Optimal. Leaf size=100 \[ -\frac{3392}{165} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )-\frac{25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac{1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{495} x \left (14889 x^2+11497\right ) \sqrt{-x^4+x^2+2}+\frac{85942}{495} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0738041, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac{1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{495} x \left (14889 x^2+11497\right ) \sqrt{-x^4+x^2+2}-\frac{3392}{165} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{85942}{495} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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 \left (2+x^2-x^4\right )^{3/2} \, dx &=-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac{1}{11} \int \left (-589-920 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx\\ &=\frac{1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}+\frac{1}{231} \int \left (23044+34741 x^2\right ) \sqrt{2+x^2-x^4} \, dx\\ &=\frac{1}{495} x \left (11497+14889 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac{\int \frac{-530362-601594 x^2}{\sqrt{2+x^2-x^4}} \, dx}{3465}\\ &=\frac{1}{495} x \left (11497+14889 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac{2 \int \frac{-530362-601594 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{3465}\\ &=\frac{1}{495} x \left (11497+14889 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac{6784}{165} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx+\frac{85942}{495} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx\\ &=\frac{1}{495} x \left (11497+14889 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac{25}{11} x \left (2+x^2-x^4\right )^{5/2}+\frac{85942}{495} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{3392}{165} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [B] time = 0.008, size = 193, normalized size = 1.9 \begin{align*} -{\frac{25\,{x}^{9}}{11}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{470\,{x}^{7}}{99}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{112\,{x}^{5}}{9}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{21404\,{x}^{3}}{495}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{10627\,x}{495}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{37883\,\sqrt{2}}{495}\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{42971\,\sqrt{2}}{495}\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{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{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^{8} + 45 \, x^{6} - 71 \, x^{4} - 189 \, x^{2} - 98\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 \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \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{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{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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