Optimal. Leaf size=47 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
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Rubi [A] time = 0.0224933, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 47, 63, 208} \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^2}} x \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{1}{2} \sqrt{a+\frac{b}{x^2}} x^2-\frac{1}{4} b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{2} \sqrt{a+\frac{b}{x^2}} x^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^2}}\right )\\ &=\frac{1}{2} \sqrt{a+\frac{b}{x^2}} x^2+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0263612, size = 58, normalized size = 1.23 \[ \frac{1}{2} x \sqrt{a+\frac{b}{x^2}} \left (\frac{b \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a} \sqrt{a x^2+b}}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 62, normalized size = 1.3 \begin{align*}{\frac{x}{2}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( x\sqrt{a{x}^{2}+b}\sqrt{a}+b\ln \left ( x\sqrt{a}+\sqrt{a{x}^{2}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{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 = 1.51477, size = 292, normalized size = 6.21 \begin{align*} \left [\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} + \sqrt{a} b \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right )}{4 \, a}, \frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - \sqrt{-a} b \arctan \left (\frac{\sqrt{-a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right )}{2 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.93149, size = 41, normalized size = 0.87 \begin{align*} \frac{\sqrt{b} x \sqrt{\frac{a x^{2}}{b} + 1}}{2} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.71551, size = 70, normalized size = 1.49 \begin{align*} \frac{b \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{4 \, \sqrt{a}} + \frac{1}{2} \,{\left (\sqrt{a x^{2} + b} x - \frac{b \log \left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\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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