Optimal. Leaf size=171 \[ \frac{\sqrt [4]{5} \left (2+\sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{\sqrt{x^4+5}}+\frac{4 \sqrt{x^4+5} x}{x^2+\sqrt{5}}-\frac{\left (2-x^2\right ) \sqrt{x^4+5}}{x}-\frac{4 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
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Rubi [A] time = 0.0670011, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1272, 1198, 220, 1196} \[ \frac{4 \sqrt{x^4+5} x}{x^2+\sqrt{5}}-\frac{\left (2-x^2\right ) \sqrt{x^4+5}}{x}+\frac{\sqrt [4]{5} \left (2+\sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}}-\frac{4 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Rule 1272
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \sqrt{5+x^4}}{x^2} \, dx &=-\frac{\left (2-x^2\right ) \sqrt{5+x^4}}{x}-\frac{2}{3} \int \frac{-15-6 x^2}{\sqrt{5+x^4}} \, dx\\ &=-\frac{\left (2-x^2\right ) \sqrt{5+x^4}}{x}-\left (4 \sqrt{5}\right ) \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx+\left (2 \left (5+2 \sqrt{5}\right )\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=-\frac{\left (2-x^2\right ) \sqrt{5+x^4}}{x}+\frac{4 x \sqrt{5+x^4}}{\sqrt{5}+x^2}-\frac{4 \sqrt [4]{5} \left (\sqrt{5}+x^2\right ) \sqrt{\frac{5+x^4}{\left (\sqrt{5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{5+x^4}}+\frac{\sqrt [4]{5} \left (2+\sqrt{5}\right ) \left (\sqrt{5}+x^2\right ) \sqrt{\frac{5+x^4}{\left (\sqrt{5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0226249, size = 53, normalized size = 0.31 \[ 3 \sqrt{5} x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right )-\frac{2 \sqrt{5} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 167, normalized size = 1. \begin{align*} x\sqrt{{x}^{4}+5}+{\frac{2\,\sqrt{5}}{5\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}-2\,{\frac{\sqrt{{x}^{4}+5}}{x}}+{\frac{{\frac{4\,i}{5}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \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} + 5}{\left (3 \, x^{2} + 2\right )}}{x^{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} + 5}{\left (3 \, x^{2} + 2\right )}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.83293, size = 78, normalized size = 0.46 \begin{align*} \frac{3 \sqrt{5} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 x \Gamma \left (\frac{3}{4}\right )} \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} + 5}{\left (3 \, x^{2} + 2\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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