3.2608 \(\int \frac{x^{-1+5 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=71 \[ \frac{16 \log \left (b x^n+2\right )}{b^5 n}-\frac{8 x^n}{b^4 n}+\frac{2 x^{2 n}}{b^3 n}-\frac{2 x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

[Out]

(-8*x^n)/(b^4*n) + (2*x^(2*n))/(b^3*n) - (2*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n)
+ (16*Log[2 + b*x^n])/(b^5*n)

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Rubi [A]  time = 0.092397, antiderivative size = 71, 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 \[ \frac{16 \log \left (b x^n+2\right )}{b^5 n}-\frac{8 x^n}{b^4 n}+\frac{2 x^{2 n}}{b^3 n}-\frac{2 x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 5*n)/(2 + b*x^n),x]

[Out]

(-8*x^n)/(b^4*n) + (2*x^(2*n))/(b^3*n) - (2*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n)
+ (16*Log[2 + b*x^n])/(b^5*n)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{4 n}}{4 b n} - \frac{2 x^{3 n}}{3 b^{2} n} + \frac{4 \int ^{x^{n}} x\, dx}{b^{3} n} - \frac{8 x^{n}}{b^{4} n} + \frac{16 \log{\left (b x^{n} + 2 \right )}}{b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+5*n)/(2+b*x**n),x)

[Out]

x**(4*n)/(4*b*n) - 2*x**(3*n)/(3*b**2*n) + 4*Integral(x, (x, x**n))/(b**3*n) - 8
*x**n/(b**4*n) + 16*log(b*x**n + 2)/(b**5*n)

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Mathematica [A]  time = 0.0355076, size = 54, normalized size = 0.76 \[ \frac{b x^n \left (3 b^3 x^{3 n}-8 b^2 x^{2 n}+24 b x^n-96\right )+192 \log \left (b x^n+2\right )}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 5*n)/(2 + b*x^n),x]

[Out]

(b*x^n*(-96 + 24*b*x^n - 8*b^2*x^(2*n) + 3*b^3*x^(3*n)) + 192*Log[2 + b*x^n])/(1
2*b^5*n)

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Maple [A]  time = 0.041, size = 78, normalized size = 1.1 \[ -8\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{4}n}}+2\,{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{3}n}}-{\frac{2\, \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,bn}}+16\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+5*n)/(2+b*x^n),x)

[Out]

-8/b^4/n*exp(n*ln(x))+2/b^3/n*exp(n*ln(x))^2-2/3/b^2/n*exp(n*ln(x))^3+1/4/b/n*ex
p(n*ln(x))^4+16/b^5/n*ln(2+b*exp(n*ln(x)))

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Maxima [A]  time = 1.43926, size = 85, normalized size = 1.2 \[ \frac{3 \, b^{3} x^{4 \, n} - 8 \, b^{2} x^{3 \, n} + 24 \, b x^{2 \, n} - 96 \, x^{n}}{12 \, b^{4} n} + \frac{16 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + 2),x, algorithm="maxima")

[Out]

1/12*(3*b^3*x^(4*n) - 8*b^2*x^(3*n) + 24*b*x^(2*n) - 96*x^n)/(b^4*n) + 16*log((b
*x^n + 2)/b)/(b^5*n)

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Fricas [A]  time = 0.227474, size = 74, normalized size = 1.04 \[ \frac{3 \, b^{4} x^{4 \, n} - 8 \, b^{3} x^{3 \, n} + 24 \, b^{2} x^{2 \, n} - 96 \, b x^{n} + 192 \, \log \left (b x^{n} + 2\right )}{12 \, b^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + 2),x, algorithm="fricas")

[Out]

1/12*(3*b^4*x^(4*n) - 8*b^3*x^(3*n) + 24*b^2*x^(2*n) - 96*b*x^n + 192*log(b*x^n
+ 2))/(b^5*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+5*n)/(2+b*x**n),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5 \, n - 1}}{b x^{n} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5*n - 1)/(b*x^n + 2),x, algorithm="giac")

[Out]

integrate(x^(5*n - 1)/(b*x^n + 2), x)