Optimal. Leaf size=43 \[ x \sqrt{b-\frac{a}{x^2}}+\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b-\frac{a}{x^2}}}\right ) \]
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Rubi [A] time = 0.023209, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {1972, 242, 277, 217, 203} \[ x \sqrt{b-\frac{a}{x^2}}+\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{b-\frac{a}{x^2}}}\right ) \]
Antiderivative was successfully verified.
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Rule 1972
Rule 242
Rule 277
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \sqrt{\frac{-a+b x^2}{x^2}} \, dx &=\int \sqrt{b-\frac{a}{x^2}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{\sqrt{b-a x^2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{b-\frac{a}{x^2}} x+a \operatorname{Subst}\left (\int \frac{1}{\sqrt{b-a x^2}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{b-\frac{a}{x^2}} x+a \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,\frac{1}{\sqrt{b-\frac{a}{x^2}} x}\right )\\ &=\sqrt{b-\frac{a}{x^2}} x+\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a}}{\sqrt{b-\frac{a}{x^2}} x}\right )\\ \end{align*}
Mathematica [A] time = 0.0314661, size = 68, normalized size = 1.58 \[ x \sqrt{b-\frac{a}{x^2}}-\frac{\sqrt{a} x \sqrt{b-\frac{a}{x^2}} \tan ^{-1}\left (\frac{\sqrt{b x^2-a}}{\sqrt{a}}\right )}{\sqrt{b x^2-a}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 81, normalized size = 1.9 \begin{align*}{x\sqrt{{\frac{b{x}^{2}-a}{{x}^{2}}}} \left ( a\ln \left ( 2\,{\frac{\sqrt{-a}\sqrt{b{x}^{2}-a}-a}{x}} \right ) +\sqrt{-a}\sqrt{b{x}^{2}-a} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{b{x}^{2}-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.75714, size = 259, normalized size = 6.02 \begin{align*} \left [x \sqrt{\frac{b x^{2} - a}{x^{2}}} + \frac{1}{2} \, \sqrt{-a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{-a} x \sqrt{\frac{b x^{2} - a}{x^{2}}} - 2 \, a}{x^{2}}\right ), x \sqrt{\frac{b x^{2} - a}{x^{2}}} + \sqrt{a} \arctan \left (\frac{\sqrt{a} x \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{2} - a}\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{- a + b x^{2}}{x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22661, size = 86, normalized size = 2. \begin{align*} -{\left (\sqrt{a} \arctan \left (\frac{\sqrt{b x^{2} - a}}{\sqrt{a}}\right ) - \sqrt{b x^{2} - a}\right )} \mathrm{sgn}\left (x\right ) +{\left (\sqrt{a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{a}}\right ) - \sqrt{-a}\right )} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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