3.4.25 \(\int \frac {1}{a x^n+b x^n} \, dx\) [325]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^{1-n}}{(1-n) (a+b)} \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1-n}}{(a+b) (1-n)} \end {gather*}

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 = 0.02, size = 19, normalized size = 0.95

method	result	size
gosper	\(-\frac {x \,x^{-n}}{\left (-1+n \right ) \left (a +b \right )}\)	\(19\)
risch	\(-\frac {x \,x^{-n}}{\left (-1+n \right ) \left (a +b \right )}\)	\(19\)
norman	\(-\frac {x \,{\mathrm e}^{-n \ln \left (x \right )}}{a n +b n -a -b}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x}{{\left (a {\left (n - 1\right )} + b {\left (n - 1\right )}\right )} x^{n}} \end {gather*}

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 = 1.33, size = 22, normalized size = 1.10 \begin {gather*} -\frac {x}{{\left ({\left (a + b\right )} n - a - b\right )} x^{n}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (10) = 20\).
time = 0.26, size = 32, normalized size = 1.60 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 5.18, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x^{1-n}}{\left (a+b\right )\,\left (n-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^n + b*x^n),x)

[Out]

-x^(1 - n)/((a + b)*(n - 1))

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