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

Optimal. Leaf size=39 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0317599, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.98728, size = 29, normalized size = 0.74 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0265096, size = 41, normalized size = 1.05 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{b x^2}{a}\right )}{a (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^2),x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^2 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^m/(b*x^2 + a), x)

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Sympy [A]  time = 4.48813, size = 88, normalized size = 2.26 \[ \frac{m x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} + \frac{x x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{1}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(b*x**2+a),x)

[Out]

m*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(4*a*
gamma(m/2 + 3/2)) + x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamm
a(m/2 + 1/2)/(4*a*gamma(m/2 + 3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^2 + a), x)