Optimal. Leaf size=36 \[ -\frac{\sqrt{\frac{b x^2}{a}+1} \text{EllipticF}\left (\sin ^{-1}(x),-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]
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Rubi [A] time = 0.0248665, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 421, 419} \[ -\frac{\sqrt{\frac{b x^2}{a}+1} F\left (\sin ^{-1}(x)|-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\sqrt{1-x^2}}{\left (-1+x^2\right ) \sqrt{a+b x^2}} \, dx &=-\int \frac{1}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx\\ &=-\frac{\sqrt{1+\frac{b x^2}{a}} \int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{b x^2}{a}}} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{\sqrt{1+\frac{b x^2}{a}} F\left (\sin ^{-1}(x)|-\frac{b}{a}\right )}{\sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0420609, size = 37, normalized size = 1.03 \[ -\frac{\sqrt{\frac{a+b x^2}{a}} \text{EllipticF}\left (\sin ^{-1}(x),-\frac{b}{a}\right )}{\sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 35, normalized size = 1. \begin{align*} -{\sqrt{{\frac{b{x}^{2}+a}{a}}}{\it EllipticF} \left ( x,\sqrt{-{\frac{b}{a}}} \right ){\frac{1}{\sqrt{b{x}^{2}+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{-x^{2} + 1}}{\sqrt{b x^{2} + a}{\left (x^{2} - 1\right )}}\,{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{-x^{2} + 1}}{b x^{4} +{\left (a - b\right )} x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.7869, size = 19, normalized size = 0.53 \begin{align*} \begin{cases} - \frac{F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{b}{a}\right )}{\sqrt{a}} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \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} + 1}}{\sqrt{b x^{2} + a}{\left (x^{2} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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