Optimal. Leaf size=28 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
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Rubi [A] time = 0.0529034, antiderivative size = 28, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
[In] Int[1/(x*Sqrt[a + b*x^n]),x]
[Out]
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Rubi in Sympy [A] time = 5.68451, size = 26, normalized size = 0.93 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{\sqrt{a} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.0214613, size = 28, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*Sqrt[a + b*x^n]),x]
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Maple [A] time = 0., size = 23, normalized size = 0.8 \[ -2\,{\frac{1}{n\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(a+b*x^n)^(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^n + a)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.240346, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right )}{\sqrt{a} n}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right )}{\sqrt{-a} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^n + a)*x),x, algorithm="fricas")
[Out]
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Sympy [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/x/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x^n + a)*x),x, algorithm="giac")
[Out]