Optimal. Leaf size=49 \[ -\frac{\left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{3}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x} \]
[Out]
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Rubi [A] time = 0.0673001, antiderivative size = 58, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{x \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^n]/x^2,x]
[Out]
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Rubi in Sympy [A] time = 6.71517, size = 44, normalized size = 0.9 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{x \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0712452, size = 73, normalized size = 1.49 \[ \frac{2 \left (a+b x^n\right )-a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{(n-2) x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^n]/x^2,x]
[Out]
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Maple [F] time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt{a+b{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(1/2)/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x^2,x, algorithm="maxima")
[Out]
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Fricas [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(sqrt(b*x^n + a)/x^2,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((a+b*x**n)**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x^2,x, algorithm="giac")
[Out]