Optimal. Leaf size=91 \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{2}\right ),-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]
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Rubi [A] time = 0.0577295, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {423, 426, 424, 421, 419} \[ \frac{(c+4 d) \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}} \]
Antiderivative was successfully verified.
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Rule 423
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{4-x^2}}{\sqrt{c+d x^2}} \, dx &=-\frac{\int \frac{\sqrt{c+d x^2}}{\sqrt{4-x^2}} \, dx}{d}-\frac{(-c-4 d) \int \frac{1}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx}{d}\\ &=-\frac{\sqrt{c+d x^2} \int \frac{\sqrt{1+\frac{d x^2}{c}}}{\sqrt{4-x^2}} \, dx}{d \sqrt{1+\frac{d x^2}{c}}}-\frac{\left ((-c-4 d) \sqrt{1+\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{4-x^2} \sqrt{1+\frac{d x^2}{c}}} \, dx}{d \sqrt{c+d x^2}}\\ &=-\frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{1+\frac{d x^2}{c}}}+\frac{(c+4 d) \sqrt{1+\frac{d x^2}{c}} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0380096, size = 60, normalized size = 0.66 \[ \frac{2 \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|-\frac{c}{4 d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{c+d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 78, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( c{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) +4\,{\it EllipticF} \left ( x/2,2\,\sqrt{-{\frac{d}{c}}} \right ) d-c{\it EllipticE} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) \right ) \sqrt{{\frac{d{x}^{2}+c}{c}}}{\frac{1}{\sqrt{d{x}^{2}+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{-x^{2} + 4}}{\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{-x^{2} + 4}}{\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{- \left (x - 2\right ) \left (x + 2\right )}}{\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{-x^{2} + 4}}{\sqrt{d x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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