3.2605 \(\int \frac{x^{-1-3 n}}{a+b x^n} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0360553, antiderivative size = 76, 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} \[ -\frac{b^2 x^{-n}}{a^3 n}+\frac{b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac{b^3 \log (x)}{a^4}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{x^{-3 n}}{3 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)/(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0738045, size = 62, normalized size = 0.82 \[ -\frac{a x^{-3 n} \left (2 a^2-3 a b x^n+6 b^2 x^{2 n}\right )-6 b^3 \log \left (a+b x^n\right )+6 b^3 n \log (x)}{6 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)/(a + b*x^n),x]

[Out]

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

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Maple [A]  time = 0.02, size = 88, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{1}{3\,an}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }+{\frac{{b}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975223, size = 96, normalized size = 1.26 \begin{align*} -\frac{b^{3} \log \left (x\right )}{a^{4}} + \frac{b^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} - \frac{6 \, b^{2} x^{2 \, n} - 3 \, a b x^{n} + 2 \, a^{2}}{6 \, a^{3} n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.01885, size = 159, normalized size = 2.09 \begin{align*} -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 6 \, b^{3} x^{3 \, n} \log \left (b x^{n} + a\right ) + 6 \, a b^{2} x^{2 \, n} - 3 \, a^{2} b x^{n} + 2 \, a^{3}}{6 \, a^{4} n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*b^3*n*x^(3*n)*log(x) - 6*b^3*x^(3*n)*log(b*x^n + a) + 6*a*b^2*x^(2*n) - 3*a^2*b*x^n + 2*a^3)/(a^4*n*x^
(3*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)/(a+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^{-3 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)/(a+b*x^n),x, algorithm="giac")

[Out]

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