Optimal. Leaf size=762 \[ -\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}} \]
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Rubi [A] time = 1.45, antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2528, 2524, 2418, 2394, 2393, 2391} \[ -\frac {n \text {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \text {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}} \]
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^2} \, dx &=\int \left (\frac {\sqrt {-d} \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}\\ &=\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \int \frac {(b+2 c x) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{a+b x+c x^2} \, dx}{2 \sqrt {-d} \sqrt {e}}+\frac {n \int \frac {(b+2 c x) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{a+b x+c x^2} \, dx}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \int \left (\frac {2 c \log \left (\sqrt {-d}-\sqrt {e} x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (\sqrt {-d}-\sqrt {e} x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{2 \sqrt {-d} \sqrt {e}}+\frac {n \int \left (\frac {2 c \log \left (\sqrt {-d}+\sqrt {e} x\right )}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {2 c \log \left (\sqrt {-d}+\sqrt {e} x\right )}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(c n) \int \frac {\log \left (\sqrt {-d}-\sqrt {e} x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {-d} \sqrt {e}}-\frac {(c n) \int \frac {\log \left (\sqrt {-d}-\sqrt {e} x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {-d} \sqrt {e}}+\frac {(c n) \int \frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {-d} \sqrt {e}}+\frac {(c n) \int \frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{\sqrt {-d} \sqrt {e}}\\ &=-\frac {n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \int \frac {\log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {n \int \frac {\log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {n \int \frac {\log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {n \int \frac {\log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}\\ &=-\frac {n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{x} \, dx,x,\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{x} \, dx,x,\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{x} \, dx,x,\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{-2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{x} \, dx,x,\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}\\ &=-\frac {n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 626, normalized size = 0.82 \[ \frac {-n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )-n \text {Li}_2\left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )+n \text {Li}_2\left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 \sqrt {-d} c+\left (\sqrt {b^2-4 a c}-b\right ) \sqrt {e}}\right )+n \text {Li}_2\left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )-n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )-n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )+n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}-b-2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}-b\right )+2 c \sqrt {-d}}\right )+n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )-2 c \sqrt {-d}}\right )+\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )}{2 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.20, size = 610, normalized size = 0.80 \[ \text {result too large to display} \]
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 \[ \int \frac {\ln \left (g\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{e\,x^2+d} \,d x \]
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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