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

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

[Out]

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

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Rubi [A]  time = 0.0799551, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{c^2 \log \left (b+c x^n\right )}{b^3 n}+\frac{c^2 \log (x)}{b^3}+\frac{c x^{-n}}{b^2 n}-\frac{x^{-2 n}}{2 b n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.7116, size = 51, normalized size = 0.89 \[ - \frac{x^{- 2 n}}{2 b n} + \frac{c x^{- n}}{b^{2} n} + \frac{c^{2} \log{\left (x^{n} \right )}}{b^{3} n} - \frac{c^{2} \log{\left (b + c x^{n} \right )}}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0432396, size = 46, normalized size = 0.81 \[ -\frac{x^{-2 n} \left (2 c^2 x^{2 n} \log \left (b x^{-n}+c\right )+b \left (b-2 c x^n\right )\right )}{2 b^3 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.034, size = 69, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{c{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}-{\frac{1}{2\,bn}}+{\frac{{c}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{3}}} \right ) }-{\frac{{c}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{b}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.74915, size = 76, normalized size = 1.33 \[ \frac{c^{2} \log \left (x\right )}{b^{3}} + \frac{{\left (2 \, c x^{n} - b\right )} x^{-2 \, n}}{2 \, b^{2} n} - \frac{c^{2} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290453, size = 80, normalized size = 1.4 \[ \frac{2 \, c^{2} n x^{2 \, n} \log \left (x\right ) - 2 \, c^{2} x^{2 \, n} \log \left (c x^{n} + b\right ) + 2 \, b c x^{n} - b^{2}}{2 \, b^{3} n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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