Optimal. Leaf size=65 \[ -542 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )-25 \sqrt{-x^4+x^2+2} x^3-\frac{625}{3} \sqrt{-x^4+x^2+2} x+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0763707, antiderivative size = 65, 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, 1679, 1180, 524, 424, 419} \[ -25 \sqrt{-x^4+x^2+2} x^3-\frac{625}{3} \sqrt{-x^4+x^2+2} x-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (7+5 x^2\right )^3}{\sqrt{2+x^2-x^4}} \, dx &=-25 x^3 \sqrt{2+x^2-x^4}-\frac{1}{5} \int \frac{-1715-4425 x^2-3125 x^4}{\sqrt{2+x^2-x^4}} \, dx\\ &=-\frac{625}{3} x \sqrt{2+x^2-x^4}-25 x^3 \sqrt{2+x^2-x^4}+\frac{1}{15} \int \frac{11395+19525 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=-\frac{625}{3} x \sqrt{2+x^2-x^4}-25 x^3 \sqrt{2+x^2-x^4}+\frac{2}{15} \int \frac{11395+19525 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=-\frac{625}{3} x \sqrt{2+x^2-x^4}-25 x^3 \sqrt{2+x^2-x^4}-1084 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{3905}{3} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=-\frac{625}{3} x \sqrt{2+x^2-x^4}-25 x^3 \sqrt{2+x^2-x^4}+\frac{3905}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-542 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.112961, size = 97, normalized size = 1.49 \[ \frac{-10089 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )+150 x^7+1100 x^5-1550 x^3+7810 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-2500 x}{6 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 142, normalized size = 2.2 \begin{align*} -25\,{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2}-{\frac{625\,x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2279\,\sqrt{2}}{6}\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{3905\,\sqrt{2}}{6}\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 \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + 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{{\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt{-x^{4} + x^{2} + 2}}{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 \frac{\left (5 x^{2} + 7\right )^{3}}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\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 \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{-x^{4} + x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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