Optimal. Leaf size=20 \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right ),-\frac{2}{5}\right )}{\sqrt{5}} \]
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Rubi [A] time = 0.0091487, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {419} \[ \frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-5 x^2} \sqrt{1+x^2}} \, dx &=\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0065552, size = 20, normalized size = 1. \[ \frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right ),-\frac{2}{5}\right )}{\sqrt{5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 19, normalized size = 1. \begin{align*}{\frac{\sqrt{5}}{5}{\it EllipticF} \left ({\frac{x\sqrt{10}}{2}},{\frac{i}{5}}\sqrt{10} \right ) } \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{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}\,{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{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}{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 = 4.56159, size = 36, normalized size = 1.8 \begin{align*} \begin{cases} \frac{\sqrt{5} F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | - \frac{2}{5}\right )}{5} & \text{for}\: x > - \frac{\sqrt{10}}{5} \wedge x < \frac{\sqrt{10}}{5} \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{x^{2} + 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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