Optimal. Leaf size=17 \[ \frac{1}{5} b x^4 \sqrt{b x^2} \]
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Rubi [A] time = 0.0015975, antiderivative size = 17, 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} \[ \frac{1}{5} b x^4 \sqrt{b x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (b x^2\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b x^2}\right ) \int x^4 \, dx}{x}\\ &=\frac{1}{5} b x^4 \sqrt{b x^2}\\ \end{align*}
Mathematica [A] time = 0.0011324, size = 16, normalized size = 0.94 \[ \frac{1}{5} x^2 \left (b x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 13, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{5} \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 [A] time = 0.9646, size = 16, normalized size = 0.94 \begin{align*} \frac{\left (b x^{2}\right )^{\frac{5}{2}}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28163, size = 31, normalized size = 1.82 \begin{align*} \frac{1}{5} \, \sqrt{b x^{2}} b x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.604684, size = 15, normalized size = 0.88 \begin{align*} \frac{b^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15152, size = 14, normalized size = 0.82 \begin{align*} \frac{1}{5} \, b^{\frac{3}{2}} x^{5} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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