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

Optimal. Leaf size=20 \[ \frac{x^{1-n}}{(1-n) (a+b)} \]

[Out]

x^(1 - n)/((a + b)*(1 - n))

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Rubi [A]  time = 0.0058103, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6, 12, 30} \[ \frac{x^{1-n}}{(1-n) (a+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^n + b*x^n)^(-1),x]

[Out]

x^(1 - n)/((a + b)*(1 - n))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0038062, size = 20, normalized size = 1. \[ \frac{x^{1-n}}{(1-n) (a+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^n + b*x^n)^(-1),x]

[Out]

x^(1 - n)/((a + b)*(1 - n))

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \begin{align*} -{\frac{x}{ \left ( -1+n \right ){x}^{n} \left ( a+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(-1+n)/(x^n)/(a+b)

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Maxima [A]  time = 1.07105, size = 28, normalized size = 1.4 \begin{align*} -\frac{x}{{\left (a{\left (n - 1\right )} + b{\left (n - 1\right )}\right )} x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/((a*(n - 1) + b*(n - 1))*x^n)

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Fricas [A]  time = 0.872954, size = 41, normalized size = 2.05 \begin{align*} -\frac{x}{{\left ({\left (a + b\right )} n - a - b\right )} x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(((a + b)*n - a - b)*x^n)

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Sympy [A]  time = 0.839191, size = 32, normalized size = 1.6 \begin{align*} \begin{cases} - \frac{x}{a n x^{n} - a x^{n} + b n x^{n} - b x^{n}} & \text{for}\: n \neq 1 \\\frac{\log{\left (x \right )}}{a + b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x/(a*n*x**n - a*x**n + b*n*x**n - b*x**n), Ne(n, 1)), (log(x)/(a + b), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(a*x^n + b*x^n), x)