3.2622 \(\int \frac{x^{-1-2 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=53 \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]

[Out]

-1/(4*n*x^(2*n)) + b/(4*n*x^n) + (b^2*Log[x])/8 - (b^2*Log[2 + b*x^n])/(8*n)

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Rubi [A]  time = 0.0233912, antiderivative size = 53, 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} \[ -\frac{b^2 \log \left (b x^n+2\right )}{8 n}+\frac{1}{8} b^2 \log (x)+\frac{b x^{-n}}{4 n}-\frac{x^{-2 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)/(2 + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0413474, size = 42, normalized size = 0.79 \[ -\frac{b^2 \log \left (b x^n+2\right )-b^2 n \log (x)+x^{-2 n} \left (2-2 b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)/(2 + b*x^n),x]

[Out]

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

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Maple [A]  time = 0.017, size = 59, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{8}}-{\frac{1}{4\,n}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) }-{\frac{{b}^{2}\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{8\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971497, size = 63, normalized size = 1.19 \begin{align*} \frac{1}{8} \, b^{2} \log \left (x\right ) - \frac{b^{2} \log \left (\frac{b x^{n} + 2}{b}\right )}{8 \, n} + \frac{b x^{n} - 1}{4 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.05591, size = 112, normalized size = 2.11 \begin{align*} \frac{b^{2} n x^{2 \, n} \log \left (x\right ) - b^{2} x^{2 \, n} \log \left (b x^{n} + 2\right ) + 2 \, b x^{n} - 2}{8 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b^2*n*x^(2*n)*log(x) - b^2*x^(2*n)*log(b*x^n + 2) + 2*b*x^n - 2)/(n*x^(2*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**(-1-2*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^{-2 \, n - 1}}{b x^{n} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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