Optimal. Leaf size=89 \[ \frac{\sqrt{c} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}} \]
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Rubi [A] time = 0.054621, antiderivative size = 89, 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 = {427, 426, 424} \[ \frac{\sqrt{c} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{d x^2-c}} \]
Antiderivative was successfully verified.
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Rule 427
Rule 426
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{a-b x^2}}{\sqrt{-c+d x^2}} \, dx &=\frac{\sqrt{1-\frac{d x^2}{c}} \int \frac{\sqrt{a-b x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{\sqrt{-c+d x^2}}\\ &=\frac{\left (\sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{1-\frac{b x^2}{a}}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{\sqrt{1-\frac{b x^2}{a}} \sqrt{-c+d x^2}}\\ &=\frac{\sqrt{c} \sqrt{a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0474373, size = 89, normalized size = 1. \[ \frac{\sqrt{a-b x^2} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a-b x^2}{a}} \sqrt{d x^2-c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 166, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}-ad{x}^{2}-bc{x}^{2}+ac \right ) d} \left ( -a{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d+bc{\it EllipticF} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) -bc{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \right ) \sqrt{-b{x}^{2}+a}\sqrt{d{x}^{2}-c}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}{\frac{1}{\sqrt{{\frac{b}{a}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-b x^{2} + a}}{\sqrt{d x^{2} - c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-b x^{2} + a}}{\sqrt{d x^{2} - c}}, x\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{\sqrt{a - b x^{2}}}{\sqrt{- c + d 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 x^{2} + a}}{\sqrt{d x^{2} - c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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