Optimal. Leaf size=18 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b} \]
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Rubi [A] time = 0.0030336, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x \sqrt{a+b x^2} \, dx &=\frac{\left (a+b x^2\right )^{3/2}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0028413, size = 18, normalized size = 1. \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.2608, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47137, size = 34, normalized size = 1.89 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.160883, size = 39, normalized size = 2.17 \begin{align*} \begin{cases} \frac{a \sqrt{a + b x^{2}}}{3 b} + \frac{x^{2} \sqrt{a + b x^{2}}}{3} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} 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 = 2.48041, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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