3.2699 \(\int \left (a+b x^n\right )^{-3-\frac{1}{n}} \, dx\)

Optimal. Leaf size=96 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0949738, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac{2 n x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a (2 n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-3 - n^(-1)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.6053, size = 80, normalized size = 0.83 \[ \frac{x \left (a + b x^{n}\right )^{-2 - \frac{1}{n}}}{a \left (2 n + 1\right )} + \frac{2 n x \left (a + b x^{n}\right )^{-1 - \frac{1}{n}}}{a^{2} \left (n + 1\right ) \left (2 n + 1\right )} + \frac{2 n^{2} x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a^{3} \left (n + 1\right ) \left (2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(-3-1/n),x)

[Out]

x*(a + b*x**n)**(-2 - 1/n)/(a*(2*n + 1)) + 2*n*x*(a + b*x**n)**(-1 - 1/n)/(a**2*
(n + 1)*(2*n + 1)) + 2*n**2*x*(a + b*x**n)**(-1/n)/(a**3*(n + 1)*(2*n + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0513605, size = 55, normalized size = 0.57 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-3 - n^(-1)),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.15, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-3-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(-3-1/n),x)

[Out]

int((a+b*x^n)^(-3-1/n),x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 3),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(-1/n - 3), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.242118, size = 170, normalized size = 1.77 \[ \frac{2 \, b^{3} n^{2} x x^{3 \, n} + 2 \,{\left (3 \, a b^{2} n^{2} + a b^{2} n\right )} x x^{2 \, n} +{\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} +{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} x}{{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )}{\left (b x^{n} + a\right )}^{\frac{3 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 3),x, algorithm="fricas")

[Out]

(2*b^3*n^2*x*x^(3*n) + 2*(3*a*b^2*n^2 + a*b^2*n)*x*x^(2*n) + (6*a^2*b*n^2 + 5*a^
2*b*n + a^2*b)*x*x^n + (2*a^3*n^2 + 3*a^3*n + a^3)*x)/((2*a^3*n^2 + 3*a^3*n + a^
3)*(b*x^n + a)^((3*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)**(-3-1/n),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 3),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(-1/n - 3), x)