Optimal. Leaf size=21 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right ),-1\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {329, 221} \[ \frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right |-1\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 329
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {1-a^2 x^2}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right |-1\right )}{\sqrt {a}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 1.14 \[ 2 \sqrt {x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};a^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {x}}{a^{2} x^{3} - x}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 66, normalized size = 3.14 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x +1}\, \sqrt {-2 a x +2}\, \sqrt {-a x}\, \EllipticF \left (\sqrt {a x +1}, \frac {\sqrt {2}}{2}\right )}{\left (a^{2} x^{2}-1\right ) a \sqrt {x}} \]
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 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{\sqrt {x}\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.80, size = 36, normalized size = 1.71 \[ \frac {\sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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