3.2778 \(\int \frac{(c x)^{-2+n}}{a+b x^n} \, dx\)

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 = 0.0127508, 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, Rules used = {364} \[ -\frac{(c x)^{n-1} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1-n)} \]

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)))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{(c x)^{-2+n}}{a+b x^n} \, dx &=-\frac{(c x)^{-1+n} \, _2F_1\left (1,-\frac{1-n}{n};2-\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1-n)}\\ \end{align*}

Mathematica [A]  time = 0.0116366, size = 44, normalized size = 0.88 \[ \frac{x (c x)^{n-2} \, _2F_1\left (1,\frac{n-1}{n};\frac{n-1}{n}+1;-\frac{b x^n}{a}\right )}{a (n-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-2+n}}{a+b{x}^{n}}}\, dx \end{align*}

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. \begin{align*} -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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x)^(-2+n)/(a+b*x^n),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. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{n - 2}}{b x^{n} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 19.2793, size = 41, normalized size = 0.82 \begin{align*} \frac{c^{n} \Phi \left (\frac{a x^{- n} e^{i \pi }}{b}, 1, \frac{1}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{b c^{2} n^{2} x \Gamma \left (1 - \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**n*lerchphi(a*x**(-n)*exp_polar(I*pi)/b, 1, 1/n)*gamma(-1/n)/(b*c**2*n**2*x*gamma(1 - 1/n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{n - 2}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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