3.2616 \(\int \frac{x^{-1-3 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=68 \[ \frac{b^3 \log \left (b x^n+2\right )}{16 n}-\frac{1}{16} b^3 \log (x)-\frac{b^2 x^{-n}}{8 n}+\frac{b x^{-2 n}}{8 n}-\frac{x^{-3 n}}{6 n} \]

[Out]

-1/(6*n*x^(3*n)) + b/(8*n*x^(2*n)) - b^2/(8*n*x^n) - (b^3*Log[x])/16 + (b^3*Log[
2 + b*x^n])/(16*n)

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Rubi [A]  time = 0.0752581, antiderivative size = 68, 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 \[ \frac{b^3 \log \left (b x^n+2\right )}{16 n}-\frac{1}{16} b^3 \log (x)-\frac{b^2 x^{-n}}{8 n}+\frac{b x^{-2 n}}{8 n}-\frac{x^{-3 n}}{6 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)/(2 + b*x^n),x]

[Out]

-1/(6*n*x^(3*n)) + b/(8*n*x^(2*n)) - b^2/(8*n*x^n) - (b^3*Log[x])/16 + (b^3*Log[
2 + b*x^n])/(16*n)

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Rubi in Sympy [A]  time = 11.3594, size = 56, normalized size = 0.82 \[ - \frac{b^{3} \log{\left (x^{n} \right )}}{16 n} + \frac{b^{3} \log{\left (b x^{n} + 2 \right )}}{16 n} - \frac{b^{2} x^{- n}}{8 n} + \frac{b x^{- 2 n}}{8 n} - \frac{x^{- 3 n}}{6 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)/(2+b*x**n),x)

[Out]

-b**3*log(x**n)/(16*n) + b**3*log(b*x**n + 2)/(16*n) - b**2*x**(-n)/(8*n) + b*x*
*(-2*n)/(8*n) - x**(-3*n)/(6*n)

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Mathematica [A]  time = 0.0225313, size = 50, normalized size = 0.74 \[ \frac{x^{-3 n} \left (3 b^3 x^{3 n} \log \left (b+2 x^{-n}\right )-6 b^2 x^{2 n}+6 b x^n-8\right )}{48 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)/(2 + b*x^n),x]

[Out]

(-8 + 6*b*x^n - 6*b^2*x^(2*n) + 3*b^3*x^(3*n)*Log[b + 2/x^n])/(48*n*x^(3*n))

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Maple [A]  time = 0.037, size = 74, normalized size = 1.1 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{16}}-{\frac{1}{6\,n}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{8\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{8\,n}} \right ) }+{\frac{{b}^{3}\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{16\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)/(2+b*x^n),x)

[Out]

(-1/16*b^3*ln(x)*exp(n*ln(x))^3-1/6/n+1/8*b/n*exp(n*ln(x))-1/8*b^2/n*exp(n*ln(x)
)^2)/exp(n*ln(x))^3+1/16*b^3/n*ln(2+b*exp(n*ln(x)))

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Maxima [A]  time = 1.44389, size = 76, normalized size = 1.12 \[ -\frac{1}{16} \, b^{3} \log \left (x\right ) + \frac{b^{3} \log \left (\frac{b x^{n} + 2}{b}\right )}{16 \, n} - \frac{{\left (3 \, b^{2} x^{2 \, n} - 3 \, b x^{n} + 4\right )} x^{-3 \, n}}{24 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(b*x^n + 2),x, algorithm="maxima")

[Out]

-1/16*b^3*log(x) + 1/16*b^3*log((b*x^n + 2)/b)/n - 1/24*(3*b^2*x^(2*n) - 3*b*x^n
+ 4)*x^(-3*n)/n

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Fricas [A]  time = 0.228238, size = 82, normalized size = 1.21 \[ -\frac{3 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 3 \, b^{3} x^{3 \, n} \log \left (b x^{n} + 2\right ) + 6 \, b^{2} x^{2 \, n} - 6 \, b x^{n} + 8}{48 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(b*x^n + 2),x, algorithm="fricas")

[Out]

-1/48*(3*b^3*n*x^(3*n)*log(x) - 3*b^3*x^(3*n)*log(b*x^n + 2) + 6*b^2*x^(2*n) - 6
*b*x^n + 8)/(n*x^(3*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**(-1-3*n)/(2+b*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-3 \, n - 1}}{b x^{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/(b*x^n + 2),x, algorithm="giac")

[Out]

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