3.2597 \(\int \frac{x^{-1-2 n}}{a+b x^n} \, dx\)

Optimal. Leaf size=57 \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]

[Out]

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

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Rubi [A]  time = 0.0799289, antiderivative size = 57, 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 \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)/(a + b*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.9972, size = 51, normalized size = 0.89 \[ - \frac{x^{- 2 n}}{2 a n} + \frac{b x^{- n}}{a^{2} n} + \frac{b^{2} \log{\left (x^{n} \right )}}{a^{3} n} - \frac{b^{2} \log{\left (a + b x^{n} \right )}}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0389531, size = 46, normalized size = 0.81 \[ -\frac{x^{-2 n} \left (2 b^2 x^{2 n} \log \left (a x^{-n}+b\right )+a \left (a-2 b x^n\right )\right )}{2 a^3 n} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a - 2*b*x^n) + 2*b^2*x^(2*n)*Log[b + a/x^n])/(2*a^3*n*x^(2*n))

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Maple [A]  time = 0.032, size = 69, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}n}}-{\frac{1}{2\,an}}+{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}} \right ) }-{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.4628, size = 76, normalized size = 1.33 \[ \frac{b^{2} \log \left (x\right )}{a^{3}} + \frac{{\left (2 \, b x^{n} - a\right )} x^{-2 \, n}}{2 \, a^{2} n} - \frac{b^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230255, size = 80, normalized size = 1.4 \[ \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 2 \, b^{2} x^{2 \, n} \log \left (b x^{n} + a\right ) + 2 \, a b x^{n} - a^{2}}{2 \, a^{3} n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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