Optimal. Leaf size=235 \[ -\frac{463 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{625 \sqrt{x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac{31 \left (x^2+2\right ) x}{56 \sqrt{x^4+3 x^2+2}}+\frac{\left (11 x^2+20\right ) x}{36 \sqrt{x^4+3 x^2+2}}+\frac{31 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{375 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.429144, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {1228, 1178, 1189, 1099, 1135, 1223, 1716, 1214, 1456, 539} \[ \frac{625 \sqrt{x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac{31 \left (x^2+2\right ) x}{56 \sqrt{x^4+3 x^2+2}}+\frac{\left (11 x^2+20\right ) x}{36 \sqrt{x^4+3 x^2+2}}-\frac{463 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{31 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{375 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1228
Rule 1178
Rule 1189
Rule 1099
Rule 1135
Rule 1223
Rule 1716
Rule 1214
Rule 1456
Rule 539
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2}} \, dx &=\int \left (\frac{14+5 x^2}{36 \left (2+3 x^2+x^4\right )^{3/2}}-\frac{25}{6 \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}}-\frac{25}{36 \left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}}\right ) \, dx\\ &=\frac{1}{36} \int \frac{14+5 x^2}{\left (2+3 x^2+x^4\right )^{3/2}} \, dx-\frac{25}{36} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx-\frac{25}{6} \int \frac{1}{\left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (20+11 x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}-\frac{1}{72} \int \frac{26+22 x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{25}{504} \int \frac{62+70 x^2+25 x^4}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx-\frac{25}{72} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{125}{144} \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{x \left (20+11 x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}-\frac{25 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{72 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{1}{504} \int \frac{-175-125 x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{11}{36} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{13}{36} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{325}{504} \int \frac{1}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx+\frac{\left (125 \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}{144 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{11 x \left (2+x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{x \left (20+11 x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{504 \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 )}{18 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{17 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{24 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{125 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{504 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}-\frac{125}{504} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx-\frac{325 \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx}{1008}-\frac{25}{72} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{1625 \int \frac{2+2 x^2}{\left (7+5 x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx}{2016}\\ &=-\frac{31 x \left (2+x^2\right )}{56 \sqrt{2+3 x^2+x^4}}+\frac{x \left (20+11 x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}+\frac{31 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{463 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{125 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{504 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}+\frac{\left (1625 \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}{2016 \sqrt{2+3 x^2+x^4}}\\ &=-\frac{31 x \left (2+x^2\right )}{56 \sqrt{2+3 x^2+x^4}}+\frac{x \left (20+11 x^2\right )}{36 \sqrt{2+3 x^2+x^4}}+\frac{625 x \sqrt{2+3 x^2+x^4}}{504 \left (7+5 x^2\right )}+\frac{31 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{2+3 x^2+x^4}}-\frac{463 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{2+3 x^2+x^4}}+\frac{375 \left (2+x^2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.274219, size = 208, normalized size = 0.89 \[ \frac{182 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+3255 x^5+10157 x^3+651 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1125 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1575 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+7490 x}{1176 \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 185, normalized size = 0.8 \begin{align*}{\frac{625\,x}{2520\,{x}^{2}+3528}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-2\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}} \left ( -{\frac{11\,{x}^{3}}{72}}-{\frac{5\,x}{18}} \right ) }+{{\frac{13\,i}{168}}\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{31\,i}{112}}\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{75\,i}{392}}\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}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{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{\sqrt{x^{4} + 3 \, x^{2} + 2}}{25 \, x^{12} + 220 \, x^{10} + 794 \, x^{8} + 1504 \, x^{6} + 1577 \, x^{4} + 868 \, x^{2} + 196}, 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}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{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{1}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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