Optimal. Leaf size=197 \[ \frac{10 \sqrt [4]{5} \left (7+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right ),\frac{1}{2}\right )}{7 \sqrt{x^4+5}}+\frac{1}{21} x \left (7 x^2+6\right ) \left (x^4+5\right )^{3/2}+\frac{2}{7} x \left (7 x^2+10\right ) \sqrt{x^4+5}+\frac{20 x \sqrt{x^4+5}}{x^2+\sqrt{5}}-\frac{20 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
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Rubi [A] time = 0.0817294, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1177, 1198, 220, 1196} \[ \frac{1}{21} x \left (7 x^2+6\right ) \left (x^4+5\right )^{3/2}+\frac{2}{7} x \left (7 x^2+10\right ) \sqrt{x^4+5}+\frac{20 x \sqrt{x^4+5}}{x^2+\sqrt{5}}+\frac{10 \sqrt [4]{5} \left (7+2 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{7 \sqrt{x^4+5}}-\frac{20 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{\sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Rule 1177
Rule 1198
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac{1}{21} x \left (6+7 x^2\right ) \left (5+x^4\right )^{3/2}+\frac{1}{21} \int \left (180+210 x^2\right ) \sqrt{5+x^4} \, dx\\ &=\frac{2}{7} x \left (10+7 x^2\right ) \sqrt{5+x^4}+\frac{1}{21} x \left (6+7 x^2\right ) \left (5+x^4\right )^{3/2}+\frac{1}{315} \int \frac{9000+6300 x^2}{\sqrt{5+x^4}} \, dx\\ &=\frac{2}{7} x \left (10+7 x^2\right ) \sqrt{5+x^4}+\frac{1}{21} x \left (6+7 x^2\right ) \left (5+x^4\right )^{3/2}-\left (20 \sqrt{5}\right ) \int \frac{1-\frac{x^2}{\sqrt{5}}}{\sqrt{5+x^4}} \, dx+\frac{1}{7} \left (20 \left (10+7 \sqrt{5}\right )\right ) \int \frac{1}{\sqrt{5+x^4}} \, dx\\ &=\frac{20 x \sqrt{5+x^4}}{\sqrt{5}+x^2}+\frac{2}{7} x \left (10+7 x^2\right ) \sqrt{5+x^4}+\frac{1}{21} x \left (6+7 x^2\right ) \left (5+x^4\right )^{3/2}-\frac{20 \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{10 \sqrt [4]{5} \left (7+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 )}{7 \sqrt{5+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0129397, size = 49, normalized size = 0.25 \[ 5 \sqrt{5} x \left (x^2 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};-\frac{x^4}{5}\right )+2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-\frac{x^4}{5}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 192, normalized size = 1. \begin{align*}{\frac{{x}^{7}}{3}\sqrt{{x}^{4}+5}}+{\frac{11\,{x}^{3}}{3}\sqrt{{x}^{4}+5}}+{\frac{4\,i}{\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}^{5}}{7}\sqrt{{x}^{4}+5}}+{\frac{30\,x}{7}\sqrt{{x}^{4}+5}}+{\frac{8\,\sqrt{5}}{7\,\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{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}\,{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 ({\left (3 \, x^{6} + 2 \, x^{4} + 15 \, x^{2} + 10\right )} \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 = 2.92852, size = 158, normalized size = 0.8 \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 )}}{4 \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 )}}{2 \Gamma \left (\frac{9}{4}\right )} + \frac{15 \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{5 \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 )} \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{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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