3.2604 \(\int \frac{x^{-1-2 n}}{a+b x^n} \, dx\)

Optimal. Leaf size=57 \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]

[Out]

-1/(2*a*n*x^(2*n)) + b/(a^2*n*x^n) + (b^2*Log[x])/a^3 - (b^2*Log[a + b*x^n])/(a^3*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0293391, antiderivative size = 57, 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} \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)/(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0542028, size = 49, normalized size = 0.86 \[ \frac{-2 b^2 \log \left (a+b x^n\right )+a x^{-2 n} \left (2 b x^n-a\right )+2 b^2 n \log (x)}{2 a^3 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)/(a + b*x^n),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 69, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}n}}-{\frac{1}{2\,an}}+{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}} \right ) }-{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.964899, size = 78, normalized size = 1.37 \begin{align*} \frac{b^{2} \log \left (x\right )}{a^{3}} - \frac{b^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} + \frac{2 \, b x^{n} - a}{2 \, a^{2} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.04845, size = 128, normalized size = 2.25 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 2 \, b^{2} x^{2 \, n} \log \left (b x^{n} + a\right ) + 2 \, a b x^{n} - a^{2}}{2 \, a^{3} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)/(a+b*x**n),x)

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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