Optimal. Leaf size=34 \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]
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Rubi [A] time = 0.0285417, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2176, 2194} \[ \frac{2 x \sqrt{e^{a+b x}}}{b}-\frac{4 \sqrt{e^{a+b x}}}{b^2} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \sqrt{e^{a+b x}} x \, dx &=\frac{2 \sqrt{e^{a+b x}} x}{b}-\frac{2 \int \sqrt{e^{a+b x}} \, dx}{b}\\ &=-\frac{4 \sqrt{e^{a+b x}}}{b^2}+\frac{2 \sqrt{e^{a+b x}} x}{b}\\ \end{align*}
Mathematica [A] time = 0.0101217, size = 21, normalized size = 0.62 \[ \frac{2 (b x-2) \sqrt{e^{a+b x}}}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 0.6 \begin{align*} 2\,{\frac{ \left ( bx-2 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0455, size = 32, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (b x e^{\left (\frac{1}{2} \, a\right )} - 2 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5013, size = 50, normalized size = 1.47 \begin{align*} \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.094429, size = 26, normalized size = 0.76 \begin{align*} \begin{cases} \frac{\left (2 b x - 4\right ) \sqrt{e^{a + b x}}}{b^{2}} & \text{for}\: b^{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 = 1.2432, size = 26, normalized size = 0.76 \begin{align*} \frac{2 \,{\left (b x - 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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