Optimal. Leaf size=19 \[ -\frac{1}{2 b x \sqrt{b x^2}} \]
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Rubi [A] time = 0.0016886, antiderivative size = 19, 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} \[ -\frac{1}{2 b x \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{1}{x^3} \, dx}{b \sqrt{b x^2}}\\ &=-\frac{1}{2 b x \sqrt{b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0001464, size = 14, normalized size = 0.74 \[ -\frac{x}{2 \left (b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 11, normalized size = 0.6 \begin{align*} -{\frac{x}{2} \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 = 0.998931, size = 11, normalized size = 0.58 \begin{align*} -\frac{1}{2 \, b^{\frac{3}{2}} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48086, size = 38, normalized size = 2. \begin{align*} -\frac{\sqrt{b x^{2}}}{2 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.636939, size = 15, normalized size = 0.79 \begin{align*} - \frac{x}{2 b^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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