Optimal. Leaf size=28 \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]
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Rubi [A] time = 0.0260666, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {515, 23, 8} \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 515
Rule 23
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{b-\frac{a}{x^2}} x}{\sqrt{a-b x^2}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int \frac{\sqrt{-a+b x^2}}{\sqrt{a-b x^2}} \, dx}{\sqrt{-a+b x^2}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int 1 \, dx}{\sqrt{a-b x^2}}\\ &=\frac{\sqrt{b-\frac{a}{x^2}} x^2}{\sqrt{a-b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0081602, size = 28, normalized size = 1. \[ \frac{x^2 \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 42, normalized size = 1.5 \begin{align*} -{\frac{{x}^{2}}{b{x}^{2}-a}\sqrt{{\frac{b{x}^{2}-a}{{x}^{2}}}}\sqrt{-b{x}^{2}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.1588, size = 9, normalized size = 0.32 \begin{align*} -i \, \sqrt{x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44236, size = 77, normalized size = 2.75 \begin{align*} -\frac{\sqrt{-b x^{2} + a} x^{2} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b x^{2} - a} \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 \frac{x \sqrt{- \frac{a}{x^{2}} + b}}{\sqrt{a - b x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b - \frac{a}{x^{2}}} x}{\sqrt{-b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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