Optimal. Leaf size=16 \[ \frac{1}{3} x^2 \sqrt{b x^2} \]
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Rubi [A] time = 0.0016844, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{1}{3} x^2 \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \sqrt{b x^2} \, dx &=\frac{\sqrt{b x^2} \int x^2 \, dx}{x}\\ &=\frac{1}{3} x^2 \sqrt{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0010001, size = 16, normalized size = 1. \[ \frac{1}{3} x^2 \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{3}\sqrt{b{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985859, size = 16, normalized size = 1. \begin{align*} \frac{\left (b x^{2}\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 = 1.22889, size = 28, normalized size = 1.75 \begin{align*} \frac{1}{3} \, \sqrt{b x^{2}} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.213405, size = 15, normalized size = 0.94 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{x^{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16882, size = 14, normalized size = 0.88 \begin{align*} \frac{1}{3} \, \sqrt{b} x^{3} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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