Optimal. Leaf size=91 \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}} \]
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Rubi [A] time = 0.0500475, antiderivative size = 91, 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 = {427, 426, 424} \[ \frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{b x^2-a} \sqrt{\frac{d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 427
Rule 426
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{-c-d x^2}}{\sqrt{-a+b x^2}} \, dx &=\frac{\sqrt{1-\frac{b x^2}{a}} \int \frac{\sqrt{-c-d x^2}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{\sqrt{-a+b x^2}}\\ &=\frac{\left (\sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2}\right ) \int \frac{\sqrt{1+\frac{d x^2}{c}}}{\sqrt{1-\frac{b x^2}{a}}} \, dx}{\sqrt{-a+b x^2} \sqrt{1+\frac{d x^2}{c}}}\\ &=\frac{\sqrt{a} \sqrt{1-\frac{b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )|-\frac{a d}{b c}\right )}{\sqrt{b} \sqrt{-a+b x^2} \sqrt{1+\frac{d x^2}{c}}}\\ \end{align*}
Mathematica [A] time = 0.0418449, size = 91, normalized size = 1. \[ \frac{\sqrt{\frac{a-b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{\frac{b}{a}} \sqrt{b x^2-a} \sqrt{\frac{c+d x^2}{c}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 110, normalized size = 1.2 \begin{align*}{\frac{c}{bd{x}^{4}-ad{x}^{2}+bc{x}^{2}-ac}\sqrt{-d{x}^{2}-c}\sqrt{b{x}^{2}-a}\sqrt{-{\frac{b{x}^{2}-a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ){\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{-d x^{2} - c}}{\sqrt{b x^{2} - a}}\,{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{-d x^{2} - c}}{\sqrt{b x^{2} - a}}, 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{- c - d x^{2}}}{\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{-d x^{2} - c}}{\sqrt{b x^{2} - a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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