Optimal. Leaf size=121 \[ -\frac{3199778 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{5005}-\frac{125}{13} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{7825}{143} \left (-x^4+x^2+2\right )^{5/2} x+\frac{\left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2} x}{3003}+\frac{\left (5712051 x^2+2512273\right ) \sqrt{-x^4+x^2+2} x}{15015}+\frac{31072528 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{15015} \]
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Rubi [A] time = 0.101206, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1206, 1679, 1176, 1180, 524, 424, 419} \[ -\frac{125}{13} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{7825}{143} \left (-x^4+x^2+2\right )^{5/2} x+\frac{\left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2} x}{3003}+\frac{\left (5712051 x^2+2512273\right ) \sqrt{-x^4+x^2+2} x}{15015}-\frac{3199778 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}+\frac{31072528 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{15015} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^3 \left (2+x^2-x^4\right )^{3/2} \, dx &=-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac{1}{13} \int \left (-4459-10305 x^2-7825 x^4\right ) \left (2+x^2-x^4\right )^{3/2} \, dx\\ &=-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}+\frac{1}{143} \int \left (64699+160305 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx\\ &=\frac{x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac{\int \left (-2649774-5712051 x^2\right ) \sqrt{2+x^2-x^4} \, dx}{3003}\\ &=\frac{x \left (2512273+5712051 x^2\right ) \sqrt{2+x^2-x^4}}{15015}+\frac{x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}+\frac{\int \frac{64419582+93217584 x^2}{\sqrt{2+x^2-x^4}} \, dx}{45045}\\ &=\frac{x \left (2512273+5712051 x^2\right ) \sqrt{2+x^2-x^4}}{15015}+\frac{x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}+\frac{2 \int \frac{64419582+93217584 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{45045}\\ &=\frac{x \left (2512273+5712051 x^2\right ) \sqrt{2+x^2-x^4}}{15015}+\frac{x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac{6399556 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{5005}+\frac{31072528 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{15015}\\ &=\frac{x \left (2512273+5712051 x^2\right ) \sqrt{2+x^2-x^4}}{15015}+\frac{x \left (33792+374045 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{3003}-\frac{7825}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac{125}{13} x^3 \left (2+x^2-x^4\right )^{5/2}+\frac{31072528 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{15015}-\frac{3199778 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}\\ \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 [A] time = 0.007, size = 210, normalized size = 1.7 \begin{align*} -{\frac{436307\,x}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{65248\,{x}^{5}}{273}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{125\,{x}^{11}}{13}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{5757461\,{x}^{3}}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{5075\,{x}^{9}}{143}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{5890\,{x}^{7}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{15536264\,\sqrt{2}}{15015}\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}}}}+{\frac{10736597\,\sqrt{2}}{15015}\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}}}} \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 )}^{3}\,{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 (125 \, x^{10} + 400 \, x^{8} - 40 \, x^{6} - 1442 \, x^{4} - 1813 \, x^{2} - 686\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 )^{3}\, 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 )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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