\(\int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx\) [352]

Optimal result

Integrand size = 18, antiderivative size = 234 \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=-\frac {x}{c}+\frac {x \log (x)}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2} \]

[Out]

-x/c+x*ln(x)/c-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))/c^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))/c^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))/c^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))/c^2

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2404, 2332, 2354, 2438} \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\log (x) \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c^2}-\frac {\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c^2}-\frac {x}{c}+\frac {x \log (x)}{c} \]

[In]

Int[(x^2*Log[x])/(a + b*x + c*x^2),x]

[Out]

-(x/c) + (x*Log[x])/c - ((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*c^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*c^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*c^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*c^2)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.85 \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=-\frac {x}{c}+\frac {x \log (x)}{c}-\frac {a \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {a \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {a \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {a \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(542\) vs. \(2(212)=424\).

Time = 1.10 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.32

method	result	size
default	\(\frac {\ln \left (x \right ) x -x}{c}+\frac {-\frac {\ln \left (x \right ) \left (\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b +2 \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a c -\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}+\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b -2 \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a c +\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}\right )}{2 c \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b +2 \operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a c -\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}+\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b -2 \operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a c +\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c \sqrt {-4 c a +b^{2}}}}{c}\)	\(543\)
risch	\(\frac {x \ln \left (x \right )}{c}-\frac {x}{c}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}+\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}\)	\(593\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int { \frac {x^{2} \log \left (x\right )}{c x^{2} + b x + a} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int \frac {x^{2} \log {\left (x \right )}}{a + b x + c x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(x^2*log(x)/(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

Giac [F]

\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int { \frac {x^{2} \log \left (x\right )}{c x^{2} + b x + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int \frac {x^2\,\ln \left (x\right )}{c\,x^2+b\,x+a} \,d x \]

[In]

int((x^2*log(x))/(a + b*x + c*x^2),x)

[Out]

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