Optimal. Leaf size=46 \[ 3 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+x \sqrt{-x^4+x^2+2} \left (x^2+2\right )+7 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0460943, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1176, 1180, 524, 424, 419} \[ x \sqrt{-x^4+x^2+2} \left (x^2+2\right )+3 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+7 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (7+5 x^2\right ) \sqrt{2+x^2-x^4} \, dx &=x \left (2+x^2\right ) \sqrt{2+x^2-x^4}-\frac{1}{15} \int \frac{-150-105 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=x \left (2+x^2\right ) \sqrt{2+x^2-x^4}-\frac{2}{15} \int \frac{-150-105 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=x \left (2+x^2\right ) \sqrt{2+x^2-x^4}+6 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+7 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=x \left (2+x^2\right ) \sqrt{2+x^2-x^4}+7 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+3 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.0809684, size = 94, normalized size = 2.04 \[ \frac{-12 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-x^7-x^5+4 x^3+7 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+4 x}{\sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 141, normalized size = 3.1 \begin{align*}{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2}+2\,x\sqrt{-{x}^{4}+{x}^{2}+2}+5\,{\frac{\sqrt{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ( 1/2\,x\sqrt{2},i\sqrt{2} \right ) }{\sqrt{-{x}^{4}+{x}^{2}+2}}}-{\frac{7\,\sqrt{2}}{2}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{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 \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}\,{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 (\sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}, 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 \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )\, 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 \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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