Optimal. Leaf size=87 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]
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Rubi [A] time = 0.0524197, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {421, 419} \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x^2} \sqrt{c-d x^2}} \, dx &=\frac{\sqrt{1-\frac{d x^2}{c}} \int \frac{1}{\sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}}} \, dx}{\sqrt{c-d x^2}}\\ &=\frac{\left (\sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}}} \, dx}{\sqrt{a+b x^2} \sqrt{c-d x^2}}\\ &=\frac{\sqrt{c} \sqrt{1+\frac{b x^2}{a}} \sqrt{1-\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0649762, size = 89, normalized size = 1.02 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{\frac{c-d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (x \sqrt{-\frac{b}{a}}\right ),-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 106, normalized size = 1.2 \begin{align*} -{\frac{1}{bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac}{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}+c}{\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{1}{\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}}{b d x^{4} -{\left (b c - a d\right )} x^{2} - a 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{1}{\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{1}{\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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