Optimal. Leaf size=17 \[ \frac{1}{4} b x^3 \sqrt{b x^2} \]
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Rubi [A] time = 0.0015696, antiderivative size = 17, 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} \[ \frac{1}{4} b x^3 \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (b x^2\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b x^2}\right ) \int x^3 \, dx}{x}\\ &=\frac{1}{4} b x^3 \sqrt{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0005967, size = 14, normalized size = 0.82 \[ \frac{1}{4} x \left (b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 11, normalized size = 0.7 \begin{align*}{\frac{x}{4} \left ( b{x}^{2} \right ) ^{{\frac{3}{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 = 1.18293, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{4} \, \sqrt{b x^{2}} b x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.473972, size = 14, normalized size = 0.82 \begin{align*} \frac{b^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19765, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{4} \, b^{\frac{3}{2}} x^{4} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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