3.2689 \(\int \frac{x^m}{\sqrt{a+b x^{2-m}}} \, dx\)

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]

(x^(1 + m)*Sqrt[a + b*x^(2 - m)]*Hypergeometric2F1[1, (4 + m)/(2*(2 - m)), 3/(2
- m), -((b*x^(2 - m))/a)])/(a*(1 + m))

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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]

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

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Rubi in Sympy [A]  time = 8.40574, 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]

x**(m + 1)*sqrt(a + b*x**(-m + 2))*hyper((1/2, -(m + 1)/(m - 2)), (-3/(m - 2),),
-b*x**(-m + 2)/a)/(a*sqrt(1 + b*x**(-m + 2)/a)*(m + 1))

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Mathematica [A]  time = 0.229026, 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]

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

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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]

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

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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]

crash

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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(-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]

Timed 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]

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