Optimal. Leaf size=53 \[ -\frac{8 x \sqrt{e^{a+b x}}}{b^2}+\frac{16 \sqrt{e^{a+b x}}}{b^3}+\frac{2 x^2 \sqrt{e^{a+b x}}}{b} \]
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Rubi [A] time = 0.0635734, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2176, 2194} \[ -\frac{8 x \sqrt{e^{a+b x}}}{b^2}+\frac{16 \sqrt{e^{a+b x}}}{b^3}+\frac{2 x^2 \sqrt{e^{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int \sqrt{e^{a+b x}} x^2 \, dx &=\frac{2 \sqrt{e^{a+b x}} x^2}{b}-\frac{4 \int \sqrt{e^{a+b x}} x \, dx}{b}\\ &=-\frac{8 \sqrt{e^{a+b x}} x}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^2}{b}+\frac{8 \int \sqrt{e^{a+b x}} \, dx}{b^2}\\ &=\frac{16 \sqrt{e^{a+b x}}}{b^3}-\frac{8 \sqrt{e^{a+b x}} x}{b^2}+\frac{2 \sqrt{e^{a+b x}} x^2}{b}\\ \end{align*}
Mathematica [A] time = 0.0126865, size = 29, normalized size = 0.55 \[ \frac{2 \left (b^2 x^2-4 b x+8\right ) \sqrt{e^{a+b x}}}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 0.5 \begin{align*} 2\,{\frac{ \left ({x}^{2}{b}^{2}-4\,bx+8 \right ) \sqrt{{{\rm e}^{bx+a}}}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02316, size = 49, normalized size = 0.92 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} e^{\left (\frac{1}{2} \, a\right )} - 4 \, b x e^{\left (\frac{1}{2} \, a\right )} + 8 \, e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (\frac{1}{2} \, b x\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47572, size = 66, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} - 4 \, b x + 8\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.103519, size = 34, normalized size = 0.64 \begin{align*} \begin{cases} \frac{\left (2 b^{2} x^{2} - 8 b x + 16\right ) \sqrt{e^{a + b x}}}{b^{3}} & \text{for}\: b^{3} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22556, size = 36, normalized size = 0.68 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} - 4 \, b x + 8\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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