3.2609 \(\int \frac{x^{-1+4 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=56 \[ -\frac{8 \log \left (b x^n+2\right )}{b^4 n}+\frac{4 x^n}{b^3 n}-\frac{x^{2 n}}{b^2 n}+\frac{x^{3 n}}{3 b n} \]

[Out]

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

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Rubi [A]  time = 0.0726141, antiderivative size = 56, 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 \[ -\frac{8 \log \left (b x^n+2\right )}{b^4 n}+\frac{4 x^n}{b^3 n}-\frac{x^{2 n}}{b^2 n}+\frac{x^{3 n}}{3 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)/(2 + b*x^n),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3 n}}{3 b n} - \frac{2 \int ^{x^{n}} x\, dx}{b^{2} n} + \frac{4 x^{n}}{b^{3} n} - \frac{8 \log{\left (b x^{n} + 2 \right )}}{b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(2+b*x**n),x)

[Out]

x**(3*n)/(3*b*n) - 2*Integral(x, (x, x**n))/(b**2*n) + 4*x**n/(b**3*n) - 8*log(b
*x**n + 2)/(b**4*n)

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Mathematica [A]  time = 0.0279009, size = 43, normalized size = 0.77 \[ \frac{b x^n \left (b^2 x^{2 n}-3 b x^n+12\right )-24 \log \left (b x^n+2\right )}{3 b^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)/(2 + b*x^n),x]

[Out]

(b*x^n*(12 - 3*b*x^n + b^2*x^(2*n)) - 24*Log[2 + b*x^n])/(3*b^4*n)

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Maple [A]  time = 0.034, size = 63, normalized size = 1.1 \[ 4\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}}-{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,bn}}-8\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)/(2+b*x^n),x)

[Out]

4/b^3/n*exp(n*ln(x))-1/b^2/n*exp(n*ln(x))^2+1/3/b/n*exp(n*ln(x))^3-8/b^4/n*ln(2+
b*exp(n*ln(x)))

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Maxima [A]  time = 1.44734, size = 70, normalized size = 1.25 \[ \frac{b^{2} x^{3 \, n} - 3 \, b x^{2 \, n} + 12 \, x^{n}}{3 \, b^{3} n} - \frac{8 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="maxima")

[Out]

1/3*(b^2*x^(3*n) - 3*b*x^(2*n) + 12*x^n)/(b^3*n) - 8*log((b*x^n + 2)/b)/(b^4*n)

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Fricas [A]  time = 0.22594, size = 59, normalized size = 1.05 \[ \frac{b^{3} x^{3 \, n} - 3 \, b^{2} x^{2 \, n} + 12 \, b x^{n} - 24 \, \log \left (b x^{n} + 2\right )}{3 \, b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="fricas")

[Out]

1/3*(b^3*x^(3*n) - 3*b^2*x^(2*n) + 12*b*x^n - 24*log(b*x^n + 2))/(b^4*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+4*n)/(2+b*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{b x^{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + 2),x, algorithm="giac")

[Out]

integrate(x^(4*n - 1)/(b*x^n + 2), x)