Optimal. Leaf size=237 \[ \frac{81 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{7840 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{11 \sqrt{x^4+3 x^2+2} x}{2352 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac{11 \left (x^2+2\right ) x}{11760 \sqrt{x^4+3 x^2+2}}+\frac{11 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5880 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{1201 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{164640 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.595241, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1228, 1223, 1696, 1716, 1189, 1099, 1135, 1214, 1456, 539} \[ \frac{11 \sqrt{x^4+3 x^2+2} x}{2352 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac{11 \left (x^2+2\right ) x}{11760 \sqrt{x^4+3 x^2+2}}+\frac{81 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{7840 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{11 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5880 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{1201 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{164640 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1223
Rule 1696
Rule 1716
Rule 1189
Rule 1099
Rule 1135
Rule 1214
Rule 1456
Rule 539
Rubi steps
\begin{align*} \int \frac{\sqrt{2+3 x^2+x^4}}{\left (7+5 x^2\right )^3} \, dx &=\int \left (-\frac{6}{25 \left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}}+\frac{1}{25 \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}}+\frac{1}{25 \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}}\right ) \, dx\\ &=\frac{1}{25} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx+\frac{1}{25} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx-\frac{6}{25} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}-\frac{x \sqrt{2+3 x^2+x^4}}{84 \left (7+5 x^2\right )}+\frac{\int \frac{62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{2100}-\frac{1}{700} \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{1}{50} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{1}{20} \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{11 x \sqrt{2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac{\left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{50 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{\int \frac{2838+2310 x^2+975 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{58800}-\frac{\int \frac{-175-125 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{52500}+\frac{13 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{2100}-\frac{\left (\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}{20 \sqrt{2+3 x^2+x^4}}\\ &=\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{11 x \sqrt{2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac{\left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{50 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{\left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{70 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{\int \frac{-4725-4875 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{1470000}+\frac{1}{420} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{13 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{4200}+\frac{1}{300} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{13 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{1680}-\frac{101 \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3920}\\ &=\frac{x \left (2+x^2\right )}{420 \sqrt{2+3 x^2+x^4}}+\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{11 x \sqrt{2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}-\frac{\left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{210 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{37 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{1400 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{\left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{70 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{9 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{2800}-\frac{13 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{3920}-\frac{101 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{7840}+\frac{101 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{3136}-\frac{\left (13 \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}{1680 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{11 x \left (2+x^2\right )}{11760 \sqrt{2+3 x^2+x^4}}+\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{11 x \sqrt{2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac{11 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5880 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{81 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{7840 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{97 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{5880 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{\left (101 \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}{3136 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{11 x \left (2+x^2\right )}{11760 \sqrt{2+3 x^2+x^4}}+\frac{x \sqrt{2+3 x^2+x^4}}{28 \left (7+5 x^2\right )^2}+\frac{11 x \sqrt{2+3 x^2+x^4}}{2352 \left (7+5 x^2\right )}+\frac{11 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5880 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{81 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{7840 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{1201 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{164640 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.341789, size = 174, normalized size = 0.73 \[ \frac{-434 i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+\frac{1925 x \left (x^4+3 x^2+2\right )}{5 x^2+7}+\frac{14700 x \left (x^4+3 x^2+2\right )}{\left (5 x^2+7\right )^2}+385 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-1201 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )}{411600 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.018, size = 186, normalized size = 0.8 \begin{align*}{\frac{x}{28\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{11\,x}{11760\,{x}^{2}+16464}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{31\,i}{58800}}\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{11\,i}{23520}}\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{1201\,i}{411600}}\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{\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^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, 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{\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{\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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