3.2779 \(\int \frac{(c x)^{-3+n}}{a+b x^n} \, dx\)

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)))

________________________________________________________________________________________

Rubi [A]  time = 0.0126807, 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, Rules used = {364} \[ -\frac{(c x)^{n-2} \, _2F_1\left (1,-\frac{2-n}{n};2 \left (1-\frac{1}{n}\right );-\frac{b x^n}{a}\right )}{a c (2-n)} \]

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)))

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)^{-3+n}}{a+b x^n} \, dx &=-\frac{(c x)^{-2+n} \, _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)}\\ \end{align*}

Mathematica [A]  time = 0.0124242, size = 44, normalized size = 0.85 \[ \frac{x (c x)^{n-3} \, _2F_1\left (1,\frac{n-2}{n};\frac{n-2}{n}+1;-\frac{b x^n}{a}\right )}{a (n-2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{-3+n}}{a+b{x}^{n}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [C]  time = 93.7586, size = 44, normalized size = 0.85 \begin{align*} \frac{2 c^{n} \Phi \left (\frac{a x^{- n} e^{i \pi }}{b}, 1, \frac{2}{n}\right ) \Gamma \left (- \frac{2}{n}\right )}{b c^{3} n^{2} x^{2} \Gamma \left (1 - \frac{2}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c**n*lerchphi(a*x**(-n)*exp_polar(I*pi)/b, 1, 2/n)*gamma(-2/n)/(b*c**3*n**2*x**2*gamma(1 - 2/n))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{n - 3}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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