Optimal. Leaf size=93 \[ \frac{627857}{6} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+625 \sqrt{-x^4+x^2+2} x^3+\frac{27500}{3} \sqrt{-x^4+x^2+2} x+\frac{\left (1419793 x^2+1419985\right ) x}{18 \sqrt{-x^4+x^2+2}}-\frac{3482293}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.100598, antiderivative size = 93, 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 = {1205, 1679, 1180, 524, 424, 419} \[ 625 \sqrt{-x^4+x^2+2} x^3+\frac{27500}{3} \sqrt{-x^4+x^2+2} x+\frac{\left (1419793 x^2+1419985\right ) x}{18 \sqrt{-x^4+x^2+2}}+\frac{627857}{6} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{3482293}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1205
Rule 1679
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{\left (7+5 x^2\right )^5}{\left (2+x^2-x^4\right )^{3/2}} \, dx &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}-\frac{1}{18} \int \frac{1268722+3084793 x^2+450000 x^4+56250 x^6}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}+625 x^3 \sqrt{2+x^2-x^4}+\frac{1}{90} \int \frac{-6343610-15761465 x^2-2475000 x^4}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}+\frac{27500}{3} x \sqrt{2+x^2-x^4}+625 x^3 \sqrt{2+x^2-x^4}-\frac{1}{270} \int \frac{23980830+52234395 x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}+\frac{27500}{3} x \sqrt{2+x^2-x^4}+625 x^3 \sqrt{2+x^2-x^4}-\frac{1}{135} \int \frac{23980830+52234395 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}+\frac{27500}{3} x \sqrt{2+x^2-x^4}+625 x^3 \sqrt{2+x^2-x^4}-\frac{3482293}{18} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+\frac{627857}{3} \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{x \left (1419985+1419793 x^2\right )}{18 \sqrt{2+x^2-x^4}}+\frac{27500}{3} x \sqrt{2+x^2-x^4}+625 x^3 \sqrt{2+x^2-x^4}-\frac{3482293}{18} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{627857}{6} 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 = 0.033, size = 280, normalized size = 3. \begin{align*} 6250\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{17\,{x}^{3}}{18}}+{\frac{7\,x}{9}} \right ) }+625\,{x}^{3}\sqrt{-{x}^{4}+{x}^{2}+2}+{\frac{27500\,x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{799361\,\sqrt{2}}{18}\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{3482293\,\sqrt{2}}{36}\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}}}}+43750\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{7\,{x}^{3}}{18}}+5/9\,x \right ) }+122500\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,{x}^{3}}{18}}+x/9 \right ) }+171500\,{\frac{1/18\,{x}^{3}+2/9\,x}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+120050\,{\frac{1/9\,{x}^{3}-x/18}{\sqrt{-{x}^{4}+{x}^{2}+2}}}+33614\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ({\frac{5\,x}{36}}-1/36\,{x}^{3} \right ) } \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 )}^{5}}{{\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 (\frac{{\left (3125 \, x^{10} + 21875 \, x^{8} + 61250 \, x^{6} + 85750 \, x^{4} + 60025 \, x^{2} + 16807\right )} \sqrt{-x^{4} + x^{2} + 2}}{x^{8} - 2 \, x^{6} - 3 \, x^{4} + 4 \, x^{2} + 4}, 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 )^{5}}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\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 \frac{{\left (5 \, x^{2} + 7\right )}^{5}}{{\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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