3.2627 \(\int \frac{x^{-1+n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{b n \left (a+b x^n\right )} \]

[Out]

-(1/(b*n*(a + b*x^n)))

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Rubi [A]  time = 0.0045928, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{1}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)/(a + b*x^n)^2,x]

[Out]

-(1/(b*n*(a + b*x^n)))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0052548, size = 17, normalized size = 1. \[ -\frac{1}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)/(a + b*x^n)^2,x]

[Out]

-(1/(b*n*(a + b*x^n)))

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Maple [A]  time = 0.016, size = 24, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{an \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a/n*exp(n*ln(x))/(a+b*exp(n*ln(x)))

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Maxima [A]  time = 0.985065, size = 23, normalized size = 1.35 \begin{align*} -\frac{1}{{\left (b x^{n} + a\right )} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x^n + a)*b*n)

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Fricas [A]  time = 0.967611, size = 32, normalized size = 1.88 \begin{align*} -\frac{1}{b^{2} n x^{n} + a b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*n*x^n + a*b*n)

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Sympy [A]  time = 9.18444, size = 51, normalized size = 3. \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{x^{- n}}{b^{2} n} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{n}}{n} & \text{for}\: b = - a x^{- n} \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{x^{n}}{a^{2} n + a b n x^{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x**(-n)/(b**2*n), Eq(a, 0)), (zoo*x**n/n, Eq(b, -a*x
**(-n))), (log(x)/(a + b)**2, Eq(n, 0)), (x**n/(a**2*n + a*b*n*x**n), True))

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Giac [A]  time = 1.12172, size = 23, normalized size = 1.35 \begin{align*} -\frac{1}{{\left (b x^{n} + a\right )} b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x^n + a)*b*n)