3.2610 \(\int \frac{x^{-1+3 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=43 \[ \frac{4 \log \left (b x^n+2\right )}{b^3 n}-\frac{2 x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

[Out]

(-2*x^n)/(b^2*n) + x^(2*n)/(2*b*n) + (4*Log[2 + b*x^n])/(b^3*n)

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Rubi [A]  time = 0.0593684, antiderivative size = 43, 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 \[ \frac{4 \log \left (b x^n+2\right )}{b^3 n}-\frac{2 x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)/(2 + b*x^n),x]

[Out]

(-2*x^n)/(b^2*n) + x^(2*n)/(2*b*n) + (4*Log[2 + b*x^n])/(b^3*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{n}} x\, dx}{b n} - \frac{2 x^{n}}{b^{2} n} + \frac{4 \log{\left (b x^{n} + 2 \right )}}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x, (x, x**n))/(b*n) - 2*x**n/(b**2*n) + 4*log(b*x**n + 2)/(b**3*n)

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Mathematica [A]  time = 0.0220743, size = 33, normalized size = 0.77 \[ \frac{b x^n \left (b x^n-4\right )+8 \log \left (b x^n+2\right )}{2 b^3 n} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^n*(-4 + b*x^n) + 8*Log[2 + b*x^n])/(2*b^3*n)

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Maple [A]  time = 0.032, size = 48, normalized size = 1.1 \[ -2\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}+4\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(2+b*x^n),x)

[Out]

-2/b^2/n*exp(n*ln(x))+1/2/b/n*exp(n*ln(x))^2+4/b^3/n*ln(2+b*exp(n*ln(x)))

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Maxima [A]  time = 1.44422, size = 57, normalized size = 1.33 \[ \frac{b x^{2 \, n} - 4 \, x^{n}}{2 \, b^{2} n} + \frac{4 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225858, size = 46, normalized size = 1.07 \[ \frac{b^{2} x^{2 \, n} - 4 \, b x^{n} + 8 \, \log \left (b x^{n} + 2\right )}{2 \, b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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