Optimal. Leaf size=67 \[ -\frac{375}{32} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{20} \left (5 x^2+4\right ) \left (x^4+5\right )^{5/2}-\frac{5}{16} x^2 \left (x^4+5\right )^{3/2}-\frac{75}{32} x^2 \sqrt{x^4+5} \]
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Rubi [A] time = 0.10457, antiderivative size = 67, 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 \[ -\frac{375}{32} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{20} \left (5 x^2+4\right ) \left (x^4+5\right )^{5/2}-\frac{5}{16} x^2 \left (x^4+5\right )^{3/2}-\frac{75}{32} x^2 \sqrt{x^4+5} \]
Antiderivative was successfully verified.
[In] Int[x^3*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.05241, size = 61, normalized size = 0.91 \[ - \frac{5 x^{2} \left (x^{4} + 5\right )^{\frac{3}{2}}}{16} - \frac{75 x^{2} \sqrt{x^{4} + 5}}{32} + \frac{\left (15 x^{2} + 12\right ) \left (x^{4} + 5\right )^{\frac{5}{2}}}{60} - \frac{375 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(3*x**2+2)*(x**4+5)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0484758, size = 54, normalized size = 0.81 \[ \frac{1}{160} \left (\sqrt{x^4+5} \left (40 x^{10}+32 x^8+350 x^6+320 x^4+375 x^2+800\right )-1875 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.011, size = 58, normalized size = 0.9 \[{\frac{1}{5} \left ({x}^{4}+5 \right ) ^{{\frac{5}{2}}}}+{\frac{{x}^{10}}{4}\sqrt{{x}^{4}+5}}+{\frac{35\,{x}^{6}}{16}\sqrt{{x}^{4}+5}}+{\frac{75\,{x}^{2}}{32}\sqrt{{x}^{4}+5}}-{\frac{375}{32}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(3*x^2+2)*(x^4+5)^(3/2),x)
[Out]
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Maxima [A] time = 0.781443, size = 159, normalized size = 2.37 \[ \frac{1}{5} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{125 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{8 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{5}{2}}}{x^{10}}\right )}}{32 \,{\left (\frac{3 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{3 \,{\left (x^{4} + 5\right )}^{2}}{x^{8}} + \frac{{\left (x^{4} + 5\right )}^{3}}{x^{12}} - 1\right )}} - \frac{375}{64} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{375}{64} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267292, size = 312, normalized size = 4.66 \[ -\frac{1280 \, x^{24} + 1024 \, x^{22} + 24000 \, x^{20} + 20480 \, x^{18} + 162000 \, x^{16} + 158400 \, x^{14} + 482500 \, x^{12} + 584000 \, x^{10} + 618750 \, x^{8} + 1000000 \, x^{6} + 281250 \, x^{4} + 600000 \, x^{2} - 1875 \,{\left (32 \, x^{12} + 240 \, x^{8} + 450 \, x^{4} - 2 \,{\left (16 \, x^{10} + 80 \, x^{6} + 75 \, x^{2}\right )} \sqrt{x^{4} + 5} + 125\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (1280 \, x^{22} + 1024 \, x^{20} + 20800 \, x^{18} + 17920 \, x^{16} + 114000 \, x^{14} + 116800 \, x^{12} + 252500 \, x^{10} + 340000 \, x^{8} + 212500 \, x^{6} + 400000 \, x^{4} + 46875 \, x^{2} + 100000\right )} \sqrt{x^{4} + 5}}{160 \,{\left (32 \, x^{12} + 240 \, x^{8} + 450 \, x^{4} - 2 \,{\left (16 \, x^{10} + 80 \, x^{6} + 75 \, x^{2}\right )} \sqrt{x^{4} + 5} + 125\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 27.1352, size = 124, normalized size = 1.85 \[ \frac{x^{14}}{4 \sqrt{x^{4} + 5}} + \frac{55 x^{10}}{16 \sqrt{x^{4} + 5}} + \frac{x^{8} \sqrt{x^{4} + 5}}{5} + \frac{425 x^{6}}{32 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{3} + \frac{375 x^{2}}{32 \sqrt{x^{4} + 5}} + \frac{5 \left (x^{4} + 5\right )^{\frac{3}{2}}}{3} - \frac{10 \sqrt{x^{4} + 5}}{3} - \frac{375 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(3*x**2+2)*(x**4+5)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265655, size = 80, normalized size = 1.19 \[ \frac{1}{160} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x^{2} + 4\right )} x^{2} + 175\right )} x^{2} + 160\right )} x^{2} + 375\right )} x^{2} + 800\right )} + \frac{375}{32} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x^3,x, algorithm="giac")
[Out]