Optimal. Leaf size=74 \[ -\frac{79}{7} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )-\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0606281, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1206, 1176, 1180, 524, 424, 419} \[ -\frac{25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{21} x \left (354 x^2+275\right ) \sqrt{-x^4+x^2+2}-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4} \, dx &=-\frac{25}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac{1}{7} \int \left (-393-590 x^2\right ) \sqrt{2+x^2-x^4} \, dx\\ &=\frac{1}{21} x \left (275+354 x^2\right ) \sqrt{2+x^2-x^4}-\frac{25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{1}{105} \int \frac{9040+10225 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{1}{21} x \left (275+354 x^2\right ) \sqrt{2+x^2-x^4}-\frac{25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{2}{105} \int \frac{9040+10225 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{21} x \left (275+354 x^2\right ) \sqrt{2+x^2-x^4}-\frac{25}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac{158}{7} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{2045}{21} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=\frac{1}{21} x \left (275+354 x^2\right ) \sqrt{2+x^2-x^4}-\frac{25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{2045}{21} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{79}{7} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.0893145, size = 102, normalized size = 1.38 \[ \frac{-2949 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-75 x^9-204 x^7+304 x^5+683 x^3+2045 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+250 x}{21 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 159, normalized size = 2.2 \begin{align*}{\frac{25\,{x}^{5}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{93\,{x}^{3}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{125\,x}{21}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{904\,\sqrt{2}}{21}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{2045\,\sqrt{2}}{42}\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 )}^{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 ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{-x^{4} + 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 \sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{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 \sqrt{-x^{4} + x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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