3.2623 \(\int \frac{x^{-1-3 n}}{2+b x^n} \, dx\)

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

[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 = 0.0297424, 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, Rules used = {266, 44} \[ -\frac{b^2 x^{-n}}{8 n}+\frac{b^3 \log \left (b x^n+2\right )}{16 n}-\frac{1}{16} b^3 \log (x)+\frac{b x^{-2 n}}{8 n}-\frac{x^{-3 n}}{6 n} \]

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)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{-1-3 n}}{2+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x^4}-\frac{b}{4 x^3}+\frac{b^2}{8 x^2}-\frac{b^3}{16 x}+\frac{b^4}{16 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-3 n}}{6 n}+\frac{b x^{-2 n}}{8 n}-\frac{b^2 x^{-n}}{8 n}-\frac{1}{16} b^3 \log (x)+\frac{b^3 \log \left (2+b x^n\right )}{16 n}\\ \end{align*}

Mathematica [A]  time = 0.0532214, size = 53, normalized size = 0.78 \[ -\frac{x^{-3 n} \left (6 b^2 x^{2 n}-6 b x^n+8\right )-3 b^3 \log \left (b x^n+2\right )+3 b^3 n \log (x)}{48 n} \]

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

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Maple [A]  time = 0.02, size = 74, normalized size = 1.1 \begin{align*}{\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}} \end{align*}

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 = 0.967581, size = 78, normalized size = 1.15 \begin{align*} -\frac{1}{16} \, b^{3} \log \left (x\right ) + \frac{b^{3} \log \left (\frac{b x^{n} + 2}{b}\right )}{16 \, n} - \frac{3 \, b^{2} x^{2 \, n} - 3 \, b x^{n} + 4}{24 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.06815, size = 142, normalized size = 2.09 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)/(2+b*x^n),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. \begin{align*} \text{Timed out} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-3 \, n - 1}}{b x^{n} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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