3.2700 \(\int \left (a+b x^n\right )^{-2-\frac{1}{n}} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0405838, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-2 - n^(-1)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.77428, size = 39, normalized size = 0.78 \[ \frac{x \left (a + b x^{n}\right )^{-1 - \frac{1}{n}}}{a \left (n + 1\right )} + \frac{n x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(-2-1/n),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0410468, size = 55, normalized size = 1.1 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-2 - n^(-1)),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(-2-1/n),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 2),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(-1/n - 2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.242405, size = 92, normalized size = 1.84 \[ \frac{b^{2} n x x^{2 \, n} +{\left (2 \, a b n + a b\right )} x x^{n} +{\left (a^{2} n + a^{2}\right )} x}{{\left (a^{2} n + a^{2}\right )}{\left (b x^{n} + a\right )}^{\frac{2 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 2),x, algorithm="fricas")

[Out]

(b^2*n*x*x^(2*n) + (2*a*b*n + a*b)*x*x^n + (a^2*n + a^2)*x)/((a^2*n + a^2)*(b*x^
n + a)^((2*n + 1)/n))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 2),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(-1/n - 2), x)