Optimal. Leaf size=212 \[ \frac{\sqrt{c} \sqrt{-a-b x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right ),1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{-a-b x^2}}{\sqrt{-c-d x^2}}-\frac{\sqrt{c} \sqrt{-a-b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
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Rubi [A] time = 0.103342, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {422, 418, 492, 411} \[ \frac{x \sqrt{-a-b x^2}}{\sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{-a-b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{-a-b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
Antiderivative was successfully verified.
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Rule 422
Rule 418
Rule 492
Rule 411
Rubi steps
\begin{align*} \int \frac{\sqrt{-a-b x^2}}{\sqrt{-c-d x^2}} \, dx &=-\left (a \int \frac{1}{\sqrt{-a-b x^2} \sqrt{-c-d x^2}} \, dx\right )-b \int \frac{x^2}{\sqrt{-a-b x^2} \sqrt{-c-d x^2}} \, dx\\ &=\frac{x \sqrt{-a-b x^2}}{\sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{-a-b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+c \int \frac{\sqrt{-a-b x^2}}{\left (-c-d x^2\right )^{3/2}} \, dx\\ &=\frac{x \sqrt{-a-b x^2}}{\sqrt{-c-d x^2}}-\frac{\sqrt{c} \sqrt{-a-b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{\sqrt{c} \sqrt{-a-b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end{align*}
Mathematica [A] time = 0.0437262, size = 92, normalized size = 0.43 \[ \frac{\sqrt{-a-b x^2} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 165, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) d} \left ( -a{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d+bc{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) -bc{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) \right ) \sqrt{-b{x}^{2}-a}\sqrt{-d{x}^{2}-c}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\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{-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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