \, dx\) [356]

Optimal. Leaf size=251 \[ -\frac {1}{a x}-\frac {\log (x)}{a x}-\frac {b \log ^2(x)}{2 a^2}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.26, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2404, 2341, 2338, 2354, 2438} \begin {gather*} \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (2,-\frac {2 c x}{\sqrt {b^2-4 a c}+b}\right )}{2 a^2}+\frac {\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a^2}+\frac {\log (x) \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a^2}-\frac {b \log ^2(x)}{2 a^2}-\frac {1}{a x}-\frac {\log (x)}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[x]/(x^2*(a + b*x + c*x^2)),x]

[Out]

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

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 2341

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 2354

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 2404

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 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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 255, normalized size = 1.02 \begin {gather*} \frac {-\frac {2 a}{x}-\frac {2 a \log (x)}{x}-b \log ^2(x)+\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+\left (b+\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\left (b+\frac {-b^2+2 a c}{\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^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(551\) vs. \(2(227)=454\).
time = 0.78, size = 552, normalized size = 2.20

method	result	size
default	\(\frac {-\frac {\ln \left (x \right )}{x}-\frac {1}{x}}{a}-\frac {b \ln \left (x \right )^{2}}{2 a^{2}}+\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b -2 \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a c +\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}+\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b +2 \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a c -\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}\right )}{2 \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b -2 \dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a c +\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}+\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b +2 \dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a c -\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 \sqrt {-4 c a +b^{2}}}}{a^{2}}\)	\(552\)
risch	\(-\frac {\ln \left (x \right )}{a x}-\frac {1}{a x}-\frac {b \ln \left (x \right )^{2}}{2 a^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) c}{a \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) c}{a \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a^{2}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) c}{a \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 a^{2}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) c}{a \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{2} \sqrt {-4 c a +b^{2}}}\)	\(608\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (x\right )}{x^2\,\left (c\,x^2+b\,x+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(x^2*(a + b*x + c*x^2)),x)

[Out]

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

________________________________________________________________________________________