Integrand size = 28, antiderivative size = 782 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=-\frac {n \log \left (-\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c e-b f+\sqrt {b^2-4 a c} f-c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
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Time = 0.96 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2608, 2604, 2465, 2441, 2440, 2438} \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \operatorname {PolyLog}\left (2,-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (-\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \sqrt {b^2-4 a c}-b f-c \sqrt {e^2-4 d f}+c e}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}} \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2604
Rule 2608
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 f \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}+2 f x\right )}-\frac {2 f \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}\right ) \, dx \\ & = \frac {(2 f) \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}} \\ & = \frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {n \int \frac {(b+2 c x) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{a+b x+c x^2} \, dx}{\sqrt {e^2-4 d f}}+\frac {n \int \frac {(b+2 c x) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{a+b x+c x^2} \, dx}{\sqrt {e^2-4 d f}} \\ & = \frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {n \int \left (\frac {2 c \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{\sqrt {e^2-4 d f}}+\frac {n \int \left (\frac {2 c \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{\sqrt {e^2-4 d f}} \\ & = \frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {(2 c n) \int \frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 c n) \int \frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(2 c n) \int \frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(2 c n) \int \frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {e^2-4 d f}} \\ & = -\frac {n \log \left (-\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c e-b f+\sqrt {b^2-4 a c} f-c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}+\frac {(2 f n) \int \frac {\log \left (\frac {2 f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 \left (b-\sqrt {b^2-4 a c}\right ) f-2 c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f n) \int \frac {\log \left (\frac {2 f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 \left (b-\sqrt {b^2-4 a c}\right ) f-2 c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(2 f n) \int \frac {\log \left (\frac {2 f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 \left (b+\sqrt {b^2-4 a c}\right ) f-2 c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f n) \int \frac {\log \left (\frac {2 f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 \left (b+\sqrt {b^2-4 a c}\right ) f-2 c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}} \\ & = -\frac {n \log \left (-\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c e-b f+\sqrt {b^2-4 a c} f-c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{2 \left (b-\sqrt {b^2-4 a c}\right ) f-2 c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{2 \left (b+\sqrt {b^2-4 a c}\right ) f-2 c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{2 \left (b-\sqrt {b^2-4 a c}\right ) f-2 c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{2 \left (b+\sqrt {b^2-4 a c}\right ) f-2 c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}} \\ & = -\frac {n \log \left (-\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c e-b f+\sqrt {b^2-4 a c} f-c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Li}_2\left (-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Li}_2\left (-\frac {c \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (-\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 663, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\frac {-n \log \left (\frac {f \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{-c e+b f-\sqrt {b^2-4 a c} f+c \sqrt {e^2-4 d f}}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )-n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e-\sqrt {e^2-4 d f}+2 f x\right )+n \log \left (\frac {f \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )+n \log \left (\frac {f \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right ) \log \left (e+\sqrt {e^2-4 d f}+2 f x\right )+\log \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (e+\sqrt {e^2-4 d f}+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )-n \operatorname {PolyLog}\left (2,\frac {c \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right )-n \operatorname {PolyLog}\left (2,\frac {c \left (-e+\sqrt {e^2-4 d f}-2 f x\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (-e+\sqrt {e^2-4 d f}\right )}\right )+n \operatorname {PolyLog}\left (2,\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{\left (-b+\sqrt {b^2-4 a c}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )}\right )+n \operatorname {PolyLog}\left (2,\frac {c \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{-\left (\left (b+\sqrt {b^2-4 a c}\right ) f\right )+c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.78 (sec) , antiderivative size = 637, normalized size of antiderivative = 0.81
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\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d} \,d x } \]
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Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d} \,d x } \]
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Timed out. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx=\int \frac {\ln \left (g\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{f\,x^2+e\,x+d} \,d x \]
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