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

Optimal. Leaf size=47 \[ -\frac{(c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right )}{a c (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0487984, antiderivative size = 47, 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 \[ -\frac{(c x)^{m-1} \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right )}{a c (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.22391, size = 34, normalized size = 0.72 \[ - \frac{\left (c x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} - \frac{1}{2} \\ \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.075732, size = 59, normalized size = 1.26 \[ \frac{x (c x)^{m-2} \left (a (m+1)-b (m-1) x^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )\right )}{a^2 \left (m^2-1\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-2+m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-2+m)/(b*x^2+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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