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

Optimal. Leaf size=93 \[ -\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c^3 x^{-n}}{b^4 n}-\frac{c^4 \log \left (b+c x^n\right )}{b^5 n}+\frac{c^4 \log (x)}{b^5}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{x^{-4 n}}{4 b n} \]

[Out]

-1/(4*b*n*x^(4*n)) + c/(3*b^2*n*x^(3*n)) - c^2/(2*b^3*n*x^(2*n)) + c^3/(b^4*n*x^n) + (c^4*Log[x])/b^5 - (c^4*L
og[b + c*x^n])/(b^5*n)

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Rubi [A]  time = 0.0538467, antiderivative size = 93, 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} \[ -\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c^3 x^{-n}}{b^4 n}-\frac{c^4 \log \left (b+c x^n\right )}{b^5 n}+\frac{c^4 \log (x)}{b^5}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{x^{-4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*b*n*x^(4*n)) + c/(3*b^2*n*x^(3*n)) - c^2/(2*b^3*n*x^(2*n)) + c^3/(b^4*n*x^n) + (c^4*Log[x])/b^5 - (c^4*L
og[b + c*x^n])/(b^5*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^{-1-3 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac{x^{-1-4 n}}{b+c x^n} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^5 (b+c x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b x^5}-\frac{c}{b^2 x^4}+\frac{c^2}{b^3 x^3}-\frac{c^3}{b^4 x^2}+\frac{c^4}{b^5 x}-\frac{c^5}{b^5 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-4 n}}{4 b n}+\frac{c x^{-3 n}}{3 b^2 n}-\frac{c^2 x^{-2 n}}{2 b^3 n}+\frac{c^3 x^{-n}}{b^4 n}+\frac{c^4 \log (x)}{b^5}-\frac{c^4 \log \left (b+c x^n\right )}{b^5 n}\\ \end{align*}

Mathematica [A]  time = 0.106363, size = 75, normalized size = 0.81 \[ -\frac{b x^{-4 n} \left (-4 b^2 c x^n+3 b^3+6 b c^2 x^{2 n}-12 c^3 x^{3 n}\right )+12 c^4 \log \left (b+c x^n\right )-12 c^4 n \log (x)}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(3*b^3 - 4*b^2*c*x^n + 6*b*c^2*x^(2*n) - 12*c^3*x^(3*n)))/x^(4*n) - 12*c^4*n*Log[x] + 12*c^4*Log[b + c*x^
n])/(12*b^5*n)

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Maple [A]  time = 0.026, size = 105, normalized size = 1.1 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{c}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{b}^{4}n}}-{\frac{1}{4\,bn}}+{\frac{c{{\rm e}^{n\ln \left ( x \right ) }}}{3\,{b}^{2}n}}-{\frac{{c}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{3}n}}+{\frac{{c}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{b}^{5}}} \right ) }-{\frac{{c}^{4}\ln \left ( c{{\rm e}^{n\ln \left ( x \right ) }}+b \right ) }{{b}^{5}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3/b^4/n*exp(n*ln(x))^3-1/4/b/n+1/3*c/b^2/n*exp(n*ln(x))-1/2*c^2/b^3/n*exp(n*ln(x))^2+c^4/b^5*ln(x)*exp(n*ln
(x))^4)/exp(n*ln(x))^4-c^4/b^5/n*ln(c*exp(n*ln(x))+b)

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Maxima [A]  time = 1.00449, size = 113, normalized size = 1.22 \begin{align*} \frac{c^{4} \log \left (x\right )}{b^{5}} - \frac{c^{4} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{5} n} + \frac{12 \, c^{3} x^{3 \, n} - 6 \, b c^{2} x^{2 \, n} + 4 \, b^{2} c x^{n} - 3 \, b^{3}}{12 \, b^{4} n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^4*log(x)/b^5 - c^4*log((c*x^n + b)/c)/(b^5*n) + 1/12*(12*c^3*x^(3*n) - 6*b*c^2*x^(2*n) + 4*b^2*c*x^n - 3*b^3
)/(b^4*n*x^(4*n))

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Fricas [A]  time = 1.56316, size = 190, normalized size = 2.04 \begin{align*} \frac{12 \, c^{4} n x^{4 \, n} \log \left (x\right ) - 12 \, c^{4} x^{4 \, n} \log \left (c x^{n} + b\right ) + 12 \, b c^{3} x^{3 \, n} - 6 \, b^{2} c^{2} x^{2 \, n} + 4 \, b^{3} c x^{n} - 3 \, b^{4}}{12 \, b^{5} n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*c^4*n*x^(4*n)*log(x) - 12*c^4*x^(4*n)*log(c*x^n + b) + 12*b*c^3*x^(3*n) - 6*b^2*c^2*x^(2*n) + 4*b^3*c
*x^n - 3*b^4)/(b^5*n*x^(4*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)/(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^{-3 \, 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^(-1-3*n)/(b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

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