Optimal. Leaf size=74 \[ \frac{48}{35} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+\frac{1}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{35} x \left (3 x^2+19\right ) \sqrt{-x^4+x^2+2}+\frac{34}{35} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0516522, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {1091, 1176, 1180, 524, 424, 419} \[ \frac{1}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{35} x \left (3 x^2+19\right ) \sqrt{-x^4+x^2+2}+\frac{48}{35} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{34}{35} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1091
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (2+x^2-x^4\right )^{3/2} \, dx &=\frac{1}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{3}{7} \int \left (4+x^2\right ) \sqrt{2+x^2-x^4} \, dx\\ &=\frac{1}{35} x \left (19+3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac{1}{35} \int \frac{-82-34 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{1}{35} x \left (19+3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac{2}{35} \int \frac{-82-34 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{35} x \left (19+3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{34}{35} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+\frac{96}{35} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{35} x \left (19+3 x^2\right ) \sqrt{2+x^2-x^4}+\frac{1}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac{34}{35} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{48}{35} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.0533497, size = 102, normalized size = 1.38 \[ \frac{-75 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )+5 x^9-13 x^7-31 x^5+45 x^3+34 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+58 x}{35 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 159, normalized size = 2.2 \begin{align*} -{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{8\,{x}^{3}}{35}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{29\,x}{35}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{41\,\sqrt{2}}{35}\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{17\,\sqrt{2}}{35}\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}}\,{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 (-x^{4} + x^{2} + 2\right )}^{\frac{3}{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 (- x^{4} + x^{2} + 2\right )^{\frac{3}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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