3.2473 \(\int \frac{1}{x^2 \left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.029889, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^n)^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.08492, size = 22, normalized size = 0.65 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} 2, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0604563, size = 56, normalized size = 1.65 \[ \frac{a-(n+1) \left (a+b x^n\right ) \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{a^2 n x \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b*x^n)^2,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^2*x^2), x)