3.349 \(\int \frac{(c x)^{-1+m}}{a+b x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{(c x)^m \, _2F_1\left (1,\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right )}{a c m} \]

[Out]

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

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Rubi [A]  time = 0.0353322, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{(c x)^m \, _2F_1\left (1,\frac{m}{2};\frac{m+2}{2};-\frac{b x^2}{a}\right )}{a c m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.99646, size = 24, normalized size = 0.63 \[ \frac{\left (c x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} \\ \frac{m}{2} + 1 \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0500373, size = 58, normalized size = 1.53 \[ \frac{(c x)^m \left (a (m+2)-b m x^2 \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )\right )}{a^2 c m (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-1+m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1+m)/(b*x^2+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m - 1}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 1)/(b*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{m - 1}}{b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 1)/(b*x^2 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 67.745, size = 39, normalized size = 1.03 \[ \frac{c^{m} m x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2}\right ) \Gamma \left (\frac{m}{2}\right )}{4 a c \Gamma \left (\frac{m}{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m - 1}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 1)/(b*x^2 + a),x, algorithm="giac")

[Out]

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