Optimal. Leaf size=21 \[ \frac{x e^{a x}}{a}-\frac{e^{a x}}{a^2} \]
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Rubi [A] time = 0.0086599, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2176, 2194} \[ \frac{x e^{a x}}{a}-\frac{e^{a x}}{a^2} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{a x} x \, dx &=\frac{e^{a x} x}{a}-\frac{\int e^{a x} \, dx}{a}\\ &=-\frac{e^{a x}}{a^2}+\frac{e^{a x} x}{a}\\ \end{align*}
Mathematica [A] time = 0.0056161, size = 14, normalized size = 0.67 \[ \frac{e^{a x} (a x-1)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ){{\rm e}^{ax}}}{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942147, size = 18, normalized size = 0.86 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6796, size = 31, normalized size = 1.48 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101134, size = 19, normalized size = 0.9 \begin{align*} \begin{cases} \frac{\left (a x - 1\right ) e^{a x}}{a^{2}} & \text{for}\: a^{2} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09533, size = 18, normalized size = 0.86 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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