3.351 \(\int \frac{(c x)^{-3+m}}{a+b x^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{(c x)^{m-2} \, _2F_1\left (1,\frac{m-2}{2};\frac{m}{2};-\frac{b x^2}{a}\right )}{a c (2-m)} \]

[Out]

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

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Rubi [A]  time = 0.0476106, antiderivative size = 45, 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 \[ -\frac{(c x)^{m-2} \, _2F_1\left (1,\frac{m-2}{2};\frac{m}{2};-\frac{b x^2}{a}\right )}{a c (2-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.28221, size = 29, normalized size = 0.64 \[ - \frac{\left (c x\right )^{m - 2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{2} - 1 \\ \frac{m}{2} \end{matrix}\middle |{- \frac{b x^{2}}{a}} \right )}}{a c \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-3+m)/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0955517, size = 80, normalized size = 1.78 \[ \frac{(c x)^m \left (b^2 (m-2) m x^4 \, _2F_1\left (1,\frac{m}{2}+1;\frac{m}{2}+2;-\frac{b x^2}{a}\right )+a (m+2) \left (a m-b (m-2) x^2\right )\right )}{a^3 c^3 m \left (m^2-4\right ) x^2} \]

Antiderivative was successfully verified.

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

[Out]

((c*x)^m*(a*(2 + m)*(a*m - b*(-2 + m)*x^2) + b^2*(-2 + m)*m*x^4*Hypergeometric2F
1[1, 1 + m/2, 2 + m/2, -((b*x^2)/a)]))/(a^3*c^3*m*(-4 + m^2)*x^2)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-3+m}}{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-3+m)/(b*x^2+a),x)

[Out]

int((c*x)^(-3+m)/(b*x^2+a),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m - 3}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 3)/(b*x^2 + a),x, algorithm="maxima")

[Out]

integrate((c*x)^(m - 3)/(b*x^2 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (c x\right )^{m - 3}}{b x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 3)/(b*x^2 + a),x, algorithm="fricas")

[Out]

integral((c*x)^(m - 3)/(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)**(-3+m)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{m - 3}}{b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(m - 3)/(b*x^2 + a),x, algorithm="giac")

[Out]

integrate((c*x)^(m - 3)/(b*x^2 + a), x)