Integrand size = 15, antiderivative size = 153 \[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\frac {\log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {\log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+\frac {\operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {\operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2404, 2354, 2438} \[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\frac {\operatorname {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}-\frac {\operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+\frac {\log (x) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{\sqrt {b^2-4 a c}}-\frac {\log (x) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{\sqrt {b^2-4 a c}} \]
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Rule 2354
Rule 2404
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94 \[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\frac {\log (x) \left (\log \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )-\log \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )-\operatorname {PolyLog}\left (2,-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Time = 1.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {\ln \left (x \right ) \left (\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )-\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )\right )}{\sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )-\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )}{\sqrt {-4 c a +b^{2}}}\) | \(169\) |
risch | \(-\frac {\ln \left (x \right ) \left (\ln \left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )-\ln \left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )\right )}{\sqrt {-4 c a +b^{2}}}+\frac {\operatorname {dilog}\left (\frac {-2 x c +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right )}{\sqrt {-4 c a +b^{2}}}-\frac {\operatorname {dilog}\left (\frac {2 x c +\sqrt {-4 c a +b^{2}}+b}{b +\sqrt {-4 c a +b^{2}}}\right )}{\sqrt {-4 c a +b^{2}}}\) | \(178\) |
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\[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\int { \frac {\log \left (x\right )}{c x^{2} + b x + a} \,d x } \]
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\[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\int \frac {\log {\left (x \right )}}{a + b x + c x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\int { \frac {\log \left (x\right )}{c x^{2} + b x + a} \,d x } \]
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Timed out. \[ \int \frac {\log (x)}{a+b x+c x^2} \, dx=\int \frac {\ln \left (x\right )}{c\,x^2+b\,x+a} \,d x \]
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