3.2698 \(\int \left (a+b x^n\right )^{-4-\frac{1}{n}} \, dx\)

Optimal. Leaf size=146 \[ \frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]

[Out]

(x*(a + b*x^n)^(-3 - n^(-1)))/(a*(1 + 3*n)) + (3*n*x*(a + b*x^n)^(-2 - n^(-1)))/
(a^2*(1 + 5*n + 6*n^2)) + (6*n^2*x*(a + b*x^n)^(-1 - n^(-1)))/(a^3*(1 + n)*(1 +
2*n)*(1 + 3*n)) + (6*n^3*x)/(a^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi [A]  time = 0.206559, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-4 - n^(-1)),x]

[Out]

(x*(a + b*x^n)^(-3 - n^(-1)))/(a*(1 + 3*n)) + (3*n*x*(a + b*x^n)^(-2 - n^(-1)))/
(a^2*(1 + 5*n + 6*n^2)) + (6*n^2*x*(a + b*x^n)^(-1 - n^(-1)))/(a^3*(1 + n)*(1 +
2*n)*(1 + 3*n)) + (6*n^3*x)/(a^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*(a + b*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.7323, size = 124, normalized size = 0.85 \[ \frac{x \left (a + b x^{n}\right )^{-3 - \frac{1}{n}}}{a \left (3 n + 1\right )} + \frac{3 n x \left (a + b x^{n}\right )^{-2 - \frac{1}{n}}}{a^{2} \left (2 n + 1\right ) \left (3 n + 1\right )} + \frac{6 n^{2} x \left (a + b x^{n}\right )^{-1 - \frac{1}{n}}}{a^{3} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right )} + \frac{6 n^{3} x \left (a + b x^{n}\right )^{- \frac{1}{n}}}{a^{4} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**(-4-1/n),x)

[Out]

x*(a + b*x**n)**(-3 - 1/n)/(a*(3*n + 1)) + 3*n*x*(a + b*x**n)**(-2 - 1/n)/(a**2*
(2*n + 1)*(3*n + 1)) + 6*n**2*x*(a + b*x**n)**(-1 - 1/n)/(a**3*(n + 1)*(2*n + 1)
*(3*n + 1)) + 6*n**3*x*(a + b*x**n)**(-1/n)/(a**4*(n + 1)*(2*n + 1)*(3*n + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0677958, size = 55, normalized size = 0.38 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-4 - n^(-1)),x]

[Out]

(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[4 + n^(-1), n^(-1), 1 + n^(-1), -((b
*x^n)/a)])/(a^4*(a + b*x^n)^n^(-1))

_______________________________________________________________________________________

Maple [F]  time = 0.171, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-4-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(-4-1/n),x)

[Out]

int((a+b*x^n)^(-4-1/n),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 4),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(-1/n - 4), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.239981, size = 259, normalized size = 1.77 \[ \frac{6 \, b^{4} n^{3} x x^{4 \, n} + 6 \,{\left (4 \, a b^{3} n^{3} + a b^{3} n^{2}\right )} x x^{3 \, n} + 3 \,{\left (12 \, a^{2} b^{2} n^{3} + 7 \, a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x x^{2 \, n} +{\left (24 \, a^{3} b n^{3} + 26 \, a^{3} b n^{2} + 9 \, a^{3} b n + a^{3} b\right )} x x^{n} +{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )} x}{{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )}{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 4),x, algorithm="fricas")

[Out]

(6*b^4*n^3*x*x^(4*n) + 6*(4*a*b^3*n^3 + a*b^3*n^2)*x*x^(3*n) + 3*(12*a^2*b^2*n^3
+ 7*a^2*b^2*n^2 + a^2*b^2*n)*x*x^(2*n) + (24*a^3*b*n^3 + 26*a^3*b*n^2 + 9*a^3*b
*n + a^3*b)*x*x^n + (6*a^4*n^3 + 11*a^4*n^2 + 6*a^4*n + a^4)*x)/((6*a^4*n^3 + 11
*a^4*n^2 + 6*a^4*n + a^4)*(b*x^n + a)^((4*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)**(-4-1/n),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(-1/n - 4),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(-1/n - 4), x)