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

Optimal. Leaf size=76 \[ -\frac{c^2 x^{-n}}{b^3 n}+\frac{c^3 \log \left (b+c x^n\right )}{b^4 n}-\frac{c^3 \log (x)}{b^4}+\frac{c x^{-2 n}}{2 b^2 n}-\frac{x^{-3 n}}{3 b n} \]

[Out]

-1/(3*b*n*x^(3*n)) + c/(2*b^2*n*x^(2*n)) - c^2/(b^3*n*x^n) - (c^3*Log[x])/b^4 + (c^3*Log[b + c*x^n])/(b^4*n)

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Rubi [A]  time = 0.0471033, antiderivative size = 76, 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} \[ -\frac{c^2 x^{-n}}{b^3 n}+\frac{c^3 \log \left (b+c x^n\right )}{b^4 n}-\frac{c^3 \log (x)}{b^4}+\frac{c x^{-2 n}}{2 b^2 n}-\frac{x^{-3 n}}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0792356, size = 62, normalized size = 0.82 \[ -\frac{b x^{-3 n} \left (2 b^2-3 b c x^n+6 c^2 x^{2 n}\right )-6 c^3 \log \left (b+c x^n\right )+6 c^3 n \log (x)}{6 b^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(2*b^2 - 3*b*c*x^n + 6*c^2*x^(2*n)))/x^(3*n) + 6*c^3*n*Log[x] - 6*c^3*Log[b + c*x^n])/(6*b^4*n)

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Maple [A]  time = 0.025, size = 88, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{1}{3\,bn}}+{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{b}^{2}n}}-{\frac{{c}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{3}n}}-{\frac{{c}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{b}^{4}}} \right ) }+{\frac{{c}^{3}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{b}^{4}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/3/b/n+1/2*c/b^2/n*exp(n*ln(x))-c^2/b^3/n*exp(n*ln(x))^2-c^3/b^4*ln(x)*exp(n*ln(x))^3)/exp(n*ln(x))^3+c^3/b
^4/n*ln(c*exp(n*ln(x))+b)

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Maxima [A]  time = 1.0671, size = 96, normalized size = 1.26 \begin{align*} -\frac{c^{3} \log \left (x\right )}{b^{4}} + \frac{c^{3} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{4} n} - \frac{6 \, c^{2} x^{2 \, n} - 3 \, b c x^{n} + 2 \, b^{2}}{6 \, b^{3} n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*log(x)/b^4 + c^3*log((c*x^n + b)/c)/(b^4*n) - 1/6*(6*c^2*x^(2*n) - 3*b*c*x^n + 2*b^2)/(b^3*n*x^(3*n))

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Fricas [A]  time = 1.57293, size = 159, normalized size = 2.09 \begin{align*} -\frac{6 \, c^{3} n x^{3 \, n} \log \left (x\right ) - 6 \, c^{3} x^{3 \, n} \log \left (c x^{n} + b\right ) + 6 \, b c^{2} x^{2 \, n} - 3 \, b^{2} c x^{n} + 2 \, b^{3}}{6 \, b^{4} n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*c^3*n*x^(3*n)*log(x) - 6*c^3*x^(3*n)*log(c*x^n + b) + 6*b*c^2*x^(2*n) - 3*b^2*c*x^n + 2*b^3)/(b^4*n*x^
(3*n))

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Sympy [A]  time = 144.718, size = 73, normalized size = 0.96 \begin{align*} - \frac{x^{- 3 n}}{3 b n} + \frac{c x^{- 2 n}}{2 b^{2} n} - \frac{c^{2} x^{- n}}{b^{3} n} + \frac{c^{4} \left (\begin{cases} \frac{x^{n}}{b} & \text{for}\: c = 0 \\\frac{\log{\left (b + c x^{n} \right )}}{c} & \text{otherwise} \end{cases}\right )}{b^{4} n} - \frac{c^{3} \log{\left (x^{n} \right )}}{b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**(-3*n)/(3*b*n) + c*x**(-2*n)/(2*b**2*n) - c**2*x**(-n)/(b**3*n) + c**4*Piecewise((x**n/b, Eq(c, 0)), (log(
b + c*x**n)/c, True))/(b**4*n) - c**3*log(x**n)/(b**4*n)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-2 \, 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^(-1-2*n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

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