Optimal. Leaf size=32 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}} \]
[Out]
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Rubi [A] time = 0.0266031, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^5}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a*x^2 + b*x^5],x]
[Out]
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Rubi in Sympy [A] time = 2.32906, size = 31, normalized size = 0.97 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{5}}} \right )}}{3 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**5+a*x**2)**(1/2),x)
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Mathematica [A] time = 0.0613647, size = 54, normalized size = 1.69 \[ -\frac{2 \sqrt{x^2 \left (a+b x^3\right )} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{3 a x \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a*x^2 + b*x^5],x]
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Maple [A] time = 0.008, size = 43, normalized size = 1.3 \[ -{\frac{2\,x}{3}\sqrt{b{x}^{3}+a}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^5+a*x^2)^(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^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.223321, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{{\left (b x^{4} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{5} + a x^{2}} a}{x^{4}}\right )}{3 \, \sqrt{a}}, -\frac{2 \, \sqrt{-a} \arctan \left (\frac{a x}{\sqrt{b x^{5} + a x^{2}} \sqrt{-a}}\right )}{3 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^5 + a*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x^{2} + b x^{5}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**5+a*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222081, size = 63, normalized size = 1.97 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{3 \, \sqrt{-a}} + \frac{2 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}{\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^5 + a*x^2),x, algorithm="giac")
[Out]