Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{7}{2}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.011929, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1095, 419} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{7}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1095
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2+5 x^2-7 x^4}} \, dx &=\left (2 \sqrt{7}\right ) \int \frac{1}{\sqrt{14-14 x^2} \sqrt{4+14 x^2}} \, dx\\ &=\frac{F\left (\sin ^{-1}(x)|-\frac{7}{2}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.024769, size = 65, normalized size = 5.42 \[ -\frac{i \sqrt{1-x^2} \sqrt{7 x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{7}{2}} x\right ),-\frac{2}{7}\right )}{\sqrt{7} \sqrt{-7 x^4+5 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 43, normalized size = 3.6 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{14} \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{14\,{x}^{2}+4}{\frac{1}{\sqrt{-7\,{x}^{4}+5\,{x}^{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{-7 \, x^{4} + 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{-7 \, x^{4} + 5 \, x^{2} + 2}}{7 \, x^{4} - 5 \, x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 7 x^{4} + 5 x^{2} + 2}}\, dx \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{-7 \, x^{4} + 5 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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