Optimal. Leaf size=49 \[ \left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \text{Gamma}\left (m+1,-a-b x-c x^2\right ) \]
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Rubi [A] time = 0.199597, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {6707, 2181} \[ \left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \text{Gamma}\left (m+1,-a-b x-c x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2181
Rubi steps
\begin{align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right )^m \, dx &=\operatorname{Subst}\left (\int e^x x^m \, dx,x,a+b x+c x^2\right )\\ &=\left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \Gamma \left (1+m,-a-b x-c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0523048, size = 44, normalized size = 0.9 \[ (-a-x (b+c x))^{-m} (a+x (b+c x))^m \text{Gamma}(m+1,-a-x (b+c x)) \]
Antiderivative was successfully verified.
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Maple [F] time = 1.413, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c{x}^{2}+bx+a}} \left ( 2\,cx+b \right ) \left ( c{x}^{2}+bx+a \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (2 \, c x + b\right )}{\left (c x^{2} + b x + a\right )}^{m} e^{\left (c x^{2} + b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.732206, size = 57, normalized size = 1.16 \begin{align*} \cos \left (\pi m\right ) \Gamma \left (m + 1, -c x^{2} - b x - a\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{\left (2 \, c x + b\right )}{\left (c x^{2} + b x + a\right )}^{m} e^{\left (c x^{2} + b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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