Optimal. Leaf size=36 \[ -\frac {\sqrt {\frac {b x^2}{a}+1} F\left (\sin ^{-1}(x)|-\frac {b}{a}\right )}{\sqrt {a+b x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 419
Rule 421
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*}
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Mathematica [A] time = 0.06, size = 37, normalized size = 1.03 \[ -\frac {\sqrt {\frac {a+b x^2}{a}} F\left (\sin ^{-1}(x)|-\frac {b}{a}\right )}{\sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{2} + a} \sqrt {-x^{2} + 1}}{b x^{4} + {\left (a - b\right )} x^{2} - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {b x^{2} + a} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 0.97 \[ -\frac {\sqrt {\frac {b \,x^{2}+a}{a}}\, \EllipticF \left (x , \sqrt {-\frac {b}{a}}\right )}{\sqrt {b \,x^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {b x^{2} + a} {\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {1}{\sqrt {1-x^2}\,\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.56, size = 19, normalized size = 0.53 \[ \begin {cases} - \frac {F\left (\operatorname {asin}{\relax (x )}\middle | - \frac {b}{a}\right )}{\sqrt {a}} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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