Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}+b x^2}}\right )}{5 \sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0341457, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^3}+b x^2}}\right )}{5 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[(a + b*x^5)/x^3],x]
[Out]
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Rubi in Sympy [A] time = 2.51158, size = 29, normalized size = 0.91 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{\frac{a}{x^{3}} + b x^{2}}} \right )}}{5 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((b*x**5+a)/x**3)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0378524, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\frac{a+b x^5}{x^3}}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[1/Sqrt[(a + b*x^5)/x^3],x]
[Out]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{\frac{b{x}^{5}+a}{{x}^{3}}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((b*x^5+a)/x^3)^(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(1/sqrt((b*x^5 + a)/x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.701436, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b} - 4 \,{\left (2 \, b^{2} x^{9} + a b x^{4}\right )} \sqrt{\frac{b x^{5} + a}{x^{3}}}\right )}{10 \, \sqrt{b}}, -\frac{\sqrt{-b} \arctan \left (\frac{2 \, \sqrt{-b} x^{4} \sqrt{\frac{b x^{5} + a}{x^{3}}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt((b*x^5 + a)/x^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x**5+a)/x**3)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt((b*x^5 + a)/x^3),x, algorithm="giac")
[Out]