3.2603 \(\int \frac{x^{-1-n}}{a+b x^n} \, dx\)

Optimal. Leaf size=38 \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

[Out]

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

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Rubi [A]  time = 0.0215037, antiderivative size = 38, 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} \[ \frac{b \log \left (a+b x^n\right )}{a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)/(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0271071, size = 32, normalized size = 0.84 \[ -\frac{-b \log \left (a+b x^n\right )+a x^{-n}+b n \log (x)}{a^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)/(a + b*x^n),x]

[Out]

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

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Maple [A]  time = 0.014, size = 50, normalized size = 1.3 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( -{\frac{1}{an}}-{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}} \right ) }+{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{2}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975883, size = 57, normalized size = 1.5 \begin{align*} -\frac{b \log \left (x\right )}{a^{2}} + \frac{b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{2} n} - \frac{1}{a n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.07272, size = 78, normalized size = 2.05 \begin{align*} -\frac{b n x^{n} \log \left (x\right ) - b x^{n} \log \left (b x^{n} + a\right ) + a}{a^{2} n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 57.1692, size = 48, normalized size = 1.26 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{b} & \text{for}\: a = 0 \wedge n = 0 \\- \frac{x^{- 2 n}}{2 b n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\- \frac{x^{- n}}{a n} + \frac{b \log{\left (x^{- n} + \frac{b}{a} \right )}}{a^{2} n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((log(x)/b, Eq(a, 0) & Eq(n, 0)), (-x**(-2*n)/(2*b*n), Eq(a, 0)), (log(x)/(a + b), Eq(n, 0)), (-x**(-
n)/(a*n) + b*log(x**(-n) + b/a)/(a**2*n), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(-n - 1)/(b*x^n + a), x)