3.2628 \(\int \frac{x^{-1+n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]

[Out]

-1/(2*b*n*(a + b*x^n)^2)

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Rubi [A]  time = 0.0200066, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)/(a + b*x^n)^3,x]

[Out]

-1/(2*b*n*(a + b*x^n)^2)

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Rubi in Sympy [A]  time = 2.45218, size = 15, normalized size = 0.79 \[ - \frac{1}{2 b n \left (a + b x^{n}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/(a+b*x**n)**3,x)

[Out]

-1/(2*b*n*(a + b*x**n)**2)

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Mathematica [A]  time = 0.0106049, size = 19, normalized size = 1. \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*n*(a + b*x^n)^2)

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Maple [A]  time = 0.039, size = 20, normalized size = 1.1 \[ -{\frac{1}{2\,bn \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)/(a+b*x^n)^3,x)

[Out]

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

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Maxima [A]  time = 1.44062, size = 23, normalized size = 1.21 \[ -\frac{1}{2 \,{\left (b x^{n} + a\right )}^{2} b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((b*x^n + a)^2*b*n)

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Fricas [A]  time = 0.214794, size = 42, normalized size = 2.21 \[ -\frac{1}{2 \,{\left (b^{3} n x^{2 \, n} + 2 \, a b^{2} n x^{n} + a^{2} b n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 87.276, size = 109, normalized size = 5.74 \[ \begin{cases} \frac{\log{\left (x \right )}}{b^{3}} & \text{for}\: a = 0 \wedge n = 0 \\- \frac{x^{- 2 n}}{2 b^{3} n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{3}} & \text{for}\: n = 0 \\\frac{2 a x^{n}}{2 a^{4} n + 4 a^{3} b n x^{n} + 2 a^{2} b^{2} n x^{2 n}} + \frac{b x^{2 n}}{2 a^{4} n + 4 a^{3} b n x^{n} + 2 a^{2} b^{2} n x^{2 n}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+n)/(a+b*x**n)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.213305, size = 23, normalized size = 1.21 \[ -\frac{1}{2 \,{\left (b x^{n} + a\right )}^{2} b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((b*x^n + a)^2*b*n)