3.2619 \(\int \frac{x^{-1+2 n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]

[Out]

a/(b^2*n*(a + b*x^n)) + Log[a + b*x^n]/(b^2*n)

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Rubi [A]  time = 0.0593578, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

a/(b^2*n*(a + b*x^n)) + Log[a + b*x^n]/(b^2*n)

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Rubi in Sympy [A]  time = 8.4168, size = 26, normalized size = 0.79 \[ \frac{a}{b^{2} n \left (a + b x^{n}\right )} + \frac{\log{\left (a + b x^{n} \right )}}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)/(a+b*x**n)**2,x)

[Out]

a/(b**2*n*(a + b*x**n)) + log(a + b*x**n)/(b**2*n)

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Mathematica [A]  time = 0.0288903, size = 33, normalized size = 1. \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

a/(b^2*n*(a + b*x^n)) + Log[a + b*x^n]/(b^2*n)

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Maple [A]  time = 0.027, size = 38, normalized size = 1.2 \[{\frac{a}{{b}^{2}n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 1.4447, size = 53, normalized size = 1.61 \[ \frac{a}{b^{3} n x^{n} + a b^{2} n} + \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219493, size = 49, normalized size = 1.48 \[ \frac{{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{b^{3} n x^{n} + a b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+2*n)/(a+b*x**n)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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