3.2772 \(\int \frac{(c x)^{-3+n}}{a+b x^n} \, dx\)

Optimal. Leaf size=52 \[ -\frac{(c x)^{n-2} \, _2F_1\left (1,-\frac{2-n}{n};2 \left (1-\frac{1}{n}\right );-\frac{b x^n}{a}\right )}{a c (2-n)} \]

[Out]

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

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Rubi [A]  time = 0.0509951, antiderivative size = 52, 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 \[ -\frac{(c x)^{n-2} \, _2F_1\left (1,-\frac{2-n}{n};2 \left (1-\frac{1}{n}\right );-\frac{b x^n}{a}\right )}{a c (2-n)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.98069, size = 31, normalized size = 0.6 \[ - \frac{\left (c x\right )^{n - 2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 2}{n} \\ 2 - \frac{2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (- n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0340331, size = 47, normalized size = 0.9 \[ \frac{x^{-n-2} (c x)^n \left (\, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )-1\right )}{2 b c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-3+n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-3+n)/(a+b*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -a c^{n} \int \frac{1}{b^{2} c^{3} x^{3} x^{n} + a b c^{3} x^{3}}\,{d x} - \frac{c^{n - 3}}{2 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*c^n*integrate(1/(b^2*c^3*x^3*x^n + a*b*c^3*x^3), x) - 1/2*c^(n - 3)/(b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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