3.2478 \(\int \frac{1}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0053188, 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, Rules used = {245} \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^(-2),x]

[Out]

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

Rule 245

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

Rubi steps

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

Mathematica [A]  time = 0.0019092, size = 24, normalized size = 1. \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^(-2),x]

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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^{n} + a^{2} n}\,{d x} + \frac{x}{a b n x^{n} + a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.24888, size = 257, normalized size = 10.71 \begin{align*} \frac{n x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{n x \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{b n x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{b x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} 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/(a+b*x**n)**2,x)

[Out]

n*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/
n))) + n*x*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - x*lerchphi(b*x**n*exp_polar(I
*pi)/a, 1, 1/n)*gamma(1/n)/(a*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) + b*n*x*x**n*lerchphi(b*x*
*n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*x**n*gamma(1 + 1/n))) - b*x*x**
n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamma(1/n)/(a**2*(a*n**3*gamma(1 + 1/n) + b*n**3*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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(-2), x)