Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{5}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0070182, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{5}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x^2} \sqrt{2+5 x^2}} \, dx &=\frac{F\left (\sin ^{-1}(x)|-\frac{5}{2}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0050385, size = 12, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{5}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 14, normalized size = 1.2 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{10} \right ) \sqrt{2}}{2}} \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{1}{\sqrt{5 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{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{5 \, x^{2} + 2} \sqrt{-x^{2} + 1}}{5 \, x^{4} - 3 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.44315, size = 19, normalized size = 1.58 \begin{align*} \begin{cases} \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{5}{2}\right )}{2} & \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{1}{\sqrt{5 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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