Optimal. Leaf size=142 \[ \frac{97 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{25}{3} \sqrt{x^4+3 x^2+2} x+\frac{20 \left (x^2+2\right ) x}{\sqrt{x^4+3 x^2+2}}-\frac{20 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.0543072, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1206, 1189, 1099, 1135} \[ \frac{25}{3} \sqrt{x^4+3 x^2+2} x+\frac{20 \left (x^2+2\right ) x}{\sqrt{x^4+3 x^2+2}}+\frac{97 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{20 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \frac{\left (7+5 x^2\right )^2}{\sqrt{2+3 x^2+x^4}} \, dx &=\frac{25}{3} x \sqrt{2+3 x^2+x^4}+\frac{1}{3} \int \frac{97+60 x^2}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{25}{3} x \sqrt{2+3 x^2+x^4}+20 \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{97}{3} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{20 x \left (2+x^2\right )}{\sqrt{2+3 x^2+x^4}}+\frac{25}{3} x \sqrt{2+3 x^2+x^4}-\frac{20 \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{97 \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{2} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0879792, size = 104, normalized size = 0.73 \[ \frac{-37 i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+25 x \left (x^4+3 x^2+2\right )-60 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )}{3 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 121, normalized size = 0.9 \begin{align*}{\frac{25\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{97\,i}{6}}\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}}}}+{10\,i\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 \frac{{\left (5 \, x^{2} + 7\right )}^{2}}{\sqrt{x^{4} + 3 \, x^{2} + 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{25 \, x^{4} + 70 \, x^{2} + 49}{\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 \frac{\left (5 x^{2} + 7\right )^{2}}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, 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{{\left (5 \, x^{2} + 7\right )}^{2}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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