Optimal. Leaf size=39 \[ \frac{x \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2}+\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]
[Out]
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Rubi [A] time = 0.0321292, antiderivative size = 48, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 3.75639, size = 41, normalized size = 1.05 \[ \frac{x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0309481, size = 49, normalized size = 1.26 \[ \frac{x \sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{\sqrt{a+b{x}^{n}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^n + a),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(1/sqrt(b*x^n + a),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/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*x^n + a),x, algorithm="giac")
[Out]