3.2688 \(\int \frac{x^m}{\sqrt{a+b x^{-2+m}}} \, dx\)

Optimal. Leaf size=65 \[ \frac{x^{m+1} \sqrt{a+b x^{m-2}} \, _2F_1\left (1,-\frac{3 m}{2 (2-m)};\frac{1-2 m}{2-m};-\frac{b x^{m-2}}{a}\right )}{a (m+1)} \]

[Out]

(x^(1 + m)*Sqrt[a + b*x^(-2 + m)]*Hypergeometric2F1[1, (-3*m)/(2*(2 - m)), (1 -
2*m)/(2 - m), -((b*x^(-2 + m))/a)])/(a*(1 + m))

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Rubi [A]  time = 0.10332, antiderivative size = 80, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{m+1} \sqrt{\frac{b x^{m-2}}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{m+1}{2-m};\frac{1-2 m}{2-m};-\frac{b x^{m-2}}{a}\right )}{(m+1) \sqrt{a+b x^{m-2}}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(x^(1 + m)*Sqrt[1 + (b*x^(-2 + m))/a]*Hypergeometric2F1[1/2, -((1 + m)/(2 - m)),
(1 - 2*m)/(2 - m), -((b*x^(-2 + m))/a)])/((1 + m)*Sqrt[a + b*x^(-2 + m)])

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Rubi in Sympy [A]  time = 8.56817, size = 60, normalized size = 0.92 \[ \frac{x^{m + 1} \sqrt{a + b x^{m - 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 1}{m - 2} \\ \frac{2 m - 1}{m - 2} \end{matrix}\middle |{- \frac{b x^{m - 2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{m - 2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a+b*x**(-2+m))**(1/2),x)

[Out]

x**(m + 1)*sqrt(a + b*x**(m - 2))*hyper((1/2, (m + 1)/(m - 2)), ((2*m - 1)/(m -
2),), -b*x**(m - 2)/a)/(a*sqrt(1 + b*x**(m - 2)/a)*(m + 1))

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Mathematica [A]  time = 0.199285, size = 110, normalized size = 1.69 \[ \frac{2 x \left (6 a x^2 \sqrt{\frac{a x^{2-m}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{m-8}{2 (m-2)};\frac{3 (m-4)}{2 (m-2)};-\frac{a x^{2-m}}{b}\right )+(m-8) \left (a x^2+b x^m\right )\right )}{b (m-8) (m+4) \sqrt{a+b x^{m-2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(2*x*((-8 + m)*(a*x^2 + b*x^m) + 6*a*x^2*Sqrt[1 + (a*x^(2 - m))/b]*Hypergeometri
c2F1[1/2, (-8 + m)/(2*(-2 + m)), (3*(-4 + m))/(2*(-2 + m)), -((a*x^(2 - m))/b)])
)/(b*(-8 + m)*(4 + m)*Sqrt[a + b*x^(-2 + m)])

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Maple [F]  time = 0.137, 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]

int(x^m/(a+b*x^(-2+m))^(1/2),x)

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Maxima [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="maxima")

[Out]

integrate(x^m/sqrt(b*x^(m - 2) + a), x)

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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]

Exception raised: TypeError

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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(x**m/(a+b*x**(-2+m))**(1/2),x)

[Out]

Exception raised: TypeError

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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]

integrate(x^m/sqrt(b*x^(m - 2) + a), x)