Optimal. Leaf size=189 \[ -\frac{\left (2-9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{30 \sqrt [4]{5} \sqrt{x^4+5}}+\frac{3 \sqrt{x^4+5} x}{5 \left (x^2+\sqrt{5}\right )}-\frac{3 \sqrt{x^4+5}}{5 x}-\frac{2 \sqrt{x^4+5}}{15 x^3}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5^{3/4} \sqrt{x^4+5}} \]
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Rubi [A] time = 0.0860867, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1282, 1198, 220, 1196} \[ \frac{3 \sqrt{x^4+5} x}{5 \left (x^2+\sqrt{5}\right )}-\frac{3 \sqrt{x^4+5}}{5 x}-\frac{2 \sqrt{x^4+5}}{15 x^3}-\frac{\left (2-9 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{30 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{5^{3/4} \sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Rule 1282
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{2+3 x^2}{x^4 \sqrt{5+x^4}} \, dx &=-\frac{2 \sqrt{5+x^4}}{15 x^3}-\frac{1}{15} \int \frac{-45+2 x^2}{x^2 \sqrt{5+x^4}} \, dx\\ &=-\frac{2 \sqrt{5+x^4}}{15 x^3}-\frac{3 \sqrt{5+x^4}}{5 x}+\frac{1}{75} \int \frac{-10+45 x^2}{\sqrt{5+x^4}} \, dx\\ &=-\frac{2 \sqrt{5+x^4}}{15 x^3}-\frac{3 \sqrt{5+x^4}}{5 x}-\frac{3 \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx}{\sqrt{5}}+\frac{1}{15} \left (-2+9 \sqrt{5}\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=-\frac{2 \sqrt{5+x^4}}{15 x^3}-\frac{3 \sqrt{5+x^4}}{5 x}+\frac{3 x \sqrt{5+x^4}}{5 \left (\sqrt{5}+x^2\right )}-\frac{3 \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 )}{5^{3/4} \sqrt{5+x^4}}-\frac{\left (2-9 \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 )}{30 \sqrt [4]{5} \sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0255959, size = 54, normalized size = 0.29 \[ -\frac{9 x^2 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{x^4}{5}\right )+2 \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};-\frac{x^4}{5}\right )}{3 \sqrt{5} x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 170, normalized size = 0.9 \begin{align*} -{\frac{3}{5\,x}\sqrt{{x}^{4}+5}}+{\frac{{\frac{3\,i}{25}}}{\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}}}}-{\frac{2}{15\,{x}^{3}}\sqrt{{x}^{4}+5}}-{\frac{2\,\sqrt{5}}{375\,\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}}}} \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{3 \, x^{2} + 2}{\sqrt{x^{4} + 5} x^{4}}\,{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^{8} + 5 \, x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.89739, size = 80, normalized size = 0.42 \begin{align*} \frac{3 \sqrt{5} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{20 x \Gamma \left (\frac{3}{4}\right )} + \frac{\sqrt{5} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{10 x^{3} \Gamma \left (\frac{1}{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{3 \, x^{2} + 2}{\sqrt{x^{4} + 5} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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