Optimal. Leaf size=16 \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]
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Rubi [A] time = 0.0023042, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {191} \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]
Antiderivative was successfully verified.
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Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx &=\frac{\sqrt{a+\frac{b}{x^2}} x}{a}\\ \end{align*}
Mathematica [A] time = 0.0057822, size = 16, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 28, normalized size = 1.8 \begin{align*}{\frac{a{x}^{2}+b}{ax}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99024, size = 19, normalized size = 1.19 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50766, size = 36, normalized size = 2.25 \begin{align*} \frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.689915, size = 17, normalized size = 1.06 \begin{align*} \frac{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20908, size = 38, normalized size = 2.38 \begin{align*} -\frac{\sqrt{b} \mathrm{sgn}\left (x\right )}{a} + \frac{\sqrt{a x^{2} + b}}{a \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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