Optimal. Leaf size=31 \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]
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Rubi [A] time = 0.0351915, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 515
Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{b-\frac{a}{x^2}} x^2}{\sqrt{a-b x^2}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int \frac{x \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 x \, dx}{\sqrt{a-b x^2}}\\ &=\frac{\sqrt{b-\frac{a}{x^2}} x^3}{2 \sqrt{a-b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0095304, size = 31, normalized size = 1. \[ \frac{x^3 \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 31, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{2}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.18261, size = 7, normalized size = 0.23 \begin{align*} -\frac{1}{2} i \, x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46176, size = 82, normalized size = 2.65 \begin{align*} -\frac{\sqrt{-b x^{2} + a} x^{3} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{2 \,{\left (b x^{2} - a\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 \frac{x^{2} \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 [A] time = 1.10978, size = 53, normalized size = 1.71 \begin{align*} -\frac{{\left (b x^{2} - a\right )} i \mathrm{sgn}\left (b x^{2} - a\right ) \mathrm{sgn}\left (x\right )}{2 \, b} + \frac{a i \mathrm{sgn}\left (a\right ) \mathrm{sgn}\left (x\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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