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

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

[Out]

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

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Rubi [A]  time = 0.0421494, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1584, 266, 44} \[ -\frac{c^2 \log \left (b+c x^n\right )}{b^3 n}+\frac{c^2 \log (x)}{b^3}+\frac{c x^{-n}}{b^2 n}-\frac{x^{-2 n}}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)/(b*x^n + c*x^(2*n)),x]

[Out]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A]  time = 0.0652911, size = 49, normalized size = 0.86 \[ \frac{-2 c^2 \log \left (b+c x^n\right )+b x^{-2 n} \left (2 c x^n-b\right )+2 c^2 n \log (x)}{2 b^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 69, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{c{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}-{\frac{1}{2\,bn}}+{\frac{{c}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{3}}} \right ) }-{\frac{{c}^{2}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{b}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.962629, size = 78, normalized size = 1.37 \begin{align*} \frac{c^{2} \log \left (x\right )}{b^{3}} - \frac{c^{2} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{3} n} + \frac{2 \, c x^{n} - b}{2 \, b^{2} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76319, size = 128, normalized size = 2.25 \begin{align*} \frac{2 \, c^{2} n x^{2 \, n} \log \left (x\right ) - 2 \, c^{2} x^{2 \, n} \log \left (c x^{n} + b\right ) + 2 \, b c x^{n} - b^{2}}{2 \, b^{3} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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