Optimal. Leaf size=219 \[ \frac{300}{77} \sqrt{x^4+5} x+\frac{40 \sqrt{x^4+5} x}{3 \left (x^2+\sqrt{5}\right )}+\frac{10 \sqrt [4]{5} \left (154-45 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{231 \sqrt{x^4+5}}-\frac{40 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt{x^4+5}}+\frac{1}{99} \left (27 x^2+22\right ) \left (x^4+5\right )^{3/2} x^3+\frac{2}{231} \left (135 x^2+154\right ) \sqrt{x^4+5} x^3 \]
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Rubi [A] time = 0.29666, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{300}{77} \sqrt{x^4+5} x+\frac{40 \sqrt{x^4+5} x}{3 \left (x^2+\sqrt{5}\right )}+\frac{10 \sqrt [4]{5} \left (154-45 \sqrt{5}\right ) \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{231 \sqrt{x^4+5}}-\frac{40 \sqrt [4]{5} \left (x^2+\sqrt{5}\right ) \sqrt{\frac{x^4+5}{\left (x^2+\sqrt{5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt [4]{5}}\right )|\frac{1}{2}\right )}{3 \sqrt{x^4+5}}+\frac{1}{99} \left (27 x^2+22\right ) \left (x^4+5\right )^{3/2} x^3+\frac{2}{231} \left (135 x^2+154\right ) \sqrt{x^4+5} x^3 \]
Antiderivative was successfully verified.
[In] Int[x^2*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
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Rubi in Sympy [A] time = 25.8658, size = 219, normalized size = 1. \[ \frac{x^{3} \left (27 x^{2} + 22\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{99} + \frac{2 x^{3} \left (135 x^{2} + 154\right ) \sqrt{x^{4} + 5}}{231} + \frac{300 x \sqrt{x^{4} + 5}}{77} + \frac{40 x \sqrt{x^{4} + 5}}{3 \left (x^{2} + \sqrt{5}\right )} - \frac{40 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{3 \sqrt{x^{4} + 5}} + \frac{10 \sqrt [4]{5} \sqrt{\frac{x^{4} + 5}{\left (\frac{\sqrt{5} x^{2}}{5} + 1\right )^{2}}} \left (- 135 \sqrt{5} + 462\right ) \left (\frac{\sqrt{5} x^{2}}{5} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{5^{\frac{3}{4}} x}{5} \right )}\middle | \frac{1}{2}\right )}{693 \sqrt{x^{4} + 5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(3*x**2+2)*(x**4+5)**(3/2),x)
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Mathematica [C] time = 0.230519, size = 110, normalized size = 0.5 \[ \frac{1}{693} \left (\frac{x \left (189 x^{12}+154 x^{10}+2700 x^8+2464 x^6+11475 x^4+8470 x^2+13500\right )}{\sqrt{x^4+5}}+60 \sqrt [4]{-5} \left (45 \sqrt{5}+154 i\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )-9240 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac{1}{5}} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
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Maple [C] time = 0.017, size = 204, normalized size = 0.9 \[{\frac{2\,{x}^{7}}{9}\sqrt{{x}^{4}+5}}+{\frac{22\,{x}^{3}}{9}\sqrt{{x}^{4}+5}}+{\frac{{\frac{8\,i}{3}}}{\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) -{\it EllipticE} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,{x}^{9}}{11}\sqrt{{x}^{4}+5}}+{\frac{195\,{x}^{5}}{77}\sqrt{{x}^{4}+5}}+{\frac{300\,x}{77}\sqrt{{x}^{4}+5}}-{\frac{60\,\sqrt{5}}{77\,\sqrt{i\sqrt{5}}}\sqrt{25-5\,i\sqrt{5}{x}^{2}}\sqrt{25+5\,i\sqrt{5}{x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{5}\sqrt{i\sqrt{5}}}{5}},i \right ){\frac{1}{\sqrt{{x}^{4}+5}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(3*x^2+2)*(x^4+5)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^2,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (3 \, x^{8} + 2 \, x^{6} + 15 \, x^{4} + 10 \, x^{2}\right )} \sqrt{x^{4} + 5}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^2,x, algorithm="fricas")
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Sympy [A] time = 8.01579, size = 160, normalized size = 0.73 \[ \frac{3 \sqrt{5} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} + \frac{\sqrt{5} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{11}{4}\right )} + \frac{15 \sqrt{5} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt{5} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(3*x**2+2)*(x**4+5)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^2,x, algorithm="giac")
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