Optimal. Leaf size=782 \[ -\frac {n \text {Li}_2\left (-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {Li}_2\left (-\frac {c \left (e+2 f x-\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {Li}_2\left (-\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f-c \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (-\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \sqrt {b^2-4 a c}-b f-c \sqrt {e^2-4 d f}+c e}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}} \]
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Rubi [A] time = 1.51, antiderivative size = 782, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2528, 2524, 2418, 2394, 2393, 2391} \[ -\frac {n \text {PolyLog}\left (2,-\frac {c \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \text {PolyLog}\left (2,-\frac {c \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {PolyLog}\left (2,-\frac {c \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \text {PolyLog}\left (2,-\frac {c \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (-\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \sqrt {b^2-4 a c}-b f-c \sqrt {e^2-4 d f}+c e}\right )}{\sqrt {e^2-4 d f}}-\frac {n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (b-\sqrt {b^2-4 a c}\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}}-\frac {\log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {e^2-4 d f}} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x+f x^2} \, dx &=\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.85, size = 663, normalized size = 0.85 \[ \frac {-n \text {Li}_2\left (\frac {c \left (-e-2 f x+\sqrt {e^2-4 d f}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) f+c \left (\sqrt {e^2-4 d f}-e\right )}\right )-n \text {Li}_2\left (\frac {c \left (-e-2 f x+\sqrt {e^2-4 d f}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) f+c \left (\sqrt {e^2-4 d f}-e\right )}\right )+n \text {Li}_2\left (\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{\left (\sqrt {b^2-4 a c}-b\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )}\right )+n \text {Li}_2\left (\frac {c \left (e+2 f x+\sqrt {e^2-4 d f}\right )}{c \left (e+\sqrt {e^2-4 d f}\right )-\left (b+\sqrt {b^2-4 a c}\right ) f}\right )-n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{-f \sqrt {b^2-4 a c}+b f+c \sqrt {e^2-4 d f}+c (-e)}\right )-n \log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )+c \left (\sqrt {e^2-4 d f}-e\right )}\right )+n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}-b-2 c x\right )}{f \left (\sqrt {b^2-4 a c}-b\right )+c \left (\sqrt {e^2-4 d f}+e\right )}\right )+n \log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (\frac {f \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{f \left (\sqrt {b^2-4 a c}+b\right )-c \left (\sqrt {e^2-4 d f}+e\right )}\right )+\log \left (-\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt {e^2-4 d f}+e+2 f x\right ) \log \left (g (a+x (b+c x))^n\right )}{\sqrt {e^2-4 d f}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{f x^{2} + e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.17, size = 764, normalized size = 0.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (g\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{f\,x^2+e\,x+d} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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