Optimal. Leaf size=87 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]
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Rubi [A] time = 0.0534156, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {427, 426, 424} \[ \frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]
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 = 0.0563871, size = 87, normalized size = 1. \[ \frac{\sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|-\frac{b c}{a d}\right )}{\sqrt{\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 106, normalized size = 1.2 \begin{align*}{\frac{a}{-bd{x}^{4}-ad{x}^{2}+bc{x}^{2}+ac}\sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}+c}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{d}{c}}}}}} \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}}{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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