Optimal. Leaf size=102 \[ -\frac{263 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{226576}-\frac{12525 \sqrt{-x^4+x^2+2} x}{453152 \left (5 x^2+7\right )}-\frac{25 \sqrt{-x^4+x^2+2} x}{952 \left (5 x^2+7\right )^2}-\frac{2505 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{453152}+\frac{58915 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3172064} \]
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Rubi [A] time = 0.191768, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1223, 1696, 1716, 1180, 524, 424, 419, 1212, 537} \[ -\frac{12525 \sqrt{-x^4+x^2+2} x}{453152 \left (5 x^2+7\right )}-\frac{25 \sqrt{-x^4+x^2+2} x}{952 \left (5 x^2+7\right )^2}-\frac{263 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{226576}-\frac{2505 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{453152}+\frac{58915 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3172064} \]
Antiderivative was successfully verified.
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Rule 1223
Rule 1696
Rule 1716
Rule 1180
Rule 524
Rule 424
Rule 419
Rule 1212
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+x^2-x^4}} \, dx &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}+\frac{1}{952} \int \frac{186-190 x^2+25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+x^2-x^4}} \, dx\\ &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac{12525 x \sqrt{2+x^2-x^4}}{453152 \left (7+5 x^2\right )}+\frac{\int \frac{37698-32690 x^2-12525 x^4}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{453152}\\ &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac{12525 x \sqrt{2+x^2-x^4}}{453152 \left (7+5 x^2\right )}-\frac{\int \frac{75775+62625 x^2}{\sqrt{2+x^2-x^4}} \, dx}{11328800}+\frac{58915 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+x^2-x^4}} \, dx}{453152}\\ &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac{12525 x \sqrt{2+x^2-x^4}}{453152 \left (7+5 x^2\right )}-\frac{\int \frac{75775+62625 x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{5664400}+\frac{58915 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2} \left (7+5 x^2\right )} \, dx}{226576}\\ &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac{12525 x \sqrt{2+x^2-x^4}}{453152 \left (7+5 x^2\right )}+\frac{58915 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3172064}-\frac{263 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx}{113288}-\frac{2505 \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx}{453152}\\ &=-\frac{25 x \sqrt{2+x^2-x^4}}{952 \left (7+5 x^2\right )^2}-\frac{12525 x \sqrt{2+x^2-x^4}}{453152 \left (7+5 x^2\right )}-\frac{2505 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{453152}-\frac{263 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{226576}+\frac{58915 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{3172064}\\ \end{align*}
Mathematica [C] time = 0.405084, size = 108, normalized size = 1.06 \[ \frac{56287 i \sqrt{2} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )+\frac{350 x \left (2505 x^6+1478 x^4-8993 x^2-7966\right )}{\left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2}}-35070 i \sqrt{2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-58915 i \sqrt{2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )}{6344128} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 189, normalized size = 1.9 \begin{align*} -{\frac{25\,x}{952\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{12525\,x}{2265760\,{x}^{2}+3172064}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{263\,\sqrt{2}}{453152}\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{2505\,\sqrt{2}}{906304}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{58915\,\sqrt{2}}{3172064}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}},-{\frac{10}{7}},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 \frac{1}{\sqrt{-x^{4} + x^{2} + 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 (-\frac{\sqrt{-x^{4} + x^{2} + 2}}{125 \, x^{10} + 400 \, x^{8} - 40 \, x^{6} - 1442 \, x^{4} - 1813 \, x^{2} - 686}, 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{1}{\sqrt{- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \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 \frac{1}{\sqrt{-x^{4} + x^{2} + 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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