Optimal. Leaf size=67 \[ \frac{x^{m+1} \sqrt{a+b x^{2-m}} \, _2F_1\left (1,\frac{m+4}{2 (2-m)};\frac{3}{2-m};-\frac{b x^{2-m}}{a}\right )}{a (m+1)} \]
[Out]
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Rubi [A] time = 0.105769, antiderivative size = 81, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{m+1} \sqrt{\frac{b x^{2-m}}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2-m};\frac{3}{2-m};-\frac{b x^{2-m}}{a}\right )}{(m+1) \sqrt{a+b x^{2-m}}} \]
Antiderivative was successfully verified.
[In] Int[x^m/Sqrt[a + b*x^(2 - m)],x]
[Out]
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Rubi in Sympy [A] time = 8.40574, size = 60, normalized size = 0.9 \[ \frac{x^{m + 1} \sqrt{a + b x^{- m + 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - \frac{m + 1}{m - 2} \\ - \frac{3}{m - 2} \end{matrix}\middle |{- \frac{b x^{- m + 2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{- m + 2}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(a+b*x**(2-m))**(1/2),x)
[Out]
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Mathematica [A] time = 0.229026, size = 79, normalized size = 1.18 \[ \frac{x^{m+1} \sqrt{\frac{b x^{2-m}}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2-m};-\frac{3}{m-2};-\frac{b x^{2-m}}{a}\right )}{(m+1) \sqrt{a+b x^{2-m}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/Sqrt[a + b*x^(2 - m)],x]
[Out]
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Maple [F] time = 0.074, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{a+b{x}^{2-m}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(a+b*x^(2-m))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{crash} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(a + b*x^(2 - m)),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(x^m/sqrt(b*x^(-m + 2) + 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**m/(a+b*x**(2-m))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{-m + 2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(b*x^(-m + 2) + a),x, algorithm="giac")
[Out]