Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}-b x^2}}\right )}{\sqrt{b} n} \]
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Rubi [A] time = 0.043375, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}-b x^2}}\right )}{\sqrt{b} n} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[x^(2 - n)*(a - b*x^n)],x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{- n + 2} \left (a - b x^{n}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**(2-n)*(a-b*x**n))**(1/2),x)
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Mathematica [A] time = 0.120053, size = 76, normalized size = 2. \[ \frac{2 x^{1-\frac{n}{2}} \sqrt{a-b x^n} \tan ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a-b x^n}}\right )}{\sqrt{b} n \sqrt{x^2 \left (a x^{-n}-b\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[x^(2 - n)*(a - b*x^n)],x]
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Maple [F] time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{{x}^{2-n} \left ( a-b{x}^{n} \right ) }}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^(2-n)*(a-b*x^n))^(1/2),x)
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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^(-n + 2)),x, algorithm="maxima")
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Fricas [A] time = 0.251685, size = 1, normalized size = 0.03 \[ \left [-\frac{\sqrt{-b} \log \left (-\frac{2 \, \sqrt{-b} b x x^{n} - a \sqrt{-b} x + 2 \, b x^{n} \sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}{x}\right )}{b n}, \frac{2 \, \arctan \left (\frac{\sqrt{b} x}{\sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}\right )}{\sqrt{b} n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(b*x^n - a)*x^(-n + 2)),x, algorithm="fricas")
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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/(x**(2-n)*(a-b*x**n))**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (b x^{n} - a\right )} x^{-n + 2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(b*x^n - a)*x^(-n + 2)),x, algorithm="giac")
[Out]