Optimal. Leaf size=201 \[ \frac{2 \sqrt [4]{5} \left (27+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{3 \sqrt{x^4+5}}-\frac{\left (10-9 x^2\right ) \left (x^4+5\right )^{3/2}}{15 x^3}-\frac{2 \left (27-2 x^2\right ) \sqrt{x^4+5}}{3 x}+\frac{36 x \sqrt{x^4+5}}{x^2+\sqrt{5}}-\frac{36 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
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Rubi [A] time = 0.0859568, antiderivative size = 201, 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 = {1272, 1198, 220, 1196} \[ -\frac{\left (10-9 x^2\right ) \left (x^4+5\right )^{3/2}}{15 x^3}-\frac{2 \left (27-2 x^2\right ) \sqrt{x^4+5}}{3 x}+\frac{36 x \sqrt{x^4+5}}{x^2+\sqrt{5}}+\frac{2 \sqrt [4]{5} \left (27+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt{x^4+5}}-\frac{36 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
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 ) \left (5+x^4\right )^{3/2}}{x^4} \, dx &=-\frac{\left (10-9 x^2\right ) \left (5+x^4\right )^{3/2}}{15 x^3}-\frac{2}{5} \int \frac{\left (-45-10 x^2\right ) \sqrt{5+x^4}}{x^2} \, dx\\ &=-\frac{2 \left (27-2 x^2\right ) \sqrt{5+x^4}}{3 x}-\frac{\left (10-9 x^2\right ) \left (5+x^4\right )^{3/2}}{15 x^3}+\frac{4}{15} \int \frac{50+135 x^2}{\sqrt{5+x^4}} \, dx\\ &=-\frac{2 \left (27-2 x^2\right ) \sqrt{5+x^4}}{3 x}-\frac{\left (10-9 x^2\right ) \left (5+x^4\right )^{3/2}}{15 x^3}-\left (36 \sqrt{5}\right ) \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx+\frac{1}{3} \left (4 \left (10+27 \sqrt{5}\right )\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=-\frac{2 \left (27-2 x^2\right ) \sqrt{5+x^4}}{3 x}+\frac{36 x \sqrt{5+x^4}}{\sqrt{5}+x^2}-\frac{\left (10-9 x^2\right ) \left (5+x^4\right )^{3/2}}{15 x^3}-\frac{36 \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{2 \sqrt [4]{5} \left (27+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 )}{3 \sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0268571, size = 54, normalized size = 0.27 \[ -\frac{5 \sqrt{5} \left (9 x^2 \, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};-\frac{x^4}{5}\right )+2 \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};-\frac{x^4}{5}\right )\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 192, normalized size = 1. \begin{align*} -{\frac{10}{3\,{x}^{3}}\sqrt{{x}^{4}+5}}+{\frac{2\,x}{3}\sqrt{{x}^{4}+5}}+{\frac{8\,\sqrt{5}}{15\,\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}}}}-15\,{\frac{\sqrt{{x}^{4}+5}}{x}}+{\frac{3\,{x}^{3}}{5}\sqrt{{x}^{4}+5}}+{\frac{{\frac{36\,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{{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{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{{\left (3 \, x^{6} + 2 \, x^{4} + 15 \, x^{2} + 10\right )} \sqrt{x^{4} + 5}}{x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.77657, size = 163, normalized size = 0.81 \begin{align*} \frac{3 \sqrt{5} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} + \frac{\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 )}}{2 \Gamma \left (\frac{5}{4}\right )} + \frac{15 \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 )}}{4 x \Gamma \left (\frac{3}{4}\right )} + \frac{5 \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 )}}{2 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{{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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