Optimal. Leaf size=198 \[ -\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}} \]
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Rubi [A] time = 0.129412, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {423, 427, 426, 424, 421, 419} \[ -\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{d} \sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}} \]
Antiderivative was successfully verified.
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Rule 423
Rule 427
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{-c+d x^2}}{\sqrt{-a-b x^2}} \, dx &=-\frac{d \int \frac{\sqrt{-a-b x^2}}{\sqrt{-c+d x^2}} \, dx}{b}-\frac{(b c+a d) \int \frac{1}{\sqrt{-a-b x^2} \sqrt{-c+d x^2}} \, dx}{b}\\ &=-\frac{\left (d \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{\sqrt{-a-b x^2}}{\sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{-c+d x^2}}-\frac{\left ((b c+a d) \sqrt{1-\frac{d x^2}{c}}\right ) \int \frac{1}{\sqrt{-a-b x^2} \sqrt{1-\frac{d x^2}{c}}} \, dx}{b \sqrt{-c+d x^2}}\\ &=-\frac{\left (d \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}{b \sqrt{1+\frac{b x^2}{a}} \sqrt{-c+d x^2}}-\frac{\left ((b c+a d) \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}{b \sqrt{-a-b x^2} \sqrt{-c+d x^2}}\\ &=-\frac{\sqrt{c} \sqrt{d} \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 )}{b \sqrt{1+\frac{b x^2}{a}} \sqrt{-c+d x^2}}-\frac{\sqrt{c} (b c+a d) \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 )}{b \sqrt{d} \sqrt{-a-b x^2} \sqrt{-c+d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0452211, size = 93, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{-a-b x^2} \sqrt{\frac{c-d x^2}{c}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 167, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}-bc{x}^{2}-ac \right ) b}\sqrt{d{x}^{2}-c}\sqrt{-b{x}^{2}-a}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}} \left ( ad{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) +c{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) b-ad{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \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{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{-b x^{2} - a} \sqrt{d x^{2} - c}}{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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