Optimal. Leaf size=38 \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \]
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Rubi [A] time = 0.0956658, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6707, 2176, 2194} \[ e^{a+b x+c x^2} \left (a+b x+c x^2\right )-e^{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{a+b x+c x^2} (b+2 c x) \left (a+b x+c x^2\right ) \, dx &=\operatorname{Subst}\left (\int e^x x \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \left (a+b x+c x^2\right )-\operatorname{Subst}\left (\int e^x \, dx,x,a+b x+c x^2\right )\\ &=-e^{a+b x+c x^2}+e^{a+b x+c x^2} \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0288192, size = 23, normalized size = 0.61 \[ e^{a+x (b+c x)} \left (a+b x+c x^2-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 24, normalized size = 0.6 \begin{align*} \left ( c{x}^{2}+bx+a-1 \right ){{\rm e}^{c{x}^{2}+bx+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.37013, size = 676, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.825417, size = 58, normalized size = 1.53 \begin{align*}{\left (c x^{2} + b x + a - 1\right )} e^{\left (c x^{2} + b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.135125, size = 22, normalized size = 0.58 \begin{align*} \left (a + b x + c x^{2} - 1\right ) e^{a + b x + c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19335, size = 59, normalized size = 1.55 \begin{align*} \frac{{\left (c^{2}{\left (2 \, x + \frac{b}{c}\right )}^{2} - b^{2} + 4 \, a c - 4 \, c\right )} e^{\left (c x^{2} + b x + a\right )}}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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