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

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

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Rubi [A]  time = 0.01, 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^{1-3 n}}{(1-3 n) (a+b)^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1-3 n}}{(1-3 n) (a+b)^3} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a x^n+b x^n\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*x^n + b*x^n)^(-3),x]

[Out]

Defer[IntegrateAlgebraic][(a*x^n + b*x^n)^(-3), x]

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fricas [B]  time = 0.42, size = 52, normalized size = 2.60 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{n} + b x^{n}\right )}^{3}}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.04, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x \,x^{-3 n}}{\left (3 n -1\right ) \left (a +b \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.40, size = 53, normalized size = 2.65 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 5.12, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x^{1-3\,n}}{{\left (a+b\right )}^3\,\left (3\,n-1\right )} \end {gather*}

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)/((a + b)^3*(3*n - 1))

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sympy [A]  time = 1.35, size = 119, normalized size = 5.95 \begin {gather*} \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 {\relax (x )}}{a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}} & \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)**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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