Optimal. Leaf size=14 \[ \frac{1}{2} x \sqrt{b x^2} \]
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Rubi [A] time = 0.0014066, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{1}{2} x \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{b x^2} \, dx &=\frac{\sqrt{b x^2} \int x \, dx}{x}\\ &=\frac{1}{2} x \sqrt{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0002802, size = 14, normalized size = 1. \[ \frac{1}{2} x \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}\sqrt{b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59678, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \, \sqrt{b x^{2}} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.191509, size = 14, normalized size = 1. \begin{align*} \frac{\sqrt{b} x \sqrt{x^{2}}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1581, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{b} x^{2} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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