Optimal. Leaf size=61 \[ \frac{x^{4-2 n} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{8}{n}-3\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 a (2-n)} \]
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Rubi [A] time = 0.0795379, antiderivative size = 72, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{4-2 n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-2 \left (1-\frac{2}{n}\right );\frac{4}{n}-1;-\frac{b x^n}{a}\right )}{2 (2-n) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x^(3 - 2*n)/Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 8.14306, size = 54, normalized size = 0.89 \[ \frac{x^{- 2 n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, -2 + \frac{4}{n} \\ - \frac{n - 4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \sqrt{1 + \frac{b x^{n}}{a}} \left (- n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3-2*n)/(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.418328, size = 116, normalized size = 1.9 \[ \frac{x^4 \left (b^2 \left (3 n^2-32 n+64\right ) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )+8 x^{-2 n} \left (a+b x^n\right ) \left (b (3 n-8) x^n-2 a (n-4)\right )\right )}{32 a^2 (n-4) (n-2) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3 - 2*n)/Sqrt[a + b*x^n],x]
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Maple [F] time = 0.061, size = 0, normalized size = 0. \[ \int{{x}^{3-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3-2*n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-2*n + 3)/sqrt(b*x^n + a),x, algorithm="maxima")
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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(x^(-2*n + 3)/sqrt(b*x^n + a),x, algorithm="fricas")
[Out]
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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(x**(3-2*n)/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-2*n + 3)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]