Optimal. Leaf size=49 \[ \left (-a-b x-c x^2\right )^{-m} \left (a+b x+c x^2\right )^m \Gamma \left (m+1,-c x^2-b x-a\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 2181
Rule 6707
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*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.90 \[ (-a-x (b+c x))^{-m} (a+x (b+c x))^m \Gamma (m+1,-a-x (b+c x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 23, normalized size = 0.47 \[ \cos \left (\pi m\right ) \Gamma \left (m + 1, -c x^{2} - b x - a\right ) \]
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 \[ \int {\left (2 \, c x + b\right )} {\left (c x^{2} + b x + a\right )}^{m} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{m} {\mathrm e}^{c \,x^{2}+b x +a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (2 \, c x + b\right )} {\left (c x^{2} + b x + a\right )}^{m} e^{\left (c x^{2} + b x + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 49, normalized size = 1.00 \[ \frac {\Gamma \left (m+1,-c\,x^2-b\,x-a\right )\,{\left (c\,x^2+b\,x+a\right )}^m}{{\left (-c\,x^2-b\,x-a\right )}^m} \]
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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