Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0651626, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/Sqrt[a + b*x^5],x]
[Out]
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Rubi in Sympy [A] time = 6.80623, size = 29, normalized size = 0.91 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a + b x^{5}}} \right )}}{5 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(b*x**5+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0604771, size = 32, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/Sqrt[a + b*x^5],x]
[Out]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(b*x^5+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x^5 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.690606, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-4 \,{\left (2 \, b^{2} x^{7} + a b x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{x} -{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b}\right )}{10 \, \sqrt{b}}, \frac{\arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right )}{5 \, \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x^5 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.80554, size = 24, normalized size = 0.75 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a}} \right )}}{5 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(b*x**5+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23552, size = 55, normalized size = 1.72 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{b + \frac{a}{x^{5}}}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} + \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(b*x^5 + a),x, algorithm="giac")
[Out]