Optimal. Leaf size=78 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac{5}{16} \left (9 x^2+16\right ) \sqrt{x^4+5} \]
[Out]
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Rubi [A] time = 0.197324, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac{5}{16} \left (9 x^2+16\right ) \sqrt{x^4+5} \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x^2)*(5 + x^4)^(3/2))/x,x]
[Out]
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Rubi in Sympy [A] time = 16.1055, size = 71, normalized size = 0.91 \[ \frac{\left (9 x^{2} + 8\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{24} + \frac{\left (45 x^{2} + 80\right ) \sqrt{x^{4} + 5}}{16} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} - 5 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3*x**2+2)*(x**4+5)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.100323, size = 68, normalized size = 0.87 \[ -5 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{48} \sqrt{x^4+5} \left (18 x^6+16 x^4+225 x^2+320\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x^2)*(5 + x^4)^(3/2))/x,x]
[Out]
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Maple [A] time = 0.021, size = 75, normalized size = 1. \[{\frac{{x}^{4}}{3}\sqrt{{x}^{4}+5}}+{\frac{20}{3}\sqrt{{x}^{4}+5}}-5\,\sqrt{5}{\it Artanh} \left ({\frac{\sqrt{5}}{\sqrt{{x}^{4}+5}}} \right ) +{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5}}+{\frac{75\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}+{\frac{225}{16}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3*x^2+2)*(x^4+5)^(3/2)/x,x)
[Out]
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Maxima [A] time = 0.785344, size = 186, normalized size = 2.38 \[ \frac{1}{3} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} + \frac{5}{2} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + 5 \, \sqrt{x^{4} + 5} + \frac{75 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{5 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{16 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289188, size = 354, normalized size = 4.54 \[ -\frac{144 \, x^{16} + 128 \, x^{14} + 2880 \, x^{12} + 3520 \, x^{10} + 15300 \, x^{8} + 20800 \, x^{6} + 22500 \, x^{4} + 32000 \, x^{2} + 675 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 240 \,{\left (4 \, \sqrt{5}{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (8 \, x^{8} + 40 \, x^{4} + 25\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (144 \, x^{14} + 128 \, x^{12} + 2520 \, x^{10} + 3200 \, x^{8} + 9450 \, x^{6} + 13200 \, x^{4} + 5625 \, x^{2} + 8000\right )} \sqrt{x^{4} + 5}}{48 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 22.7554, size = 114, normalized size = 1.46 \[ \frac{3 x^{10}}{8 \sqrt{x^{4} + 5}} + \frac{105 x^{6}}{16 \sqrt{x^{4} + 5}} + \frac{375 x^{2}}{16 \sqrt{x^{4} + 5}} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{3} + 5 \sqrt{x^{4} + 5} + \frac{5 \sqrt{5} \log{\left (x^{4} \right )}}{2} - 5 \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x**2+2)*(x**4+5)**(3/2)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)/x,x, algorithm="giac")
[Out]