3.2615 \(\int \frac{x^{-1-2 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=53 \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]

[Out]

-1/(4*n*x^(2*n)) + b/(4*n*x^n) + (b^2*Log[x])/8 - (b^2*Log[2 + b*x^n])/(8*n)

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Rubi [A]  time = 0.0639633, antiderivative size = 53, 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 \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)/(2 + b*x^n),x]

[Out]

-1/(4*n*x^(2*n)) + b/(4*n*x^n) + (b^2*Log[x])/8 - (b^2*Log[2 + b*x^n])/(8*n)

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Rubi in Sympy [A]  time = 9.38569, size = 44, normalized size = 0.83 \[ \frac{b^{2} \log{\left (x^{n} \right )}}{8 n} - \frac{b^{2} \log{\left (b x^{n} + 2 \right )}}{8 n} + \frac{b x^{- n}}{4 n} - \frac{x^{- 2 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(x**n)/(8*n) - b**2*log(b*x**n + 2)/(8*n) + b*x**(-n)/(4*n) - x**(-2*n)/
(4*n)

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Mathematica [A]  time = 0.0236503, size = 39, normalized size = 0.74 \[ -\frac{x^{-2 n} \left (b^2 x^{2 n} \log \left (b+2 x^{-n}\right )-2 b x^n+2\right )}{8 n} \]

Antiderivative was successfully verified.

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

[Out]

-(2 - 2*b*x^n + b^2*x^(2*n)*Log[b + 2/x^n])/(8*n*x^(2*n))

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Maple [A]  time = 0.031, size = 59, normalized size = 1.1 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{8}}-{\frac{1}{4\,n}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) }-{\frac{{b}^{2}\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{8\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)/(2+b*x^n),x)

[Out]

(1/8*b^2*ln(x)*exp(n*ln(x))^2-1/4/n+1/4*b/n*exp(n*ln(x)))/exp(n*ln(x))^2-1/8*b^2
/n*ln(2+b*exp(n*ln(x)))

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Maxima [A]  time = 1.44246, size = 61, normalized size = 1.15 \[ \frac{1}{8} \, b^{2} \log \left (x\right ) - \frac{b^{2} \log \left (\frac{b x^{n} + 2}{b}\right )}{8 \, n} + \frac{{\left (b x^{n} - 1\right )} x^{-2 \, n}}{4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*log(x) - 1/8*b^2*log((b*x^n + 2)/b)/n + 1/4*(b*x^n - 1)*x^(-2*n)/n

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Fricas [A]  time = 0.228343, size = 68, normalized size = 1.28 \[ \frac{b^{2} n x^{2 \, n} \log \left (x\right ) - b^{2} x^{2 \, n} \log \left (b x^{n} + 2\right ) + 2 \, b x^{n} - 2}{8 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(b^2*n*x^(2*n)*log(x) - b^2*x^(2*n)*log(b*x^n + 2) + 2*b*x^n - 2)/(n*x^(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)/(2+b*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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