Optimal. Leaf size=21 \[ \frac {x e^{a x}}{a}-\frac {e^{a x}}{a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 0.67 \[ \frac {e^{a x} (a x-1)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 13, normalized size = 0.62 \[ \frac {{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 13, normalized size = 0.62 \[ \frac {{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 14, normalized size = 0.67 \[ \frac {\left (a x -1\right ) {\mathrm e}^{a x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 13, normalized size = 0.62 \[ \frac {{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 13, normalized size = 0.62 \[ \frac {{\mathrm {e}}^{a\,x}\,\left (a\,x-1\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 19, normalized size = 0.90 \[ \begin {cases} \frac {\left (a x - 1\right ) e^{a x}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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