Optimal. Leaf size=180 \[ \frac{\left (2-3 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{20 \sqrt [4]{5} \sqrt{x^4+5}}-\frac{3 \sqrt{x^4+5} x}{10 \left (x^2+\sqrt{5}\right )}+\frac{\left (3 x^2+2\right ) x}{10 \sqrt{x^4+5}}+\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{2\ 5^{3/4} \sqrt{x^4+5}} \]
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Rubi [A] time = 0.0614915, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1179, 1198, 220, 1196} \[ -\frac{3 \sqrt{x^4+5} x}{10 \left (x^2+\sqrt{5}\right )}+\frac{\left (3 x^2+2\right ) x}{10 \sqrt{x^4+5}}+\frac{\left (2-3 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{20 \sqrt [4]{5} \sqrt{x^4+5}}+\frac{3 \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{2\ 5^{3/4} \sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Rule 1179
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{2+3 x^2}{\left (5+x^4\right )^{3/2}} \, dx &=\frac{x \left (2+3 x^2\right )}{10 \sqrt{5+x^4}}-\frac{1}{10} \int \frac{-2+3 x^2}{\sqrt{5+x^4}} \, dx\\ &=\frac{x \left (2+3 x^2\right )}{10 \sqrt{5+x^4}}+\frac{3 \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx}{2 \sqrt{5}}-\frac{1}{10} \left (-2+3 \sqrt{5}\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=\frac{x \left (2+3 x^2\right )}{10 \sqrt{5+x^4}}-\frac{3 x \sqrt{5+x^4}}{10 \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 )}{2\ 5^{3/4} \sqrt{5+x^4}}+\frac{\left (2-3 \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 )}{20 \sqrt [4]{5} \sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0256387, size = 66, normalized size = 0.37 \[ \frac{1}{25} x \left (\sqrt{5} x^2 \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{x^4}{5}\right )+\sqrt{5} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right )+\frac{5}{\sqrt{x^4+5}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 168, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{3}}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{{\frac{3\,i}{50}}}{\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{x}{5}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{\sqrt{5}}{125\,\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}{{\left (x^{4} + 5\right )}^{\frac{3}{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^{8} + 10 \, x^{4} + 25}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.25619, size = 73, normalized size = 0.41 \begin{align*} \frac{3 \sqrt{5} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{100 \Gamma \left (\frac{7}{4}\right )} + \frac{\sqrt{5} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{50 \Gamma \left (\frac{5}{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}{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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