Optimal. Leaf size=19 \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]
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Rubi [A] time = 0.0017707, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac{1}{3 b x^2 \sqrt{b x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x \left (b x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{1}{x^4} \, dx}{b \sqrt{b x^2}}\\ &=-\frac{1}{3 b x^2 \sqrt{b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0024179, size = 17, normalized size = 0.89 \[ -\frac{b x^2}{3 \left (b x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 10, normalized size = 0.5 \begin{align*} -{\frac{1}{3} \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 = 0.991847, size = 11, normalized size = 0.58 \begin{align*} -\frac{1}{3 \, b^{\frac{3}{2}} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43212, size = 38, normalized size = 2. \begin{align*} -\frac{\sqrt{b x^{2}}}{3 \, b^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.770041, size = 15, normalized size = 0.79 \begin{align*} - \frac{1}{3 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 [A] time = 1.19822, size = 20, normalized size = 1.05 \begin{align*} -\frac{1}{3 \, \sqrt{b x^{2}} b x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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