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

Optimal. Leaf size=50 \[ \frac{x^{m+1} \sqrt{a+b x^2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0636763, antiderivative size = 63, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{(m+1) \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[a + b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.3375, size = 53, normalized size = 1.06 \[ \frac{x^{m + 1} \sqrt{a + b x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0154907, size = 63, normalized size = 1.26 \[ \frac{x^{m+1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{(m+1) \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^2+a)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

integrate(x^m/sqrt(b*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{\sqrt{b x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^2 + a),x, algorithm="fricas")

[Out]

integral(x^m/sqrt(b*x^2 + a), x)

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Sympy [A]  time = 2.9253, size = 53, normalized size = 1.06 \[ \frac{x x^{m} \Gamma \left (\frac{m}{2} + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*x**m*gamma(m/2 + 1/2)*hyper((1/2, m/2 + 1/2), (m/2 + 3/2,), b*x**2*exp_polar(I
*pi)/a)/(2*sqrt(a)*gamma(m/2 + 3/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^2 + a),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^2 + a), x)