3.2771 \(\int \frac{(c x)^{-2+n}}{a+b x^n} \, dx\)

Optimal. Leaf size=50 \[ -\frac{(c x)^{n-1} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.0506085, antiderivative size = 50, 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 \[ -\frac{(c x)^{n-1} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1-n)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.93389, size = 31, normalized size = 0.62 \[ - \frac{\left (c x\right )^{n - 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n - 1}{n} \\ 2 - \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a c \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0342763, size = 44, normalized size = 0.88 \[ \frac{x^{-n-1} (c x)^n \left (\, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )-1\right )}{b c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \[ \int{\frac{ \left ( cx \right ) ^{-2+n}}{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-2+n)/(a+b*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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