3.2476 \(\int \frac{1}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0185046, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^(-3),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.7985, size = 19, normalized size = 0.79 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 0.085951, size = 71, normalized size = 2.96 \[ \frac{x \left (\left (2 n^2-3 n+1\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{a \left (a (3 n-1)+b (2 n-1) x^n\right )}{\left (a+b x^n\right )^2}\right )}{2 a^3 n^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^(-3),x]

[Out]

(x*((a*(a*(-1 + 3*n) + b*(-1 + 2*n)*x^n))/(a + b*x^n)^2 + (1 - 3*n + 2*n^2)*Hype
rgeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)]))/(2*a^3*n^2)

_______________________________________________________________________________________

Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{-3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^n)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (2 \, n^{2} - 3 \, n + 1\right )} \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n - 1\right )} x x^{n} + a{\left (3 \, n - 1\right )} x}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*n^2 - 3*n + 1)*integrate(1/2/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(b*(2*n - 1)
*x*x^n + a*(3*n - 1)*x)/(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**n)**3,x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^(-3), x)