3.2634 \(\int \frac{x^{-1+2 n}}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=24 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)^2)

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Rubi [A]  time = 0.0051769, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/(a + b*x^n)^3,x]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)^2)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx &=\frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0058874, size = 24, normalized size = 1. \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)/(a + b*x^n)^3,x]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)^2)

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Maple [A]  time = 0.025, size = 36, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-{\frac{a}{2\,{b}^{2}n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967291, size = 55, normalized size = 2.29 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="maxima")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

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Fricas [A]  time = 1.00395, size = 86, normalized size = 3.58 \begin{align*} -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="fricas")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="giac")

[Out]

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