Optimal. Leaf size=185 \[ -\frac{\sqrt [4]{5} \left (27+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{6 \sqrt{x^4+5}}+\frac{3}{5} \sqrt{x^4+5} x^3-\frac{9 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{2}{3} \sqrt{x^4+5} x+\frac{9 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
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Rubi [A] time = 0.0850326, antiderivative size = 185, 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 = {1280, 1198, 220, 1196} \[ \frac{3}{5} \sqrt{x^4+5} x^3-\frac{9 \sqrt{x^4+5} x}{x^2+\sqrt{5}}+\frac{2}{3} \sqrt{x^4+5} x-\frac{\sqrt [4]{5} \left (27+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{6 \sqrt{x^4+5}}+\frac{9 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Rule 1280
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^4 \left (2+3 x^2\right )}{\sqrt{5+x^4}} \, dx &=\frac{3}{5} x^3 \sqrt{5+x^4}-\frac{1}{5} \int \frac{x^2 \left (45-10 x^2\right )}{\sqrt{5+x^4}} \, dx\\ &=\frac{2}{3} x \sqrt{5+x^4}+\frac{3}{5} x^3 \sqrt{5+x^4}+\frac{1}{15} \int \frac{-50-135 x^2}{\sqrt{5+x^4}} \, dx\\ &=\frac{2}{3} x \sqrt{5+x^4}+\frac{3}{5} x^3 \sqrt{5+x^4}+\left (9 \sqrt{5}\right ) \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx-\frac{1}{3} \left (10+27 \sqrt{5}\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=\frac{2}{3} x \sqrt{5+x^4}+\frac{3}{5} x^3 \sqrt{5+x^4}-\frac{9 x \sqrt{5+x^4}}{\sqrt{5}+x^2}+\frac{9 \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 (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 )}{6 \sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0325322, size = 74, normalized size = 0.4 \[ \frac{1}{15} x \left (-9 \sqrt{5} x^2 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right )-10 \sqrt{5} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{x^4}{5}\right )+\left (9 x^2+10\right ) \sqrt{x^4+5}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.017, size = 168, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{3}}{5}\sqrt{{x}^{4}+5}}-{\frac{{\frac{9\,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}}}}+{\frac{2\,x}{3}\sqrt{{x}^{4}+5}}-{\frac{2\,\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}}}} \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 (3 \, x^{2} + 2\right )} x^{4}}{\sqrt{x^{4} + 5}}\,{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{3 \, x^{6} + 2 \, x^{4}}{\sqrt{x^{4} + 5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.1053, size = 75, normalized size = 0.41 \begin{align*} \frac{3 \sqrt{5} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{20 \Gamma \left (\frac{11}{4}\right )} + \frac{\sqrt{5} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{10 \Gamma \left (\frac{9}{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 (3 \, x^{2} + 2\right )} x^{4}}{\sqrt{x^{4} + 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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