Optimal. Leaf size=67 \[ \frac{3}{10} \left (x^4+5\right )^{3/2} x^4-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{8} \sqrt{x^4+5} x^2-\frac{1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2} \]
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Rubi [A] time = 0.144855, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3}{10} \left (x^4+5\right )^{3/2} x^4-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{8} \sqrt{x^4+5} x^2-\frac{1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[x^5*(2 + 3*x^2)*Sqrt[5 + x^4],x]
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Rubi in Sympy [A] time = 11.3758, size = 61, normalized size = 0.91 \[ \frac{3 x^{4} \left (x^{4} + 5\right )^{\frac{3}{2}}}{10} - \frac{5 x^{2} \sqrt{x^{4} + 5}}{8} - \frac{\left (- 30 x^{2} + 120\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{120} - \frac{25 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(3*x**2+2)*(x**4+5)**(1/2),x)
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Mathematica [A] time = 0.0345895, size = 54, normalized size = 0.81 \[ \frac{1}{2} \sqrt{x^4+5} \left (\frac{3 x^8}{5}+\frac{x^6}{2}+x^4+\frac{5 x^2}{4}-10\right )-\frac{25}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^5*(2 + 3*x^2)*Sqrt[5 + x^4],x]
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Maple [A] time = 0.028, size = 53, normalized size = 0.8 \[{\frac{{x}^{2}}{4} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{x}^{2}}{8}\sqrt{{x}^{4}+5}}-{\frac{25}{8}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{3\,{x}^{4}-10}{10} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(3*x^2+2)*(x^4+5)^(1/2),x)
[Out]
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Maxima [A] time = 0.779942, size = 138, normalized size = 2.06 \[ \frac{3}{10} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{5}{2} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - \frac{25 \,{\left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{8 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} - \frac{25}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{25}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x^5,x, algorithm="maxima")
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Fricas [A] time = 0.262343, size = 271, normalized size = 4.04 \[ -\frac{192 \, x^{20} + 160 \, x^{18} + 2000 \, x^{16} + 1800 \, x^{14} + 3500 \, x^{12} + 6750 \, x^{10} - 20000 \, x^{8} + 9375 \, x^{6} - 62500 \, x^{4} + 3125 \, x^{2} - 125 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (192 \, x^{18} + 160 \, x^{16} + 1520 \, x^{14} + 1400 \, x^{12} + 300 \, x^{10} + 3750 \, x^{8} - 17500 \, x^{6} + 3125 \, x^{4} - 25000 \, x^{2}\right )} \sqrt{x^{4} + 5} - 25000}{40 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.4333, size = 97, normalized size = 1.45 \[ \frac{x^{10}}{4 \sqrt{x^{4} + 5}} + \frac{3 x^{8} \sqrt{x^{4} + 5}}{10} + \frac{15 x^{6}}{8 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{25 x^{2}}{8 \sqrt{x^{4} + 5}} - 5 \sqrt{x^{4} + 5} - \frac{25 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(3*x**2+2)*(x**4+5)**(1/2),x)
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GIAC/XCAS [A] time = 0.264544, size = 70, normalized size = 1.04 \[ \frac{1}{40} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (6 \, x^{2} + 5\right )} x^{2} + 10\right )} x^{2} + 25\right )} x^{2} - 200\right )} + \frac{25}{8} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + 5)*(3*x^2 + 2)*x^5,x, algorithm="giac")
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