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]

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

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \[ \frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

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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fricas [A]  time = 0.75, size = 41, normalized size = 1.71 \[ -\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 36, normalized size = 1.50 \[ \frac {-\frac {{\mathrm e}^{n \ln \relax (x )}}{b n}-\frac {a}{2 b^{2} n}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 41, normalized size = 1.71 \[ -\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

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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mupad [B]  time = 1.38, size = 37, normalized size = 1.54 \[ \frac {x^{2\,n}}{2\,\left (a^3\,n+2\,a^2\,b\,n\,x^n+a\,b^2\,n\,x^{2\,n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n - 1)/(a + b*x^n)^3,x)

[Out]

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

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sympy [A]  time = 158.69, size = 61, normalized size = 2.54 \[ \begin {cases} \frac {\log {\relax (x )}}{b^{3}} & \text {for}\: a = 0 \wedge n = 0 \\- \frac {x^{- n}}{b^{3} n} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\\frac {x^{2 n}}{2 a^{3} n + 4 a^{2} b n x^{n} + 2 a b^{2} n x^{2 n}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)/b**3, Eq(a, 0) & Eq(n, 0)), (-x**(-n)/(b**3*n), Eq(a, 0)), (log(x)/(a + b)**3, Eq(n, 0)), (x
**(2*n)/(2*a**3*n + 4*a**2*b*n*x**n + 2*a*b**2*n*x**(2*n)), True))

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