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

Optimal. Leaf size=20 \[ \frac {x^{1-2 n}}{(1-2 n) (a+b)^2} \]

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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^{1-2 n}}{(1-2 n) (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^n + b*x^n)^(-2),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^{1-2 n}}{(1-2 n) (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^n + b*x^n)^(-2),x]

[Out]

x^(1 - 2*n)/((a + b)^2*(1 - 2*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 )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*x^n + b*x^n)^(-2),x]

[Out]

Defer[IntegrateAlgebraic][(a*x^n + b*x^n)^(-2), x]

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fricas [A]  time = 0.41, size = 36, normalized size = 1.80 \begin {gather*} \frac {x}{{\left (a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} n\right )} x^{2 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/((a^2 + 2*a*b + b^2 - 2*(a^2 + 2*a*b + b^2)*n)*x^(2*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 )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x^n + b*x^n)^(-2), x)

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maple [A]  time = 0.04, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x \,x^{-2 n}}{\left (2 n -1\right ) \left (a +b \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 40, normalized size = 2.00 \begin {gather*} -\frac {x}{{\left (a^{2} {\left (2 \, n - 1\right )} + 2 \, a b {\left (2 \, n - 1\right )} + b^{2} {\left (2 \, n - 1\right )}\right )} x^{2 \, n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.14, size = 21, normalized size = 1.05 \begin {gather*} -\frac {x^{1-2\,n}}{{\left (a+b\right )}^2\,\left (2\,n-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^n + b*x^n)^2,x)

[Out]

-x^(1 - 2*n)/((a + b)^2*(2*n - 1))

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sympy [A]  time = 1.03, size = 82, normalized size = 4.10 \begin {gather*} \begin {cases} - \frac {x}{2 a^{2} n x^{2 n} - a^{2} x^{2 n} + 4 a b n x^{2 n} - 2 a b x^{2 n} + 2 b^{2} n x^{2 n} - b^{2} x^{2 n}} & \text {for}\: n \neq \frac {1}{2} \\\frac {\log {\relax (x )}}{a^{2} + 2 a b + b^{2}} & \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)**2,x)

[Out]

Piecewise((-x/(2*a**2*n*x**(2*n) - a**2*x**(2*n) + 4*a*b*n*x**(2*n) - 2*a*b*x**(2*n) + 2*b**2*n*x**(2*n) - b**
2*x**(2*n)), Ne(n, 1/2)), (log(x)/(a**2 + 2*a*b + b**2), True))

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