3.2480 \(\int \frac{1}{x^2 (a+b x^n)^2} \, dx\)

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))

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Rubi [A]  time = 0.0070304, 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, Rules used = {364} \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]

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))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^2 \left (a+b x^n\right )^2} \, dx &=-\frac{\, _2F_1\left (2,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a^2 x}\\ \end{align*}

Mathematica [A]  time = 0.0035451, size = 31, normalized size = 0.91 \[ -\frac{\, _2F_1\left (2,-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^n)^2),x]

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [C]  time = 1.39887, size = 289, normalized size = 8.5 \begin{align*} - \frac{n \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{n \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{\Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{b n x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a^{2} \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} - \frac{b x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (- \frac{1}{n}\right )}{a^{2} \left (a n^{3} x \Gamma \left (1 - \frac{1}{n}\right ) + b n^{3} x x^{n} \Gamma \left (1 - \frac{1}{n}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a*(a*n**3*x*gamma(1 - 1/n) + b*n**3*x
*x**n*gamma(1 - 1/n))) - n*gamma(-1/n)/(a*(a*n**3*x*gamma(1 - 1/n) + b*n**3*x*x**n*gamma(1 - 1/n))) - lerchphi
(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a*(a*n**3*x*gamma(1 - 1/n) + b*n**3*x*x**n*gamma
(1 - 1/n))) - b*n*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n)*gamma(-1/n)/(a**2*(a*n**3*x*ga
mma(1 - 1/n) + b*n**3*x*x**n*gamma(1 - 1/n))) - b*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, exp_polar(I*pi)/n
)*gamma(-1/n)/(a**2*(a*n**3*x*gamma(1 - 1/n) + b*n**3*x*x**n*gamma(1 - 1/n)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)^2*x^2), x)