Optimal. Leaf size=16 \[ -\frac{1}{b \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0033578, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac{1}{b \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0025923, size = 16, normalized size = 1. \[ -\frac{1}{b \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.86063, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{\sqrt{b x^{2} + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30934, size = 46, normalized size = 2.88 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{b^{2} x^{2} + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.540283, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{1}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.99789, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{\sqrt{b x^{2} + a} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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