Optimal. Leaf size=168 \[ \frac{472 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}+\frac{25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{21} x \left (114 x^2+407\right ) \sqrt{x^4+3 x^2+2}+\frac{31 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}-\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.0677099, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1206, 1176, 1189, 1099, 1135} \[ \frac{25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{21} x \left (114 x^2+407\right ) \sqrt{x^4+3 x^2+2}+\frac{31 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}+\frac{472 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}-\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1176
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \left (7+5 x^2\right )^2 \sqrt{2+3 x^2+x^4} \, dx &=\frac{25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac{1}{7} \int \left (293+190 x^2\right ) \sqrt{2+3 x^2+x^4} \, dx\\ &=\frac{1}{21} x \left (407+114 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac{1}{105} \int \frac{4720+3255 x^2}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{1}{21} x \left (407+114 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+31 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{944}{21} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{31 x \left (2+x^2\right )}{\sqrt{2+3 x^2+x^4}}+\frac{1}{21} x \left (407+114 x^2\right ) \sqrt{2+3 x^2+x^4}+\frac{25}{7} x \left (2+3 x^2+x^4\right )^{3/2}-\frac{31 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2+3 x^2+x^4}}+\frac{472 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0788284, size = 114, normalized size = 0.68 \[ \frac{-293 i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+75 x^9+564 x^7+1724 x^5+2349 x^3-651 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1114 x}{21 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 155, normalized size = 0.9 \begin{align*}{\frac{25\,{x}^{5}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{113\,{x}^{3}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{557\,x}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{472\,i}{21}}\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}{2}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\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 \sqrt{x^{4} + 3 \, x^{2} + 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 ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, 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 \sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \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 \sqrt{x^{4} + 3 \, x^{2} + 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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