Optimal. Leaf size=237 \[ \frac{631 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{325 \sqrt{x^4+3 x^2+2} x}{4704 \left (5 x^2+7\right )}-\frac{25 \sqrt{x^4+3 x^2+2} x}{168 \left (5 x^2+7\right )^2}+\frac{65 \left (x^2+2\right ) x}{4704 \sqrt{x^4+3 x^2+2}}-\frac{65 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{2525 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{65856 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.249866, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1223, 1696, 1716, 1189, 1099, 1135, 1214, 1456, 539} \[ -\frac{325 \sqrt{x^4+3 x^2+2} x}{4704 \left (5 x^2+7\right )}-\frac{25 \sqrt{x^4+3 x^2+2} x}{168 \left (5 x^2+7\right )^2}+\frac{65 \left (x^2+2\right ) x}{4704 \sqrt{x^4+3 x^2+2}}+\frac{631 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{65 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{2525 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{65856 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1223
Rule 1696
Rule 1716
Rule 1189
Rule 1099
Rule 1135
Rule 1214
Rule 1456
Rule 539
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}} \, dx &=-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}+\frac{1}{168} \int \frac{74-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx\\ &=-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac{325 x \sqrt{2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}+\frac{\int \frac{2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{14112}\\ &=-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac{325 x \sqrt{2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}-\frac{\int \frac{-4725-4875 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{352800}+\frac{505 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{4704}\\ &=-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac{325 x \sqrt{2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}+\frac{3}{224} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{65 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{4704}+\frac{505 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{9408}-\frac{2525 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{18816}\\ &=\frac{65 x \left (2+x^2\right )}{4704 \sqrt{2+3 x^2+x^4}}-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac{325 x \sqrt{2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}-\frac{65 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{631 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{\left (2525 \sqrt{1+\frac{x^2}{2}} \sqrt{2+2 x^2}\right ) \int \frac{\sqrt{2+2 x^2}}{\sqrt{1+\frac{x^2}{2}} \left (7+5 x^2\right )} \, dx}{18816 \sqrt{2+3 x^2+x^4}}\\ &=\frac{65 x \left (2+x^2\right )}{4704 \sqrt{2+3 x^2+x^4}}-\frac{25 x \sqrt{2+3 x^2+x^4}}{168 \left (7+5 x^2\right )^2}-\frac{325 x \sqrt{2+3 x^2+x^4}}{4704 \left (7+5 x^2\right )}-\frac{65 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{631 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{2525 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{65856 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.334159, size = 186, normalized size = 0.78 \[ \frac{14 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )-175 x \left (65 x^6+314 x^4+487 x^2+238\right )-455 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-505 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )}{32928 \left (5 x^2+7\right )^2 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 186, normalized size = 0.8 \begin{align*} -{\frac{25\,x}{168\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{\frac{325\,x}{23520\,{x}^{2}+32928}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{{\frac{i}{4704}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{65\,i}{9408}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{505\,i}{32928}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}x\sqrt{2},{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{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} + 3 \, 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} + 3 \, x^{2} + 2}}{125 \, x^{10} + 900 \, x^{8} + 2560 \, x^{6} + 3598 \, x^{4} + 2499 \, 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} + 1\right ) \left (x^{2} + 2\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} + 3 \, 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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