Optimal. Leaf size=60 \[ \frac{3}{10} \left (x^4+5\right )^{5/2}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5} \]
[Out]
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Rubi [A] time = 0.0720775, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3}{10} \left (x^4+5\right )^{5/2}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5} \]
Antiderivative was successfully verified.
[In] Int[x*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.308, size = 54, normalized size = 0.9 \[ \frac{x^{2} \left (x^{4} + 5\right )^{\frac{3}{2}}}{4} + \frac{15 x^{2} \sqrt{x^{4} + 5}}{8} + \frac{3 \left (x^{4} + 5\right )^{\frac{5}{2}}}{10} + \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(3*x**2+2)*(x**4+5)**(3/2),x)
[Out]
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Mathematica [A] time = 0.031343, size = 56, normalized size = 0.93 \[ \frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{2} \sqrt{x^4+5} \left (\frac{3 x^8}{5}+\frac{x^6}{2}+6 x^4+\frac{25 x^2}{4}+15\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(2 + 3*x^2)*(5 + x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.019, size = 46, normalized size = 0.8 \[{\frac{{x}^{6}}{4}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{2}}{8}\sqrt{{x}^{4}+5}}+{\frac{75}{8}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{3}{10} \left ({x}^{4}+5 \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(3*x^2+2)*(x^4+5)^(3/2),x)
[Out]
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Maxima [A] time = 0.776508, size = 128, normalized size = 2.13 \[ \frac{3}{10} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} + \frac{25 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{5 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{8 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{75}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{75}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]
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 = 0.28594, size = 271, normalized size = 4.52 \[ -\frac{192 \, x^{20} + 160 \, x^{18} + 3600 \, x^{16} + 3400 \, x^{14} + 25500 \, x^{12} + 20750 \, x^{10} + 82500 \, x^{8} + 41875 \, x^{6} + 112500 \, x^{4} + 15625 \, x^{2} + 375 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (192 \, x^{18} + 160 \, x^{16} + 3120 \, x^{14} + 3000 \, x^{12} + 18300 \, x^{10} + 13750 \, x^{8} + 45000 \, x^{6} + 15625 \, x^{4} + 37500 \, x^{2}\right )} \sqrt{x^{4} + 5} + 37500}{40 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.1906, size = 109, normalized size = 1.82 \[ \frac{x^{10}}{4 \sqrt{x^{4} + 5}} + \frac{3 x^{8} \sqrt{x^{4} + 5}}{10} + \frac{35 x^{6}}{8 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{125 x^{2}}{8 \sqrt{x^{4} + 5}} + \frac{5 \left (x^{4} + 5\right )^{\frac{3}{2}}}{2} - 5 \sqrt{x^{4} + 5} + \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(3*x**2+2)*(x**4+5)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.267085, size = 70, normalized size = 1.17 \[ \frac{1}{40} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (6 \, x^{2} + 5\right )} x^{2} + 60\right )} x^{2} + 125\right )} x^{2} + 300\right )} - \frac{75}{8} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x,x, algorithm="giac")
[Out]