Optimal. Leaf size=762 \[ -\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (-\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt{-d} \sqrt{e}} \]
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Rubi [A] time = 1.45202, antiderivative size = 762, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, 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 2528
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
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*}
Mathematica [A] time = 0.940713, size = 626, normalized size = 0.82 \[ \frac{-n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )-n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )+n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}-b\right )+2 c \sqrt{-d}}\right )+n \text{PolyLog}\left (2,\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{2 c \sqrt{-d}-\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )}\right )-n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (-\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (b-\sqrt{b^2-4 a c}\right )+2 c \sqrt{-d}}\right )-n \log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )+2 c \sqrt{-d}}\right )+n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}-b-2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}-b\right )+2 c \sqrt{-d}}\right )+n \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (\sqrt{b^2-4 a c}+b+2 c x\right )}{\sqrt{e} \left (\sqrt{b^2-4 a c}+b\right )-2 c \sqrt{-d}}\right )+\log \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (g (a+x (b+c x))^n\right )}{2 \sqrt{-d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.135, size = 610, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (c x^{2} + b x + a\right )}^{n} g\right )}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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