3.327 \(\int \frac{1}{(a x^n+b x^n)^3} \, dx\)

Optimal. Leaf size=20 \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*n))

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Rubi [A]  time = 0.0089884, 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} \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^n + b*x^n)^(-3),x]

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*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}{\left (a x^n+b x^n\right )^3} \, dx &=\int \frac{x^{-3 n}}{(a+b)^3} \, dx\\ &=\frac{\int x^{-3 n} \, dx}{(a+b)^3}\\ &=\frac{x^{1-3 n}}{(a+b)^3 (1-3 n)}\\ \end{align*}

Mathematica [A]  time = 0.0032928, size = 20, normalized size = 1. \[ \frac{x^{1-3 n}}{(1-3 n) (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^n + b*x^n)^(-3),x]

[Out]

x^(1 - 3*n)/((a + b)^3*(1 - 3*n))

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Maple [A]  time = 0.002, size = 21, normalized size = 1.1 \begin{align*} -{\frac{x}{ \left ( -1+3\,n \right ) \left ({x}^{n} \right ) ^{3} \left ( a+b \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^n+b*x^n)^3,x)

[Out]

-x/(-1+3*n)/(x^n)^3/(a+b)^3

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Maxima [B]  time = 1.01413, size = 72, normalized size = 3.6 \begin{align*} -\frac{x}{{\left (a^{3}{\left (3 \, n - 1\right )} + 3 \, a^{2} b{\left (3 \, n - 1\right )} + 3 \, a b^{2}{\left (3 \, n - 1\right )} + b^{3}{\left (3 \, n - 1\right )}\right )} x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^n+b*x^n)^3,x, algorithm="maxima")

[Out]

-x/((a^3*(3*n - 1) + 3*a^2*b*(3*n - 1) + 3*a*b^2*(3*n - 1) + b^3*(3*n - 1))*x^(3*n))

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Fricas [B]  time = 0.918571, size = 112, normalized size = 5.6 \begin{align*} \frac{x}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 3 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} n\right )} x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^n+b*x^n)^3,x, algorithm="fricas")

[Out]

x/((a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*n)*x^(3*n))

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Sympy [A]  time = 1.80554, size = 119, normalized size = 5.95 \begin{align*} \begin{cases} - \frac{x}{3 a^{3} n x^{3 n} - a^{3} x^{3 n} + 9 a^{2} b n x^{3 n} - 3 a^{2} b x^{3 n} + 9 a b^{2} n x^{3 n} - 3 a b^{2} x^{3 n} + 3 b^{3} n x^{3 n} - b^{3} x^{3 n}} & \text{for}\: n \neq \frac{1}{3} \\\frac{\log{\left (x \right )}}{a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}} & \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)**3,x)

[Out]

Piecewise((-x/(3*a**3*n*x**(3*n) - a**3*x**(3*n) + 9*a**2*b*n*x**(3*n) - 3*a**2*b*x**(3*n) + 9*a*b**2*n*x**(3*
n) - 3*a*b**2*x**(3*n) + 3*b**3*n*x**(3*n) - b**3*x**(3*n)), Ne(n, 1/3)), (log(x)/(a**3 + 3*a**2*b + 3*a*b**2
+ b**3), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x^{n} + b x^{n}\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x^n+b*x^n)^3,x, algorithm="giac")

[Out]

integrate((a*x^n + b*x^n)^(-3), x)