Integrand size = 18, antiderivative size = 234 \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=-\frac {x}{c}+\frac {x \log (x)}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2} \]
-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
Time = 0.22 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.85 \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=-\frac {x}{c}+\frac {x \log (x)}{c}-\frac {a \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {a \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {a \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}+\frac {a \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{c \sqrt {b^2-4 a c}}-\frac {b \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2} \]
-(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 + S qrt[b^2 - 4*a*c] + 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/Sqr t[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)
Time = 0.50 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2804, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \int \left (\frac {\log (x)}{c}-\frac {\log (x) (a+b x)}{c \left (a+b x+c x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2}-\frac {\log (x) \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c^2}-\frac {\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c^2}-\frac {x}{c}+\frac {x \log (x)}{c}\) |
-(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)
3.4.52.3.1 Defintions of rubi rules used
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]
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 | \(\frac {x \ln \left (x \right )}{c}-\frac {x}{c}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}+\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}-\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b}{2 c^{2}}+\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) a}{c \sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 c^{2} \sqrt {-4 c a +b^{2}}}\) | \(593\) |
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*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+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))
\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int { \frac {x^{2} \log \left (x\right )}{c x^{2} + b x + a} \,d x } \]
\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int \frac {x^{2} \log {\left (x \right )}}{a + b x + c x^{2}}\, dx \]
Exception generated. \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
\[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int { \frac {x^{2} \log \left (x\right )}{c x^{2} + b x + a} \,d x } \]
Timed out. \[ \int \frac {x^2 \log (x)}{a+b x+c x^2} \, dx=\int \frac {x^2\,\ln \left (x\right )}{c\,x^2+b\,x+a} \,d x \]